Brownian Motion and Stochastic Calculus (Graduate Texts in Mathematics)

Ioannis Karatzas Steven E. Shreve Brownian Motion and Stochastic Calculus Second Edition With 10 Illustrations Spring...

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Ioannis Karatzas Steven E. Shreve

Brownian Motion and Stochastic Calculus Second Edition With 10 Illustrations

Springer-Verlag

New York Berlin Heidelberg London Paris Tokyo Hong Kong Barcelona

Ioannis Karatzas Department of Statistics Columbia University

Steven E. Shreve Department of Mathematics Carnegie Mellon University Pittsburgh, PA 15213 USA

New York, NY 10027 USA Editorial Board

J.H. Ewing

Department of Mathematics Indiana University Bloomington, Indiana 47405 USA

F.W. Gehring Department of Mathematics University of Michigan Ann Arbor, Michigan 48109 USA

P.R. Halmos Department of Mathematics University of Santa Clara Santa Clara, California 95053 USA

AMS Classifications: 60G07, 60H05

Library of Congress Cataloging-in-Publication Data Karatzas, Ioannis. Brownian motion and stochastic calculus/ Ioannis Karatzas, Steven

E. Shreve.-2nd ed. p. cm.-(Graduate texts in mathematics; 113) Includes bibliographical references and index. ISBN 0-387-97655-8 (New York).-ISBN 3-540-97655-8 (Berlin) 1. Brownian motion processes. 2. Stochastic analysis. I. Shreve, Steven E. II. Title. III. Series. QA274.75.K37 1991 530.4'75-dc20 91-22775

The present volume is the corrected softcover second edition of the previously published hardcover first edition (ISBN 0-387-96535-1). Printed on acid-free paper. © 1988, 1991 by Springer-Verlag New York, Inc. All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag, 175 Fifth Avenue, New York, New York 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc. in this publication, even

if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly by used freely by anyone. Typeset by Asco Trade Typesetting Ltd., Hong Kong. Printed and bound by R.R. Donnelley and Sons, Harrisonburg, VA. Printed in the United States of America.

98765432

1

ISBN 0-387-97655-8 Springer-Verlag New York Berlin Heidelberg ISBN 3-540-97655-8 Springer-Verlag Berlin Heidelberg New York

To Eleni and Dot

Preface

Two of the most fundamental concepts in the theory of stochastic processes are the Markov property and the martingale property.* This book is written for readers who are acquainted with both of these ideas in the discrete-time setting, and who now wish to explore stochastic processes in their continuoustime context. It has been our goal to write a systematic and thorough exposition of this subject, leading in many instances to the frontiers of knowledge. At the same time, we have endeavored to keep the mathematical prerequisites as low as possible, namely, knowledge of measure-theoretic probability and some familiarity with discrete-time processes. The vehicle we have chosen for this task is Brownian motion, which we present as the canonical example of both a Markov process and a martingale. We support this point of view by showing how, by means of stochastic integration and random time change, all continuous-path martingales and a multitude of continuous-path Markov processes can be represented in terms of Brownian motion. This approach forces us to leave aside those processes which do not have continuous paths.

Thus, the Poisson process is not a primary object of study, although it is developed in Chapter 1 to be used as a tool when we later study passage times and local time of Brownian motion. The text is organized as follows: Chapter 1 presents the basic properties of martingales, as they are used throughout the book. In particular, we generalize from the discrete to the continuous-time context the martingale convergence theorem, the optional sampling theorem, and the Doob-Meyer decomposition. The latter gives conditions under which a submartingale can be written * According to M. Loeve, "martingales, Markov dependence and stationarity are the only three dependence concepts so far isolated which are sufficiently general and sufficiently amenable to investigation, yet with a great number of deep properties" (Ann. Probab. 1 (1973), p. 6).

viii

Preface

as the sum of a martingale and an increasing process, and associates to every martingale with continuous paths a "quadratic variation process." This process is instrumental in the construction of stochastic integrals with respect to continuous martingales.

Chapter 2 contains three different constructions of Brownian motion, as well as discussions of the Markov and strong Markov properties for continuous-time processes. These properties are motivated by d-dimensional Brownian motion, but are developed in complete generality. This chapter also contains a careful discussion of the various filtrations commonly associated with Brownian motion. In Section 2.8 the strong Markov property is applied

to a study of one-dimensional Brownian motion on a half-line, and on a bounded interval with absorption and reflection at the endpoints. Many densities involving first passage times, last exit times, absorbed Brownian motion, and reflected Brownian motion are explicitly computed. Section 2.9 is devoted to a study of sample path properties of Brownian motion. Results found in most texts on this subject are included, and in addition to these, a complete proof of the Levy modulus of continuity is provided. The theory of stochastic integration with respect to continuous martingales is developed in Chapter 3. We follow a middle path between the original constructions of stochastic integrals with respect to Brownian motion and the more recent theory of stochastic integration with respect to right-continuous martingales. By avoiding discontinuous martingales, we obviate the need to introduce the concept of predictability and the associated, highly technical, measure-theoretic machinery. On the other hand, it requires little extra effort

to consider integrals with respect to continuous martingales rather than merely Brownian motion. The remainder of Chapter 3 is a testimony to the power of this more general approach; in particular, it leads to strong theorems concerning representations of continuous martingales in terms of Brownian motion (Section 3.4). In Section 3.3 we develop the chain rule for stochastic calculus, commonly known as It6"s formula. The Girsanov Theorem of Section 3.5 provides a method of changing probability measures so as to alter the drift of a stochastic process. It has become an indispensable method for constructing solutions of stochastic differential equations (Section 5.3) and is also very important in stochastic control (e.g., Section 5.8) and filtering. Local time is introduced in Sections 3.6 and 3.7, and it is shown how this concept leads to a generalization of the Ito formula to convex but not necessarily differentiable functions. Chapter 4 is a digression on the connections between Brownian motion, Laplace's equation, and the heat equation. Sharp existence and uniqueness theorems for both these equations are provided by probabilistic methods; applications to the computation of boundary crossing probabilities are discussed, and the formulas of Feynman and Kac are established. Chapter 5 returns to our main theme of stochastic integration and differential equations. In this chapter, stochastic differential equations are driven

Preface

ix

by Brownian motion and the notions of strong and weak solutions are presented. The basic Ito theory for strong solutions and some of its ramifications, including comparison and approximation results, are offered in Section 5.2,

whereas Section 5.3 studies weak solutions in the spirit of Yamada & Watanabe. Essentially equivalent to the search for a weak solution is the search for a solution to the "Martingale Problem" of Stroock & Varadhan. In the context of this martingale problem, a full discussion of existence, uniqueness, and the strong Markov property for solutions of stochastic differential equations is given in Section 5.4. For one-dimensional equations it is possible to provide a complete characterization of solutions which exist only up to an "explosion time," and this is set forth in Section 5.5. This section also

presents the recent and quite striking results of Engelbert & Schmidt concerning existence and uniqueness of solutions to one-dimensional equations. This theory makes substantial use of the local time material of Sections 3.6,

3.7 and the martingale representation results of Subsections 3.4.A,B. By analogy with Chapter 4, we discuss in Section 5.7 the connections between solutions to stochastic differential equations and elliptic and parabolic partial differential equations. Applications of many of the ideas in Chapters 3 and 5 are contained in Section 5.8, where we discuss questions of option pricing and optimal portfolio/consumption management. In particular, the Girsanov

theorem is used to remove the difference between average rates of return of different stocks, a martingale representation result provides the optimal portfolio process, and stochastic representations of solutions to partial differential equations allow us to recast the optimal portfolio and consumption

management problem in terms of two linear parabolic partial differential equations, for which explicit solutions are provided. Chapter 6 is for the most part derived from Paul Levy's profound study of Brownian excursions. Levy's intuitive work has now been formalized by such notions as filtrations, stopping times, and Poisson random measures, but the remarkable fact remains that he was able, 40 years ago and working without these tools, to penetrate into the fine structure of the Brownian path and to inspire all the subsequent research on these matters until today. In the spirit

of Levy's work, we show in Section 6.2 that when one travels along the Brownian path with a clock run by the local time, the number of excursions away from the origin that one encounters, whose duration exceeds a specified number, has a Poisson distribution. Levy's heuristic construction of Brownian motion from its excursions has been made rigorous by other authors. We do not attempt such a construction here, nor do we give a complete specification of the distribution of Brownian excursions; in the interest of intelligibility, we content ourselves with the specification of the distribution for the durations of the excursions. Sections 6.3 and 6.4 derive distributions for functionals

of Brownian motion involving its local time; we present, in particular, a Feynman-Kac result for the so-called "elastic" Brownian motion, the formulas of D. Williams and H. Taylor, and the Ray-Knight description of

Preface

x

Brownian local time. An application of this theory is given in Section 6.5, where a one-dimensional stochastic control problem of the "bang-bang" type is solved.

The writing of this book has become for us a monumental undertaking involving several-people, whose assistance we gratefully acknowledge. Foremost among these are the members of our families, Eleni, Dot, Andrea, and Matthew, whose support, encouragement, and patience made the whole en-

deavor possible. Parts of the book grew out of notes on lectures given at Columbia University over several years, and we owe much to the audiences in those courses. The inclusion of several exercises, the approaches taken to a number of theorems, and several citations of relevant literature resulted from discussions and correspondence with F. Baldursson, A. Dvoretzky, W. Fleming, 0. Kallenberg, T. Kurtz, S. Lalley, J. Lehoczky, D. Stroock, and M. Yor. We have also taken exercises from Mandl, Lanska & Vrkoe (1978), and Ethier & Kurtz (1986). As the project proceeded, G.-L. Xu, Z.-L. Ying, and Th. Zariphopoulou read large portions of the manuscript and suggested numerous corrections and improvements. Careful reading by Daniel Ocone and Manfred Schal revealed minor errors in the first printing, and these have been corrected. However, our greatest single debt of gratitude goes to Marc Yor, who read much of the near-final draft and offered substantial mathematical and editorial comments on it. The typing was done tirelessly, cheerfully, and efficiently by Stella De Vito and Doodmatie Kalicharan; they have our most sincere appreciation.

We are grateful to Sanjoy Mitter and Dimitri Bertsekas for extending to us the invitation to spend the critical initial year of this project at the Massachusetts Institute of Technology. During that time the first four chapters were essentially completed, and we were partially supported by the Army Research Office under grant DAAG-299-84-K-0005. Additional financial support was provided by the National Science Foundation under grants DMS84-16736 and DMS-84-03166 and by the Air Force Office of Scientific Research

under grants AFOSR 82-0259, AFOSR 85-0360, and AFOSR 86-0203.

Ioannis Karatzas Steven E. Shreve

Contents

Preface

Suggestions for the Reader

vii

xvii

Interdependence of the Chapters

xi x

Frequently Used Notation

xxi

CHAPTER 1

Martingales, Stopping Times, and Filtrations 1.1. Stochastic Processes and a-Fields 1.2. Stopping Times 1.3. Continuous-Time Martingales A. Fundamental inequalities B. Convergence results C. The optional sampling theorem 1.4. The Doob-Meyer Decomposition 1.5. Continuous, Square-Integrable Martingales 1.6. Solutions to Selected Problems 1.7. Notes

1 1

6 11

12 17 19 21

30 38 45

CHAPTER 2

Brownian Motion

47

2.1. Introduction 2.2. First Construction of Brownian Motion A. The consistency theorem B. The Kolmogorov-entsov theorem 2.3. Second Construction of Brownian Motion

47 49 49

2.4. The Space C[0, co), Weak Convergence, and Wiener Measure A. Weak convergence

53 56 59

60

xii

B. Tightness C. Convergence of finite-dimensional distributions D. The invariance principle and the Wiener measure 2.5. The Markov Property A. Brownian motion in several dimensions B. Markov processes and Markov families C. Equivalent formulations of the Markov property 2.6. The Strong Markov Property and the Reflection Principle A. The reflection principle B. Strong Markov processes and families C. The strong Markov property for Brownian motion 2.7. Brownian Filtrations A. Right-continuity of the augmented filtration for a strong Markov process B. A "universal" filtration C. The Blumenthal zero-one law 2.8. Computations Based on Passage Times A. Brownian motion and its running maximum B. Brownian motion on a half-line C. Brownian motion on a finite interval D. Distributions involving last exit times 2.9. The Brownian Sample Paths A. Elementary properties B. The zero set and the quadratic variation C. Local maxima and points of increase D. Nowhere differentiability E. Law of the iterated logarithm F. Modulus of continuity 2.10. Solutions to Selected Problems 2.11. Notes

Contents 61

64 66 71

72 74 75 79 79 81

84 89 90 93

94 94 95 97 97 100 103 103 104 106 109 111 114 116 126

CHAPTER 3

Stochastic Integration

128

3.1. Introduction 3.2. Construction of the Stochastic Integral A. Simple processes and approximations B. Construction and elementary properties of the integral C. A characterization of the integral D. Integration with respect to continuous, local martingales 3.3. The Change-of-Variable Formula A. The Ito rule B. Martingale characterization of Brownian motion

128 129 132 137

C. Bessel processes, questions of recurrence D. Martingale moment inequalities E. Supplementary exercises 3.4. Representations of Continuous Martingales in Terms of Brownian Motion A. Continuous local martingales as stochastic integrals with respect to Brownian motion

141 145

148 149 156 158 163 167 169

170

Contents

xiii

B. Continuous local martingales as time-changed Brownian motions C. A theorem of F. B. Knight D. Brownian martingales as stochastic integrals E. Brownian functionals as stochastic integrals 3.5. The Girsanov Theorem A. The basic result B. Proof and ramifications C. Brownian motion with drift D. The Novikov condition 3.6. Local Time and a Generalized It8 Rule for Brownian Motion A. Definition of local time and the Tanaka formula B. The Trotter existence theorem C. Reflected Brownian motion and the Skorohod equation D. A generalized Ito rule for convex functions E. The Engelbert-Schmidt zero-one law 3.7. Local Time for Continuous Semimartingales 3.8. Solutions to Selected Problems 3.9. Notes

I73 179 180 185 190 191

193 196 198

20I 203 206 210 212 215 217 226 236

CHAPTER 4

Brownian Motion and Partial Differential Equations

239

4.1. Introduction 4.2. Harmonic Functions and the Dirichlet Problem A. The mean-value property B. The Dirichlet problem C. Conditions for regularity D. Integral formulas of Poisson E. Supplementary exercises 4.3. The One-Dimensional Heat Equation A. The Tychonoff uniqueness theorem B. Nonnegative solutions of the heat equation C. Boundary crossing probabilities for Brownian motion D. Mixed initial/boundary value problems 4.4. The Formulas of Feynman and Kac A. The multidimensional formula B. The one-dimensional formula 4.5. Solutions to selected problems 4.6. Notes

239 240 241

243 247 251

253 254 255

256 262 265 267 268 271

275 278

CHAPTER 5

Stochastic Differential Equations

281

5.1. Introduction 5.2. Strong Solutions A. Definitions B. The Ito theory C. Comparison results and other refinements D. Approximations of stochastic differential equations E. Supplementary exercises

281

284 285 286 291

295 299

xiv

5.3. Weak Solutions A. Two notions of uniqueness B. Weak solutions by means of the Girsanov theorem C. A digression on regular conditional probabilities D. Results of Yamada and Watanabe on weak and strong solutions 5.4. The Martingale Problem of Stroock and Varadhan A. Some fundamental martingales B. Weak solutions and martingale problems C. Well-posedness and the strong Markov property D. Questions of existence E. Questions of uniqueness F. Supplementary exercises 5.5. A Study of the One-Dimensional Case A. The method of time change B. The method of removal of drift C. Feller's test for explosions D. Supplementary exercises 5.6. Linear Equations A. Gauss-Markov processes B. Brownian bridge C. The general, one-dimensional, linear equation D. Supplementary exercises 5.7. Connections with Partial Differential Equations A. The Dirichlet problem B. The Cauchy problem and a Feynman-Kac representation C. Supplementary exercises 5.8. Applications to Economics A. Portfolio and consumption processes B. Option pricing C. Optimal consumption and investment (general theory) D. Optimal consumption and investment (constant coefficients) 5.9. Solutions to Selected Problems 5.10. Notes

Contents

300 301

302 306 308 311

312 314 319 323 325 328 329 330 339 342 35 I

354 355 358 360 361

363 364 366 369 371 371

376 379 381

387 394

CHAPTER 6

P. Levy's Theory of Brownian Local Time

399

6.1. Introduction 6.2. Alternate Representations of Brownian Local Time A. The process of passage times B. Poisson random measures C. Subordinators D. The process of passage times revisited E. The excursion and downcrossing representations of local time 6.3. Two Independent Reflected Brownian Motions A. The positive and negative parts of a Brownian motion B. The first formula of D. Williams C. The joint density of (W(t), L(t), f (t))

399 400

400 403 405 411

414 418 418 421

423

Contents

xv

6.4. Elastic Brownian Motion A. The Feynman-Kac formulas for elastic Brownian motion B. The Ray-Knight description of local time C. The second formula of D. Williams 6.5. An Application: Transition Probabilities of Brownian Motion with Two-Valued Drift 6.6. Solutions to Selected Problems 6.7. Notes

425 426

Bibliography

447

Index

459

430 434 437 442 445


We use a hierarchical numbering system for equations and statements. The k-th equation in Section j of Chapter i is labeled (j.k) at the place where it occurs and is cited as (j.k) within Chapter i, but as (i j.k) outside Chapter i. A definition, theorem, lemma, corollary, remark, problem, exercise, or solution is a "statement," and the k-th statement in Section j of Chapter i is labeled j.k Statement at the place where it occurs, and is cited as Statement j.k within Chapter i but as Statement i.j.k outside Chapter i. This book is intended as a text and can be used in either a one-semester or a two-semester course, or as a text for a special topic seminar. The accompany-

ing figure shows dependences among sections, and in some cases among subsections. In a one-semester course, we recommend inclusion of Chapter 1 and Sections 2.1, 2.2, 2.4, 2.5, 2.6, 2.7, §2.9.A, B, E, Sections 3.2, 3.3, 5.1, 5.2, and §5.6.A, C. This material provides the basic theory of stochastic integration, including the Ito calculus and the basic existence and uniqueness results for strong solutions of stochastic differential equations. It also contains matters of interest in engineering applications, namely, Fisk-Stratonovich integrals and approximation of stochastic differential equations in §3.3.A and 5.2.D, and Gauss-Markov processes in §5.6.A. Progress through this material can

be accelerated by omitting the proof of the Doob-Meyer Decomposition Theorem 1.4.10 and the proofs in §2.4.D. The statements of Theorem 1.4.10, Theorem 2.4.20, Definition 2.4.21, and Remark 2.4.22 should, however, be retained. If possible in a one-semester course, and certainly in a two-semester course, one should include the topic of weak solutions of stochastic differential equations. This is accomplished by covering §3.4.A, B, and Sections 3.5, 5.3, and 5.4. Section 5.8 serves as an introduction to stochastic control, and so we recommend adding §3.4.C, D,.E, and Sections 5.7, and 5.8 if time permits. In either a one- or two-semester course, Section 2.8 and part or all of Chapter 4

xviii


may be included according to time and interest. The material on local time and its applications in Sections 3.6, 3.7, 5.5, and in Chapter 6 would normally be the subject of a special topic course with advanced students. The text contains about 175 "problems" and over 100 "exercises." The former are assignments to the reader to fill in details or generalize a result, and these are often quoted later in the text. We judge approximately twothirds of these problems to be nontrivial or of fundamental importance, and solutions for such problems are provided at the end of each chapter. The exercises are also often significant extensions of results developed in the text, but these will not be needed later, except perhaps in the solution of other exercises. Solutions for the exercises are not provided. There are some exercises for which the solution we know violates the dependencies among

sections shown in the figure, but such violations are pointed out in the offending exercises, usually in the form of a hint citing an earlier result.

Interdependence of the Chapters

Chapter 1 2.1

2.2

2.3

2.4

2.5

2.6

2.7

2.8

§2.9.A, B, E

§2.9.C, D, F

4.2

3.2

62

4.3

3.3

3.6

4.4

§3.4.A, B

3.7

§3.4.C, D, E

5.1

35

5.2

5.3

§5.6.A, C

5.4

§5.6.B, D

5.7

55

6.3

[ 5.8,

[ 6.4 L 6.5

1

Frequently Used Notation

I. General Notation Let a and b be real numbers. (1) A means "is defined to be."

(2) anbA min la, bl. (3) a v bA max {a, b}. (4) a+ A max{a, 0}.

(5) a- A max{ -a,0}.

II. Sets and Spaces (1) No A {0,1, 2, ... }.

(2) Q is the set of rational numbers. (3) Q+ is the set of nonnegative rational numbers. (4) Rd is the d-dimensional Euclidean space; R1 = R. (5) B, A {x e Rd; Ilx II
: The cross-variation process of X, Y in /2 (p. 31) or in i`.'" (p. 36). (6) II XL, II XII: See p. 37 for X e .412-

VII. Miscellaneous (1) mT(X, 6) A sup{ I Xs - X,I; 0 -s 0 is the sample path (realization, trajectory) of the process X associated with w. It provides

the mathematical model for a random experiment whose outcome can be observed continuously in time (e.g., the number of customers in a queue observed and recorded over a period of time, the trajectory of a molecule subjected to the random disturbances of its neighbors, the output of a communications channel operating in noise).

Let us consider two stochastic processes X and Y defined on the same probability space (S2, .97 , P). When they are regarded as functions of t and w, we would say X and Y were the same if and only if Xj(w) = Y,(w) for all t > 0 and all w E a However, in the presence of the probability measure P, we could weaken this requirement in at least three different ways to obtain three related concepts of "sameness" between two processes. We list them here.

1. Martingales, Stopping Times, and Filtrations

2

1.1 Definition. Y is a modification of X if, for every t

0, we have

P[X, = y] = 1. 1.2 Definition. X and Y have the same finite-dimensional distributions if, for < t < cc, and A e .4(Ord), we have:

any integer n > 1, real numbers 0 < t, < t2
0 we have P[Y, = P[Y, Xi; V t 0] = 0.

t=T

= PET z t] = 1, but on the other hand:

A positive result in this direction is the following.

1.5 Problem. Let Y be a modification of X, and suppose that both processes have a.s. right-continuous sample paths. Then X and Y are indistinguishable. It does not make sense to ask whether Y is a modification of X, or whether

Y and X are indistinguishable, unless X and Y are defined on the same probability space and have the same state space. However, if X and Y have the same state space but are defined on different probability spaces, we can ask whether they have the same finite-dimensional distributions.

1.2' Definition. Let X and Y be stochastic processes defined on probability spaces (11, , P) and (S, ., P), respectively, and having the same state space (Rd, g(Old. )) X and Y have the same finite-dimensional distributions if, for any integer n > 1, real numbers 0 < t, < t2 < < t < cc, and A e gi(Ora), we have

, Xt.) e A] =

Y,n)EA].

Many processes, including d-dimensional Brownian motion, are defined in terms of their finite-dimensional distributions irrespective of their probability

1.1. Stochastic Processes and o -Fields

3

space. Indeed, in Chapter 2 we will construct a standard d-dimensional Brownian motion B on a canonical probability space and then state that any process, on any probability space, which has state space (Rd, gi(Rd)) and the same finite-dimensional distributions as B, is a standard d-dimensional Brownian motion.

For technical reasons in the theory of Lebesgue integration, probability measures are defined on a-fields and random variables are assumed to be measurable with respect to these a-fields. Thus, implicit in the statement that a random process X = {Xi; 0 t < co } is a collection of (Rd, .4(Rd))-valued random variables on (fl, F), is the assumption that each X, is /.4(E4d)measurable. However, X is really a function of the pair of variables (t, w), and so, for technical reasons, it is often convenient to have some joint measurability properties.

1.6 Definition. The stochastic process X is called measurable if, for every A e MR"), the set {(t, w); Xt(co)e AI belongs to the product a-field .4([0, oo))

,F; in other words, if the mapping Xt(co): ( [0, co) x

(t,

M( [0, oo))

.F) -÷ (Rd, mold))

is measurable.

It is an immediate consequence of Fubini's theorem that the trajectories of such a process are Borel-measurable functions oft e [0, co), and provided that the components of X have defined expectations, then the same is true for the function m(t) = E Xi; here, E denotes expectation with respect to a probability measure P on (f2, Moreover, if X takes values in R and I is a subinterval of [0, co) such that jj EIX,I dt < co, then

f,

X,' dt < co a.s. P,

and

ji EX, dt = E

X, dt.

There is a very important, nontechnica reason to include a-fields in the study of stochastic processes, and that is to keep track of information. The temporal feature of a stochastic process suggests a flow of time, in which, at every moment t > 0, we can talk about a past, present, and future and can ask how much an observer of the process knows about it at present, as compared to how much he knew at some point in the past or will know at some point in the future. We equip our sample space (f2, ,F) with a filtration, i.e., a nondecreasing family {A; t > 0} of sub-a-fields of .97 for 0 s < t < co. We set d. 6-co =-- a(Ut>o Given a stochastic process, the simplest choice of a filtration is that generated by the process itself, i.e.,

A' A o-(Xs; 0 < s < t), the smallest a-field with respect to which X, is measurable for every s e [0, t].

4


We interpret A E "ix to mean that by time t, an observer of X knows whether or not A has occurred. The next two exercises illustrate this point.

1.7 Exercise. Let X be a process, every sample path of which is RCLL (i.e., right-continuous on [0, co) with finite left -hand limits on (0, co)). Let A be the event that X is continuous on [0, to). Show that A e 1.8 Exercise. Let X be a process whose sample paths are RCLL almost surely, and let A be the event that X is continuous on [0, to). Show that A can fail to be in .??-tox, but if {.";; t > 0} is a filtration satisfying .??7,' t 0, and Fto is

complete under P, then A Eo. Let {,56;; t > 0} be a filtration. We define

cr(Us 0 and Ft, nE>0,,,E to be the a-field of events immediately after t > 0. We decree .50_ A .yo and say that the filtration {..F,} is right- (left-)continuous if = (resp., 3,7,_) holds for every t > 0. The concept of measurability for a stochastic process, introduced in Definition 1.6, is a rather weak one. The introduction of a filtration {g} opens up the possibility of more interesting and useful concepts. 1.9 Definition. The stochastic process X is adapted to the filtration {.F,} if, for each t > 0, X, is an ..F- measurable random variable.

Obviously, every process X is adapted to 1,a-el. Moreover, if X is adapted to {A} and Y is a modification of X, then Y is also adapted to {,} provided that .yo contains all the P-negligible sets in Note that this requirement is not the same as saying that .F0 is complete, since some of the P-negligible sets in g" may not be in the completion of go.

1.10 Exercise. Let X be a process with every sample path LCRL (i.e., left continuous on (0, co) with finite right-hand limits on [0, co)), and let A be the event that X is continuous on [0, to]. Let X be adapted to a right-continuous filtration {.Ft}. Show that A e "o. 1.11 Definition. The stochastic process X is called progressively measurable with respect to the filtration {,;} if, for each t > 0 and A e gii(Rd), the set {(s, a)); 0 < s < t, w e SZ, Xs(w) e A} belongs to the product a-field .4([0, t]) ..Ft; in other words, if the mapping (s, co) 1-+ X s(w): ([0, t] x S2, .4([0, t]) 0 g") -+

(Rd M(Rd)) is measurable, for each t

0.

The terminology here comes from Chung & Doob (1965), which is a basic reference for this section and the next. Evidently, any progressively measurable process is measurable and adapted; the following theorem of Chung & Doob (1965) provides the extent to which the converse is true.

1.1. Stochastic Processes and a-Fields

5

1.12 Proposition. If the stochastic process X is measurable and adapted to the filtration {.9;}, then it has a progressively measurable modification.

The reader is referred to the book of Meyer (1966), p. 68, for the (lengthy, and rather demanding) proof of this result. It will be used in this text only in a tangential fashion. Nearly all processes of interest are either right- or leftcontinuous, and for them the proof of a stronger result is easier and will now be given.

1.13 Proposition. If the stochastic process X is adapted to the filtration 1.97,1 and every sample path is right-continuous or else every sample path is leftcontinuous, then X is also progressively measurable with respect to IF J.

PROOF. We treat the case of right-continuity. With t > 0, n > 1, k = 0, ..., 2" - 1, and 0 < s < t, we define: X.1")((0) = X(k+1)02^(W)

for

-kt

2"

1,

<s_ 0; X, (co) e F}.

We employ the standard convention that the infimum of the empty set is infinity. 2.6 Problem. If the set T in Example 2.5 is open, shoWthat Hr is an optional time. 2.7 Problem. If the set F in Example 2.5 is closed and the sample paths of the

process X are continuous, then 11,- is a stopping time.

Let us establish some simple properties of stopping times. 2.8 Lemma. If T is optional and B is a positive constant, then T + B is a stopping time.

PROOF. If 0 < t < 0, then {T + 0 _. t} = 0 e ...Ft. If t > 0, then

{T + 0 < t} = IT < t - 0} e ,F0_0), g Sr.,. 2.9 Lemma. If T, S are stopping times, then so are T

A

0

S, T v S, T + S.

PROOF. The first two assertions are trivial. For the third, start with the decomposition, valid for t > 0:

{T + S > t} = {T = 0,S > t} u {0 < T < t, T + S > t} u {T > t, S = 0} u {T ._ t, S > 0}.

The first, third, and fourth events in this decomposition are in ., either trivially or by virtue of Proposition 2.3. As for the second event, we rewrite it as:

U ft > T > r, S > t - r}, re Q+

0 0; (ii) T > 0, T is a stopping time. 2.11 Lemma. Let {T} n°11 be a sequence of optional times; then the random times sup Tn, n> 1

inf T, lim T, lim 'T n> 1

n-,:o

n-co

are all optional. Furthermore, if the Tn's are stopping times, then so is supn,

1 Tn.


8

PROOF. Obvious, from Corollary 2.4 and from the identities co

{ sup T,, 5 t} n = n >1

{7,, 5 t}

and

n=1

{ inf n>1

.

T < t} = U I T,, < tl.

El

n=1

How can we measure the information accumulated up to a stopping time T? In order to broach this question, let us suppose that an event A is part of this information, i.e., that the occurrence or nonoccurrence of A has been decided by time T Now if by time t one observes the value of T, which can happen only if T 5 t, then one must also be able to tell whether A has occurred. In other words, A n {T < t} and A' n {T 5 t} must both be Frmeasurable, and this must be the case for any t 0. Since

A' n IT

t} = IT

t} n (An IT

t})`,

it is enough to check only that A n IT < t} e Ft, t > 0. 2.12 Definition. Let T be a stopping time of the filtration {A}. The o--field FT of events determined prior to the stopping time T consists of those events A e F for which An {T < 4 E, for every t > 0.

2.13 Problem. Verify that FT is actually a 6 -field and T is FT-measurable. Show that if T(w) = t for some constant t > 0 and every w e n, then FT = A. 2.14 Exercise. Let T be a stopping time and S a random time such that S > T on n. If S is FT-measurable, then it is also a stopping time. 2.15 Lemma. For any two stopping times T and S, and for any A e Fs, we have A n {S < TI e FT. In particular, if S _-_ Ton Si, we have Fs FT.

PROOF. It is not hard to verify that, for every stopping time T and positive constant t, T A t is an F,- measurable random variable. With this in mind, the claim follows from the decomposition:

A n{ S 5 T} n {T5 t} = [A nIS 5 tnnIT5tInIS A t

TA tl,

which shows readily that the left-hand side is an event in A. 2.16 Lemma. Let T and S be stopping times. Then .."°-"T A s = FT n Fs, and each

of the events

IT < SI, IS < TI, IT 5 SI, IS 5 TI, IT = SI belongs to FT n Fs. PROOF. For the first claim we notice from Lemma 2.15 that FT,,s _c_ FT n Fs.

In order to establish the opposite inclusion, let us take A E .9-'s n FT and

9

1.2. Stopping Times

observe that

An{SA Tst}=An[ISstlulTstl] = [A n IS

tl] u [A n { T

t}] e F

and therefore A e Fs T From Lemma 2.15 we have {S < T} EFT, and thus {S > T} e FT. On the

other hand, consider the stopping time R = S A T, which, again by virtue of Lemma 2.15, is measurable with respect to Fr. Therefore, {S < T} = {R < T} e Fr. Interchanging the roles of S, T we see that {T > S }, {T < S} belong to Fs, and thus we have shown that both these events belong to FT n Fs. But then the same is true for their complements, and consequently also for {S = T}. 2.17 Problem. Let T, S be stopping times and Z an integrable random variable. We have

(i) E[ZIFT] = ECZ I's AT], P-a.s. on {TS S} (ii) E[E(ZIFT)I.F.s] = E[ZLFS A sr], P-a.s. Now we can start to appreciate the usefulness of the concept of stopping time in the study of stochastic processes. 2.18 Proposition. Let X = {X Ft; 0 < t < oo} be a progressively measurable

process, and let T be a stopping time of the filtration IF,I. Then the random variable XT of Definition 1.15, defined on the set {T < co} EFT, is FT-measurable, and the "stopped process" {X7. At .5'-'1; 0 S t < co } is progressively measurable. PROOF. For the first claim, one has to show that for any B e .4(W) and any 0, the event {Xr E B} n {T < t} is in F; but this event can also be written t in the form {XT Ate B} n {T < t}, and so it is sufficient to prove the progressive

measurability of the stopped process. To this end, one observes that the mapping (s, w) i- (T(w) A s, w) of [0, t] x SI

into itself is .4([0, t]) 0 Ft-measurable. Besides, by the assumption of progressive measurability, the mapping (s, w) i- Xs(w): ([0, t] x 12, .4( [0, t]) ® Ft) -. (Rd, .4(W))

is measurable, and therefore the same is true for the composite mapping (s, w)i- X Too (w): ([0, t] x f2, M( [0, t]) 0 Ft) -, (Rd, WV)).

2.19 Problem. Under the same assumptions as in Proposition 2.18, and with f(t, x): [0, Go) x Rd -R a bounded, M([l), co)) ®M(W)- measurable function, show that the process Y, = Po f(s, X3) ds; t > 0 is progressively measurable with

respect to {A}, and YT is an Fr-measurable random variable.


10

2.20 Definition. Let T be an optional time of the filtration {,F,}. The cr-field FT., of events determined immediately after the optional time T consists of those events A E gor' for which A n {T < t} e ..°?7,÷ for every t > 0.

2.21 Problem. Verify that the class ,F is indeed a o--field with respect to which T is measurable, that it coincides with {Ae,F;AnIT < tle.Vt 0}, and that if T is a stopping time (so that both .FT, 1, k > 1. Obviously Tn > 'Tn+, > T, for every n > 1. Show that each T

is a stopping time, that lim, T = T, and that for every A e FT+ we have A n { T = (k/2")} e

n, k

1.

We close this section with a statement about the set of jumps for a stochastic process whose sample paths do not admit discontinuities of the second kind.

2.25 Definition. A filtration {,F,} is said to satisfy the usual conditions if it is right-continuous and contains all the P-negligible events in

2.26 Proposition. If the process X has RCLL paths and is adapted to the filtration {.97,} which satisfies the usual conditions, then there exists a sequence {Tn}n°°.1 of stopping times of 1.Fil which exhausts the jumps of X, i.e., (2.1)

{ (t, (0) E (0, cc) X f2; X,(w)

X,_ (w)}

U {(t, co) e [0, CO) X 12; Tn(w) = t }. n=1

The proof of this result is based on the powerful "section theorems" of the general theory of processes. It can be found in Dellacherie (1972), p. 84, or Elliott (1982), p. 61. Note that our definition of the terminology "{ T},`;"_, exhausts the jumps of X" as set forth in (2.1) is a bit different from that found

1.3. Continuous-Time Martingales

11

on p. 60 of Elliott (1982). However, the proofs in the cited references justify our version of Proposition 2.26.

1.3. Continuous-Time Martingales We assume in this section that the reader is familiar with the concept and basic properties of martingales in discrete time. An excellent presentation of this material can be found in Chung (1974, §§9.3 and 9.4, pp. 319-341) and we shall cite from this source frequently. Alternative references are Ash (1972) and Billingsley (1979). The purpose of this section is to extend the discrete-time results to continuous-time martingales. The standard example of a continuous-time martingale is one-dimensional Brownian motion. This process can be regarded as the continuous-time ver-

sion of the one-dimensional symmetric random walk, as we shall see in Chapter 2. Since we have not yet introduced Brownian motion, we shall take instead the compensated Poisson process as a continuing example developed in the problems throughout this section. The compensated Poisson process is a martingale which will serve us later in the construction of Poisson random measures, a tool necessary for the treatment of passage and local times of Brownian motion. In this section we shall consider exclusively real-valued processes X = {X1; 0 < t < col on a probability space (0, P), adapted to a given filtration 1,,1 and such that El X,I < co holds for every t > 0.

3.1 Definition. The process {X Ft; 0 < t < col is said to be a submartingale (respectively, a supermartingale) if, for every 0 < s < t < cc, we have, a.s. P: E(X,I.F) X, (respectively, E(Xtig;) < X3). 0 < t < col is a martingale if it is both a subWe shall say that {X martingale and a supermartingale. 3.2 Problem. Let T1, T2, ... be a sequence of independent, exponentially distributed random variables with parameter A > 0:

P[7; E dt] = Ae't dt,

t > 0.

Let Si, = 0 and S = E7=1 Ti; n > 1. (We may think of S as the time at which the n-th customer arrives in a queue, and of the random variables T, i = 1, 2, ... as the interarrival times.) Define a continuous-time, integer-valued RCLL process (3.1)

N, = max {n > 0; S

1};

0

t < co.

(We may regard N, as the number of customers who arrive up to time t.)

(i) Show that for 0 < s < t we have


12

> tig,N] = e-1("), a.s. P. (Hint: Choose Ae AN and a nonnegative integer n. Show that there exists an event A ea(T,,...,Tn) such that A n {N, = n} = An {N, = n}, and use the independence between Tn." and the pair (S, 1A) to establish fin{r,=.}

P[S,, > t1.97,^] dP = e-A(`-s)P[A n {N, = n }].)

(ii) Show that for 0 < s < t, N, -N, is a Poisson random variable with parameter A.(t - s), independent of Fs1.1 . (Hint: With A e AN and n > 0 as before, use the result in (i) to establish

PEN, -N, < k!3 N] L{Ns=n}

= P[A n {N, = n }].

e-'(`-s)

(A(t - s))i

J=--0

j!

for every integer k > 0.)

3.3 Definition A Poisson process with intensity A > 0 is an adapted, integervalued RCLL process N = {N A; 0 < t < oo } such that N0 = 0 a.s., and for 0 < s < t, N, - N, is independent of and is Poisson distributed with mean

2(t - s).

We have demonstrated in Problem 3.2 that the process N = {N AN; 0 < t < co} of (3.1) is Poisson. Given a Poisson process N with intensity A, we define the compensated Poisson process M,

- At, A; 0 < t < co.

Note that the filtrations {A"} and {AN} agree.

3.4 Problem. Prove that a compensated Poisson process {M A; t

0} is a

martingale.

33 Remark. The reader should notice the decomposition N, = M, + A, of the (submartingale) Poisson process as the sum of the martingale M and the increasing function A, = At, t > 0. A general result along these lines, due to P. A. Meyer, will be the object of the next section (Theorem 4.10).

A. Fundamental Inequalities Consider a submartingale {X,; 0 5 t < co}, and an integrable, F,-measurable random variable X,; we recall here that ,F, = a(Uto If we also have, for every 0 5 t < co, X,

a.s. P,

1.3. Continuous-Time Martingales

13

then we say that "{X 0, lim

(c) For 0 < o-
c

3 -t < c.N./27r

,

V t > Tc.

From this we can conclude the weak law of large number for Poisson processes: (N,/t)-* A, in probability as t oo. In fact, by choosing a =r and T = 2' in Problem 3.9 (c) and us'ng ebySev's inequality, one can show

P[

sup

2^12^""

Nt t

4

> s]
1, E > 0. Then by a Borel-Cantelli argument (see Chung (1974), Theorems 4.2.1, 4.2.2), we obtain the strong law of large numbers for Poisson processes: (N,/t) = A, a.s. P.

The following result from the discrete-parameter theory will be used repeatedly in the sequel; it is contained in the proof of Theorem 9.4.7 in Chung (1974), but it deserves to be singled out and reviewed. 3.11 Problem. Let {F,,} ,;°_, be a decreasing sequence of sub-a-fields of F (i.e., n > 1} be a backward submartinVn 1), and let {X., 'n+1 g g

gale; i.e., El X ni < co, X" is F-measurable, and E(XIF,, ) X,,,, a.s. P, for E(X)> -co implies that the sequence {X n} n'°-1 every n > 1. Then 1 A is uniformly integrable.

3.12 Remark. If IX Ft; 0

t < ool is a submartingale and {t} T=1 is a nonincreasing sequence of nonnegative numbers, then {X,n, Ft.; n >_ 1} is a backward submartingale.


16

It was supposed in Theorem 3.8 that the submartingale X has rightcontinuous sample paths. It is of interest to investigate conditions under which we may assume this to be the case.

3.13 Theorem. Let X = {X A; 0 < t < co} be a submartingale, and assume the filtration {,97,} satisfies the usual conditions. Then the process X has a rightcontinuous modification if and only if the function t EX, from [0, co) to ER is right-continuous. If this right-continuous modification exists, it can be chosen so as to be RCLL and adapted to {A}, hence a submartingale with respect to {A}.

The proof of Theorem 3.13 requires the following proposition, which we establish first.

3.14 Proposition. Let X = {X ,F,; 0 < t < co} be a submartingale. We have the following: (i)

There is an event Cr e..97 with P(SI*) = 1, such that for every we Sr: the limits X,+(w) g urn Xs(w),

X,_ A lim Xs(co)

st.t seQ

stt

SEQ

exist for all t > 0 (respectively, t > 0). (ii) The limits in (i) satisfy E(Xt+1,Ft)

E(X,I.Ft4'

Xt

a.s. P, V t > 0. a.s. P, V t > 0.

(iii) {X,,, ..97,+; 0 < t < co} is a submartingale with P-almost every path RCLL. PROOF.

(i)

We wish to imitate the proof of (v), Theorem 3.8, but because we have not assumed right-continuity of sample paths, we may not use (iii) of Theorem 3.8 to argue that the events il!,'"),3 appearing in that proof have probability zero. Thus, we alter the definition slightly by considering the submartingale X evaluated only at rational times, and setting A(c3 =

E

Uto,1,Q(ot, /3; X (w)) = co},

n > 1, a
1} is a backward submartingale, with {E(Xs.)},T=1 decreasing and bounded below by E(X0). Therefore, the sequence of random variables {X,.}`°_, is uniformly integrable (Problem 3.11), and the same is of course true for {X Tn}'_, . The process is right-continuous, so XT(ow) = lim, X Tn(co) and Xs(w) = lim, Xsn(w) hold for a.e. co e D. It follows from uniform integrability that XT, Xs are integrable, and that 1, Xs dP 5 IA XT dP holds for every A a A + . 3.23 Problem. Establish the optional sampling theorem for a right-continuous

submartingale {X 5,; 0 < t < oo} and optional times S < T under either of the following two conditions: (i) T is a bounded optional time (there exists a number a > 0, such that T < a); (ii) there exists an integrable random variable Y, such that X, 5 E(YIA) a.s. P, for every t > 0.

3.24 Problem. Suppose that {X A; 0 < t < co} is a right-continuous submartingale and S < T are stopping times of {,F,}. Then (i) {XT. A t, .; 0 < < Go} is a submartingale; (ii) E[XT,,IA] > Xs," a.s. P, for every t > 0.

3.25 Problem. A submartingale of constant expectation, i.e., with E(X,) = E(X0) for every t > 0, is a martingale.

3.26 Problem. A right-continuous process X = {X ..,; 0 < t < co} with El X,I < co; 0 < t < oo is a submartingale if and only if for every pair S < T of bounded stopping times of the filtration {,F,} we have (3.2)

E(XT) > E(Xs).

3.27 Problem. Let T be a bounded stopping time of the filtration 1,,I, which satisfies the usual conditions, and define . = 'T-Ft; t > 0. Then {A} also satisfies the usual conditions. (i)

(ii)

If X = IX g",; 0 < t < col is a right-continuous submartingale, then so is )7 = {2t '' Kr+t - X T , g',; 0 < t < Op} . If g = {g .; 0 < t < oc } is a right-continuous submartingale with go = 0, a.s. P, then X = {X, A 1(t_7.) 0, ,F,; 0 < t < co} is also a submartingale.

3.28 Problem. Let Z = {Z ,5F,; 0 < t < oo} be a continuous, nonnegative martingale with Zco -4 lim, Z, = 0, a.s. P. Then for every s _. 0, b > 0:

1.4. The Doob-Meyer Decomposition 1

= -bZ,,

(i) P [sup Z,

21

a.s. on {Zs < b}.

t>s

(ii) P [sup Z, t>s

to]

= P[Z,

1

b] + -E[Zslf b

z.' 1 be an increasing sequence of right-

continuous supermartingales, such that the random variable, lim_co X:") is nonnegative and integrable for every 0 < t < co. Then there exists an RCLL

supermartingale X = {X ,F,; 0 5 t < co} which is a modification of the process

={

Ft; 0

t < co}.

1.4. The Doob-Meyer Decomposition This section is devoted to the decomposition of certain submartingales as the summation of a martingale and an increasing process (Theorem 4.10, already presaged by Remark 3.5). We develop first the necessary discrete-time results. 4.1 Definition. Consider a probability space (It P) and a random sequence {A},7=0 adapted to the discrete filtration {,},T=0. The sequence is called increasing, if for P-a.e. (c) E S/ we have 0 = Ao(w) < A,(w) < - - - , and E(A) < co holds for every n > 1.

An increasing sequence is called integrable if E(A0D) < co, where As, = A. An arbitrary random sequence { },T=0 is called predictable for the filtration {,},7_0, if for every n 1 the random variable is ,_1-measurable.

Note that if A = {A, F; n = 0, 1, ... } is predictable with E I AI < co for every n, and if {M, n = 0, 1, ...} is a bounded martingale, then the martingale transform of A by M defined by (4.1)

Yo =0 and Y "= E A(11/1 - Mk_,);

n >- 1,

k =1

is itself a martingale. This martingale transform is the discrete-time version of the stochastic integral with respect to a martingale, defined in Chapter 3. A fundamental property of such integrals is that they are martingales when parametrized by their upper limit of integration. Let us recall from Chung (1974), Theorem 9.3.2 and Exercise 9.3.9, that any


22

submartingale {X,,, ".; n = 0, 1, ... } admits the Doob decomposition X =

M + An as the summation of a martingale {M, F.} and an increasing sequence {A, ,}. It suffices for this to take Ao = 0 and A+1 = A -X + -X ki, for n 0. This increasing sequence E(X.+1 ign) = =o[E1X k+ is actually predictable, and with this proviso the Doob decomposition of a submartingale is unique. We shall try in this section to extend the Doob decomposition to suitable continuous-time submartingales. In order to motivate the developments, let us discuss the concept of predictability for stochastic sequences in some further detail. 4.2 Definition. An increasing sequence IA, F; n = 0, 1, ...1 is called natural if for every bounded martingale IM,,, .97;; n = 0, 1, ...1 we have (4.2)

E(MA) = E > Mk_i(Ak - A"), do > 1. k=1

A simple rewriting of (4.1) shows that an increasing sequence A is natural if and only if the martingale transform Y = Y1,,'=0 of A by every bounded martingale M satisfies EY = 0, n > 0. It is clear then from our discussion of martingale transforms that every predictable increasing sequence is natural. We now prove the equivalence of these two concepts. 4.3 Proposition. An increasing random sequence A is predictable if and only if

it is natural.

PROOF. Suppose that A is natural and M is a bounded martingale. With Y};,°=0 defined by (4.1), we have

E[A(M - M_1)] = EY - EY_i = 0, n > 1. It follows that

(4.3) E[M{A - E(A1,F_1) 1] = E[(M - M-,)A] + E[M_11A - E(A1,T,_1) 1]

- E[(M -

=0

for every n > 1. Let us take an arbitrary but fixed integer n > 1, and show that the random variable A is ,_1-measurable. Consider (4.3) for this fixed integer, with the martingale M given by

sgn[A - E(A1"_1)], k = n, Mk =

E(Mni,k), We obtain El An -

k > n, k = 0, 1, ..., n.

= 0, whence the desired conclusion.

1.4. The Doob-Meyer Decomposition

23

From now on we shall revert to our filtration {,51;,} parametrized by t e [0, cc) on the probability space (f2, P). Let us consider a process A = IA1; By analogy with Definitions 4.1 and 4.2, we have 0 < t < col adapted to the following: 4.4 Definition. An adapted process A is called increasing if for P-a.e. w a SI we have

(a) Ao(w) = 0 (b) t A,(w) is a nondecreasing, right-continuous function,

and E(A,) < co holds for every te [0, co). An increasing process is called integrable if E(4,0) < co, where Aco =

A,.

4.5 Definition. An increasing process A is called natural if for every bounded,

right-continuous martingale {M ,;(; 0 < t < col we have E .1

(4.4)

Ms dAs = E

1

Ms_ dA,, for every 0 < t
0, then I is right-continuous and progressively measurable. (ii) Every continuous, increasing process is natural. Indeed then, for P-a.e. E

SI we have

= 0 for every 0 < t < co,

(Ms(co) - Ms_(o.))) dA fo.o

because every path {MA)); 0 < s < col has only countably many discontinuities (Theorem 3.8(v)). (iii)

It can be shown that every natural increasing process is adapted to the filtration {,F,_ } (see Liptser & Shiryaev (1977), Theorem 3.10), provided that {,F,} satisfies the usual conditions.

4.7 Lemma. In Definition 4.5, condition (4.4) is equivalent to (4.4)'

E(M,A,) = E

Ms_ dAs. J


24

PROOF. Consider a partition H = {to, t, , ... , tk} of [0, t], with 0 = to
0). The right-continuous process {X1, A; 0 < t < co} is said to be of class D, if the family {X T} T e y is uniformly integrable; of class DL, if the family {XT} T c 9,. is uniformly integrable, for every 0 < a < oo.

4.9 Problem. Suppose X = {X1, A; 0 < t < co} is a right-continuous submartingale. Show that under any one of the following conditions, X is of class DL.

(a) X, > 0 a.s. for every t _- 0. (b) X has the special form (4.6)

X, = M, + A 0 < t < co

suggested by the Doob decomposition, where {M1, ,97,; 0 < t < co} is a martingale and {A1, A; 0 < t < co} is an increasing process. Show also that if X is a uniformly integrable martingale, then it is of class D.

The celebrated theorem which follows asserts that membership in DL is also a sufficient condition for the decomposition of the semimartingale X in the form (4.6).

4.10 Theorem (Doob-Meyer Decomposition). Let {A} satisfy the usual conditions (Definition 2.25). If the right-continuous submartingale X = {X1, A; 0 5 t < col is of class DL, then it admits the decomposition (4.6) as the summa-


25

tion of a right-continuous martingale M = {M .51;i; 0 < t < co} and an increas-

t < co}. The latter can be taken to be natural;

ing process A = {Al,

under this additional condition, the decomposition (4.6) is unique (up to indistin-

guishability). Further, if X is of class D, then M is a uniformly integrable martingale and A is integrable.

PROOF. For uniqueness, let us assume that X admits both decompositions X, = M; + A; = Ml + A;', where M' and M" are martingales and A', A" are natural increasing processes. Then {B, A A; - A," = M;' ,; 0 < t < co is a martingale (of bounded variation), and for every bounded and rightcontinuous martingale {L, .f.} we have

E[ ,(A; - A;')] = E

f

d13, = lim E E j=1

"_.co

( 0,t1

j-

[Be.) J

J-

where ri= {t(S),...,t(,")}, n > 1 is a sequence of partitions of [0, t] with = maxi j 1, we have the Doob

decomposition

= M,7, + A

,

j = 0, 1,

, 2"

where the predictable increasing sequence A(n) is given by A(,'(:!)

= AP)i + E J-1

= E E[Yr:li - Ytzo Ftz)], j = 1,...,2n. k=0

Notice also that because MI") = - A!,"), we have (4.7)

Ye?) = /1(,V) - E[A!,n]Ig"00],

j = 0, 1, ..., 2".

We now show that the sequence {4)}`°_1 is uniformly integrable. With A > 0, we define the random times


26

n.) = a A min{t } "),; A t(i'a))> A for some j, 1 < j

, r, and { TP) < a) =

} = {4 > A} E Ay. for j = 1,

We have { T71
0 as A

co.

a>1

Since the sequence { YT00}:1.1 is uniformly integrable for every c > 0, it follows from (4.10) that the sequence 1,4("))°11 is also uniformly integrable.

By the Dunford-Pettis compactness criterion (Meyer (1966), p. 20, or Dunford & Schwartz (1963), p. 294), uniform integrability of the sequence IM,")},f_1 guarantees the existence of an integrable random variable Aa, as well as of a subsequence 1/1!,nk)),`,°_i which converges to A. weakly in I):

lim E(A?k)) = E(Aa) k-.co

for every bounded random variable To simplify typography we shall assume henceforth that the preceding subsequence has been relabeled and shall denote it simply as {Anax_1. By analogy with (4.7), we define the process {A..57;,} as a right-continuous modification of (4.11)

A, = 17,

E(A.IA);

0

t

a.


27

4.11 Problem. Show that if {Aw}c°_, is a sequence of integrable random variables on a probability space (12, .F, P) which converges weakly in 1) to an

integrable random variable A, then for each 6 -field E[A(")1W] converges to E[Al] weakly in L'.

c .F, the sequence

Let n = U-=, nn. Fort E II, we have from Problem 4.11 and a comparison

of (4.7) and (4.11) that lim, E(,V)) = E(At) for every bounded random variable For s, t E n with 0 s < t < a, and any bounded and nonnegative ER(A!") - Ar)] 0, random variable we obtain E[(A, - A,)] = we get AS < A a.s. P. Because II is countable, and by selecting = for a.e. w WE SI the function ti-* At(co) is nondecreasing on II, and right-continuity

shows that it is nondecreasing on [0, a] as well. It is trivially seen that Ao = 0, a.s. P. Further, for any bounded and right-continuous martingale gt, Ftl, we have from (4.2) and Proposition 4.3: 2"

- A:740

EE

E(0A(:))

j =1 2"

= E E toni [Yon) - Ytnni] J2"

=EE

[Air - Aepi],

where we are making use of the fact that both sequences {Ai -Y

{Ar -

for t e n, are martingales. Letting n

and co one obtains by virtue

of (4.5):

E

f

sdA, = E

(0,a]

f

S

dAs,

(0,a1

as well as E

dAs, V t e [0, a], if one remembers that =E a} is also a (bounded) martingale (cf. Problem 3.24). There{SAtI fore, the process A defined in (4.11) is natural increasing, and (4.6) follows with .; 0

Mi

E [X. -

0

t

a.

Finally, if the submartingale X is of class D it is uniformly integrable, hence it possesses a last element Xco to which it converges both in 11 and almost surely as t co (Problem 3.19). The reader will have no difficulty repeating the preceding argument, with a = oo, and observing that E(A,o) < co.

Much of this book is devoted to the presentation of Brownian motion as the typical continuous martingale. To develop this theme, we must specialize the Doob-Meyer result just proved to continuous submartingales, where we discover that continuity and a bit more implies that both processes in the decomposition also turn out to be continuous. This fact will allow us to conclude that the quadratic variation process for a continuous martingale (Section 5) is itself continuous.


28

4.12 Definition. A submartingale {X .; 0 < t < oo} is called regular if for every a > 0 and every nondecreasing sequence of stopping times {

g

with T = limn T,, we have limn E(XT,.) = E(XT). 4.13 Problem. Verify that a continuous, nonnegative submartingale is regular.

4.14 Theorem. Suppose that X = {X,; 0 < t < co} is a right-continuous submartingale of class DL with respect to the filtration {A}, which satisfies the usual conditions, and let A = {A1; 0 < t < oo} be the natural increasing process in the Doob-Meyer decomposition (4.6). The process A is continuous if and only if X is regular.

PROOF. Continuity of A yields the regularity of X quite easily, by appealing

to the optional sampling theorem for bounded stopping times (Problem 3.23(i)).

Conversely, let us suppose that X is regular; then for any sequence {T}n°_, as in Definition 4.12, we have by optional sampling: limn, E(AT.) = limn E(XTn) - E(MT) = E(AT), and therefore AT(w)(w) j AT(.)(w) except for w in a P-null set which may depend on T To remove this dependence on T, let us consider the same sequence {11}n'.1 of partitions of the interval [0, a] as in the proof of Theorem 4.10, and select

a number A > 0. For each interval (tr, tp,), j = 0, 1, ..., 2" - 1 we consider a right-continuous modification of the martingale

t") < t
0 the random time

T(s) = a A inf {0

t 5 a; fir >

= a A inf {0 < t < a; 111) - (A A At) >

29


is an optional time of the right-continuous filtration {.F, }, hence a stopping time in X,' (cf. Problem 2.6 and Proposition 2.3). Further, defining for each n > I the function con( ): [0, a] -+ 11 by (p(t)= tin+)1; ti ") < t < t.11,)1, we have

(p.(7,,(E)) e tea. Because en) is decreasing in n, the increasing limit T = T(E) exists a.s., is a stopping time in 5o, and we also have

T = lim (p.(7,.(0) a.s. P. By optional sampling we obtain now 2 " -1

EN(4.1.)(,)]= E E[E(A. A ile,PV, (' T(e))I {tp)< T(e)5 ty.Vj j=1

= E[2 A Aq,.(7,n(e))],

and therefore E[(A A ilo.(T.(0)) - (A A AT,,(0)] = EN(4.1,!(,) - (2 n AT,,(0)] = E[1{7.(0 0,

P[Q > e].= P[Tn(E)< a] ,(w) - <X>.*0) + < Y>,(co) - < Y>sIcon;

0<s 0, and let 11 = {to, ti,..., t,,,}, with < t, = t, be a partition of [0, t]. Define the p-th variation 0 = to < t1 < t2 .< (p > 0) of X over the partition n to be V(P)(11) = i I Xt, - Xtk, IP. k=1

Now define the mesh of the partition 11 as 111111 = max, , (in probability); i.e., for every e > 0, ri > 0 there exists

o > 0 such that linll < (5 implies

P[I v,(2 >(n) - <x),1 > e] < ri.

The proof of Theorem 5.8 proceeds through two lemmas. The key fact employed here is that, when squaring sums of martingale increments and taking

the expectation, one can neglect the cross-product terms. More precisely, if

X e it2 and 0 <s < K hold for all s E [0, a.s. P. For any partition II = {to, tl, t,,,} as earlier we may write (see the discussion preceding Lemma 5.9 and relation (5.3)):

E(v(2)(n)- <x>1)2 =

- <X >,k_,)}]2

{(Xtk - X,k_i)2 -

= E EUX, k k=1

2 k=1

k-I

(XX,, - <X >tk_M2

EUX,k - X,k_, )4 + RX>tk - <X>tk-1)2]

2EV,(4)(11) + 2E[<X>, rni( <X>; 111111)].

As the mesh of II approaches zero, the first term on the right-hand side of this inequality converges to zero because of Lemma 5.10; the second term does as well, by the bounded convergence theorem and the sample path uniform continuity of <X>. Convergence in L2 implies convergence in probability, so this proves the theorem for martingales which are uniformly bounded. Now suppose X e Llz is not necessarily bounded. We use the technique of

localization to reduce this case to the one already studied. Let us define a sequence of stopping times (Problem 2.7) for n = 1, 2, ... by inf It > 0; 1X11 > n or <X>,

7:,

n }.

Now Xr

X,A T. is a bounded martingale relative to the filtration {.97,} - <X>t, r,,, .mot; 0 < t < co} is a bounded martingale. From the uniqueness of the Doob-Meyer decomposition, we see that (Problem 3.24), and likewise IX,2A

(5.4)

(V")>t = <X>tAT-

Therefore, for partitions II of [0, t], we have lim 11n11-0

E[

k=i

(X,k A

Xtk IA T,,)2

<X>t A 7;12 = O.

Since T. T co a.s., we have for any fixed t that P[7,, < t] = 0. These facts can be used to prove the desired convergence of v,(2)(n) to <x>, in probability.

111

1.5. Continuous, Square-Integrable Martingales

35

5.11 Problem. Let {X g'-t; 0 5 t < co} be a continuous process with the property that for each fixed t > 0 and for some p > 0, lira vt(P)(n) = L,

(in probability),

0111-0

where Li is a random variable taking values in [0, co) a.s. Show that for q > p, vi(q)(n) = 0 (in probability), and for 0 < q < limllnllo V( )(n) = GO (in probability) on the event 14 > 01.

5.12 Problem. Let X be in ./C, and T be a stopping time of IFtl. If <X> T = 0, a.s. P, then we have P[XT = 0; d0 < t < co] = 1. The conclusion to be drawn from Theorem 5.8 and Problems 5.11 and 5.12 is that for continuous, square-integrable martingales, quadratic variation is the "right" variation to study. All variations of higher order are zero, and, except in trivial cases where the martingale is a.s. constant on an initial interval, all variations of lower order are infinite with positive probability. Thus, the sample paths of continuous, square-integrable martingales are quite different from "ordinary" continuous functions. Being of unbounded first variation, they cannot be differentiable, nor is it possible to define integrals of the form Ito Ys(w) dXs(w) with respect to X e dr, in a pathwise (i.e., for every

or P-almost every w en), Lebesgue-Stieltjes sense. We shall return to this problem of the definition of stochastic integrals in Chapter 3, where we shall give Ito's construction and change-of-variable formula; the latter is the counterpart of the chain rule from classical calculus, adapted to account for the unbounded first, but bounded second, variation of such processes. It is also worth noting that for X e..11`2, the process <X>, being monotone, is its own first variation process and has quadratic variation zero. Thus, an integral of the forml Yid < X>, is defined in a pathwise, Lebesgue-Stieltjes sense (Remark 4.6 (i)).

We discuss now the cross-variation between two continuous, squareintegrable martingales.

5.13 Theorem. Let X = {X Fi; 0 < t < co} and Y = {X f7i; 0 < t < co} be members of dic2 . There is a unique (up to indistinguishability) {A}-adapted, continuous process of bounded variation {A ..Ft; 0 < t < co} satisfying Ao = 0

a.s. P, such that {X,Yi - A A; 0 < t < co} is a martingale. This process is given by the cross-variation <X, Y> of Definition 5.5. PROOF. Clearly, A = <X, Y> enjoys the stated properties (continuity is a con-

sequence of Theorem 4.14 and Problem 4.13). This shows existence of A. To prove uniqueness, suppose there exists another process B satisfying the conditions on A. Then

M A (X Y - A) - (X Y - B) = B - A is a continuous martingale with finite first variation. If we define

36


T = inf It

0:

=n },

then IM:n)

MIA T, .Ft; 0 5 t < col is a continuous, bounded (hence squareintegrable) martingale, with finite first variation on every interval [0, t]. It follows from (5.4) and Problem 5.11 that <M> t

T,, =

<M() )t = 0 a.s.,

t > 0.

Problem 5.12 shows that M(n) FE 0 a.s., and since 7,, 1. cc as n

co, we conclude

that M= 0, a.s. P. 5.14 Problem. Show that for X, Y e .//t2 and H = {to, t1,..., t,,,} a partition of

[0, a

lim E (xtk -

-

= <X, Y>,

(in probability).

11r111-0 k=1

Twice in this section we have used the technique of localization, once in the proof of Theorem 5.8 to extend a result about bounded martingales to squareintegrable ones, and again in the proof of Theorem 5.13 to apply a result about

square-integrable martingales to a continuous martingale which was not necessarily square-integrable. The next definitions and problems develop this idea formally.

5.15 Definition. Let X = {X

0 < t < col be a (continuous) process with

X0 = 0 a.s. If there exists a nondecreasing sequence { 7,,},T=1 of stopping times of such that A Xt AT , .Ft; 0 < t < co} is a martingale for each n 1

and P[lim,'T = oo] = 1, then we say that X is a (continuous) local martingale and write X e di' (respectively, X e icI if X is continuous). 5.16 Remark. Every martingale is a local martingale (cf. Problem 3.24(i)), but

the converse is not true. We shall encounter continuous, local martingales which are integrable, or even uniformly integrable, but fail to be martingales (cf. Exercises 3.3.36, 3.3.37, 3.5.18 (iii)).

5.17 Problem. Let X, Y be in .41`1". Then there is a unique (up to indistinguishability) adapted, continuous process of bounded variation <X, Y> jrioc. If X = Y, we satisfying <X, Y>0 = 0 a.s. P, such that X Y - <X, write <X> = <X, X>, and this process is nondecreasing. 5.18 Definition. We call the process <X, Y> of Problem 5.17 the cross-variation

of X and Y, in accordance with Definition 5.5. We call <X> the quadratic variation of X.

5.19 Problem. (i) A local martingale of class DL is a martingale. (ii) A nonnegative local martingale is a supermartingale. (iii) If M E ./IrI" and S is a stopping time of then E(MD < E <M)s, where Ai! -4

37

1.5. Continuous, Square-Integrable Martingales

We shall show in Theorem 3.3.16 that one-dimensional Brownian motion IB ..11,; 0 < t < col is the unique member of .11`.1" whose quadratic variation

at time t is t; i.e., B,2 - t is a martingale. We shall also show that d-dimensional Brownian motion {(Bp), ..., iv)), A; 0 < t < col is characterized by the condition , = 5,t, t > 0, where 64 is the Kronecker delta.

5.20 Exercise. Suppose X e di2 has stationary, independent increments. Then

<X>, = t(EXt),t

0.

5.21 Exercise. Employ the localization technique used in the solution of Problem 5.17 to establish the following extension of Problem 5.12: If X e dioc'1" and for some stopping time T of {Ft} we have <X>, = 0 a.s. P, then P[XT, t = 0; V 0 < t < co] = 1. In particular, every X e dr' of bounded first variation is identically equal to zero.

We close this section by imposing a metric structure on di2 and discussing the nature of both dif2 and its subspace .AZ under this metric. 5.22 Definition. For any X e 112 and 0 < t < co, we define 11X

N/E(X().

We also set IIXII

E =,

11X11,, A 1

2"

11, on [0, co) is nondecreasing, Let us observe that the function t because X2 is a submartingale. Further, IIX -Y II is a pseudo-metric on .#2, which becomes a metric if we identify indistinguishable processes. Indeed, suppose that for X, Y e.//l2 we have 11X - Y11 = 0; this implies X = Y,, a.s. P, for every n > 1, and thus X, = E(X1";,) = E(Y.1.-Fr) = Y, a.s. P, for every 0 < t < n. Since X and Y are right-continuous, they are indistinguishable (Problem 1.5).

5.23 Proposition. Under the preceding metric, //2 is a complete metric space, and 12 a closed subspace of A'2. PROOF. Let us consider a Cauchy sequence {X(")},,cc_1

`/#2: hmn,m.x MX(n) -

X(m)l1 = 0. For each fixed t, IAT1)1'_1 is Cauchy in L2(0, y, P), and so has an L2-limit X,. For 0 < s < t < cc and A e A, we have from L2-convergence and the Cauchy-Schwarz inequality that lim,E[1A(X!") - Xs)] = 0, limn E[lA(X") - X,)] = 0. Therefore, E[lAX:")] = E[1AX!")] implies E[lAXJ = E[lAX,], and X is seen to be a martingale; we can choose a right-continuous modification so that X e .4/2. We have II X( ") -X II = 0.


38

To show that 2 is closed, let {X(")},T-1 be a sequence in Jr2 with limit X in di,. We have by the first submartingale inequality of Theorem 3.8:

P[ sup Pq") - Xti >

is integrable: E <M>, < co. Then:

(i) M is a martingale, and M and the submartingale M2 are both uniMt exists a.s. P, and formly integrable; in particular, Moc, = EMS = E<M>; (ii) we may take a right-continuous modification of Z, = E(M.021.fl - Mi2; t 0, which is a potential. 5.25 Problem. Let M e .1r I" and show that for any stopping time T of 1.97,1, (5.5)

P[ max 1M,1 0 0, .5 > 0. In particular, for a sequence proln.°_1 c .A'''"c we have (5.6)

max IM(")1 t

<M(")> T

n-boo

5.26 Problem. Let IM A; 0 < t < co and IN %; 0

0.

t < ocl on (S2,3'7, P)

be continuous local martingales relative to their respective filtrations, and assume that .97,0 and are independent. With ff; A u %), show that

IM Yf; 0 < t < col, { N Ye; 0

t < co} and {M,N Yt' 0 < t < co} are

local martingales. If we define iti9, = ns>,a(ffs u Al), where

is the collec-

tion of P-negligible events in F, then the filtration {} satisfies the usual conditions, and relative to it the processes M, N and MN are still local martingales. In particular, <M, N> -a 0.

1.6. Solutions to Selected Problems 1.8. We first construct an example with A

The collection of sets of the form

{(X,,, Xr2,...)E B}, where B E ge(Rd) 0 ./(Ella) and 0 0; for CO E (1, 2), define X,(w) = 0 if t 0

co,

X,,(w) = 1. Choose t, = 2. If A E y,o, then for some B E AR) 0 AR) 0 and Choose t e (1, 2), some sequence g [0, 2], we have A = {(X,,, X,,, ...) t {t, , t2, ...}. Since to = is not in A and X,k(i) = 0, k = 1, 2, ..., we see that (0, 0, ...) B. But X,,,(co) = 0, k = 1, 2, ..., for all w E [0, 1]; we conclude that [0, 1] n A = 0, which contradicts the definition of A and the construction of X. We next show that if gl,,;`-

and ..97,0 is complete, then A Egi,o. Let N c S2

be the set on which X is not RCLL. Then

A=

(U An

NC,

n=1

where An =

n

_x>

U

m=1 qi,q,

1

1.

Qn10,101

-921 0. The inclusion is obvious, even for sets which are not open. Use rightcontinuity, and the fact that F is open, to go the other way. 2.7. (Wentzell (1981)): For x E Fe, let p(x, r) = inf ilx - yll; ye r}, and consider the nested sequence of open neighborhoods of F given by F,, = {x E ffid; p(x, F) < (1 /n) }. By virtue of Problem 2.6, the times T, n > 1, are optional. They form a nondecreasing sequence, dominated by H = Hr, with limit TA limn. Tn H, and we have the following dichotomy: On {1/ = 0}: T = 0, V n > 1. On {H > 0 }: there exists an integer k = k(w) > 1 such that = 0; V 1 n < k, and 0 < T < T,, < H;

Vn

k.

We shall show that T = H, and for this it suffices to establish T > H on > 0, T < col. On the indicated event we have, by continuity of the sample paths of X: XT = lim, XT. and Xi._ E arm g_ rn; V m > n > k. Now we can let in -> oo, to obtain X TE rn; V n > k, and thus XT E rn = r. We conclude with the desired result Tn < t }, valid for H < T The conclusion follows now from {H < t} = t > 0, and {H = 0} = (X0 E F}.

2.17. For every AET we know that A n < S) belongs to both ..97r (Lemma 2.16) and ys (Lemma 2.15), and therefore also to T ,.S = T n ,o6r.s. Consequently, SA

I{r.,s}E(Zi.Fr,,$)dP =

JAn

<S}dP Z

.1,4,{r,s}E(Z1.97r)dP =

1{T <S} E(Z1217T)dP, so (i) follows.

For claim (ii) we conclude from (i) that 1{T<S}E[E(ZVFT)13"-s] = E[1{r<s}E(ZI.FY1Fs] = E[1{T<s}E(ZISTs,,T)Vor's] = 1{r<s}E[ZI-97s,7-]+

which proves the desired result on the set IT < S}. Interchanging the roles of S and T and replacing Z by E(Z1.97T), we can also conclude from (i) that

40

1. Martingales, Stopping Times, and Filtrations 1{s< T}E[E(Z19'01Fs] = !Is< E[E(Z = !IS
1 and t 0. Obviously

{"T < t})e ,F n'e=i(A (UIT=1 {Tn < t}) = ,T+. The second claim is justified similarly, using Problem 2.22.

then, A n {T < t} = A Ae 3.2.

and thus

(i) Fix s > 0 and a nonnegative integer n. Consider the "trace" a-field g of all sets obtained by intersecting the members of Fs" with the set {Ns = n}. Consider also the similar trace a-field if of a(T,, ,7;,) on {N5 = n}. A generating family for g is the collection of sets of the form {N,, < n1, , /s1,k tl.FsN] dP firl{N,=n)

= P[{S±, >

= P[{S

n A n {S

T,,,, >

s < S,,}]

n A n {S

=ft-s P[{S > t - u} n A n {S = e- '11` -5)

s}] s}],le-1" du

P[{S + u > s} n A n

s}],le- A" du

= e-mt-s)P[{S + T+, > s} n A n {S = e-m")P[A n {N5 = Summation over n > 0 yields IA P[S,,,.+, >

dP = e-M`-s)P(A) for every

de Fs". (ii) For an arbitrary but fixed k > 1, the random variable Y.. g S -n+ k+I Sn +1 = E.721:1 'I; is independent of a(T,, , T+1); it has the gamma density P[Yke du] = [(Au)" /(k - 1)!]Ae-A" du; u > 0, for which one checks easily the

1.6. Solutions to Selected Problems

41

identity: k-1

(;1.. 0)-i

X

P[Yk > e]

3!

3=o

e';

k

1, 0 > O.

We have, as in (i),

P[N, -

5 klFsn dP

fin{N,=n}

= P[{N,

n

n+

(-) {Ns = n}]

= P[ {Sn +k +i >t }nAn {Ns =n }] = P[ {Sn +1 +Yk >t }nAn {Ns =n }]

=

f

P[{S+, + u > 1-} n A n {Ns = n}] P(Ykedu)

= P[A n {Ns = n}] P(Yk > t s

+ .1o

P[{S

,

-s)

> t - IA} n A n {Ns = n}]P(Yk E du)

= P[A n {Ns = n }] E

e-.1.0-0(.1(t - s))' 1

J =0

.1

.1t-s

+

CA(`-s-")P(YkEdU))

= P[A n (Ns = n }] E e-A0-0(Act J=0 i! Adding up over n > 0 we obtain

J.

P[N, -

k

klei.sndP = P(A) E e-m t-s)

(A(t -

J=0

for every :4 E ,Fsr I and k > 1, and both assertions follow.

for which

3.7. Let 1k; a E AI be a collection of linear functions from EI;Id supc A h5. Then for 0 < s < t we have

E[9(X,)Ifs] > E[h(X,)13r.s] = k,(Xs),

=

V a E A.

Taking the supremum over a, we obtain the submartingale inequality for 9(X). + < cc), so IIXII is a subNow II ' is convex and EllX,11 5_ E(IX11)1 + martingale. II

3.11. Thanks to the Jensen inequality (as in Proposition 3.6) we have that {X+ , n > 1} is also a backward submartingale, and so with ). > 0: ). PO XI > A] < EIXSI = -E(X)+ 2E(X+) 5 -1 + 2E(X < co. It follows that sup,, P[IXSI > )] converges to zero as ). -> co, and by the submartingale property: sfix,..,> A) J

X' dP

f

X; dP {x,;>-).}

dP.

42

1. Martingales, Stopping Times, and Filtrations Therefore, {X,;},`,°=, is a uniformly integrable sequence. On the other hand,

-fXdP > E(X) - I

0 > fn m.

f{x 0, we can certainly choose m so large that 0 5_ E(X,,) - E(X) < c/2 holds for every n > m, and for that m we select ). > 0 in such a way that PC,IdP < 2.

sup n>m

f{x^. f(x,-,> A)

(b): Uniform integrability allows us to invoke the submartingale convergence Theorem 3.15, to establish the existence of an almost sure limit Xo, for {X1; 0 < t < co} as t co, and to convert almost sure convergence into

3.19. (a)

L I- convergence. (b) (c): Let

be the L'-limit of {Xr; 0 < t < co }. For 0 < s < t and A E we have IA Xs dP < f X, dP, and letting t co we obtain the submartingale property IA Xs dP < L X, dP; 0 < s < co, A Fs. (C) (a): For 0 t < c o and A > 0, we have fix,,A} X, dP fixo. A} X0 dP, which converges to zero, uniformly in t, as A I co, because P[X, > < (1 /A)EX, < (1 / A)EX,. ...97,; 0 t < co } to obtain the 3.20. Apply Problem 3.19 to the submartingales equivalence of (a), (b), and (c). The latter obviously implies (d), which in turn

gives (a). If (d) holds, then jA Y dP = f A X, dP, V A E A. Letting t co, we obtain SA Y dP = fA Xco dP. The collection of sets A E 347. for which this equality holds

is a monotone class containing the field U,ZO A. Consequently, the equality holds for every A E Fc, which gives E( Yl..9700) = Xeo, a.s.

3.26. The necessity of (3.2) follows from the version of the optional sampling theorem for bounded stopping times (Problem 3.23 (i)). For sufficiency, consider 0 < s < t < co, A E ..Fs and define the stopping time S(co) A s lA (a)) + t lAc(a)). The condition

E(X,) > E(X s) is tantamount to the submartingale property E[X,1A] > E[Xs1A].

3.27. By assumption, each A contains the P-negligible events of .F. For the rightcontinuity of {g, }, select a sequence {t }'_i strictly decreasing to t; according to Problem 2.23, co

co

n=1

n n=1

3,--(T+0+,

and the latter agrees with ,97T+t = A under the assumption of right-continuity of 1.97,1 (Definition 2.20).

(i) With 0 < s < t < co, Problem 3.23 implies E[fCrigs]

E[XT, -XTIT+.]

X T+s

XT

Xs,

a.s. P.


43

(ii) Let S, < S2 be bounded stopping times of {A}; then for every t

{(Si - T) v 0 < t} = {Si

T + t}

0,

=

by Lemma 2.16. It develops that (S, - T) v 0 < (S2 - T) v 0 are bounded stopping times of {A} and so, according to Problem 3.26,

EXs = Eg(Ss_ T)v 0

Eg(Si- Dv 0 = EXst.

Another application of Problem 3.26 shows that X is a submartingale. 3.28. (Robbins and Siegmund (1970)): With the stopping time T= inf {t E [S, 00); Z, = b },

the process {ZT A A; 0 < t < co} is a martingale (Problem 3.24 (i)). It follows that for every A e 3,7 t > s:

ZsdP =

ZT AtdP

f

Arn{Z. c] < E[XT 1{V,. >0] < E(XT,,) < E(AT) < E(AT) co. On the other hand, we have

and (4.14) follows because '1;, T T n H,, a.s. as n < (5]

P[VT

E((5A AT)

E(As,T)

P[VT As L

thanks to (4.14), and (4.15) follows (adapted from Lenglart (1977)). From the identity F(x) = pos 1{,}dF(u), the Fubini theorem, and (4.15)' we have EF(VT) = J

A AT)

P(VT L u)dF(u)

+ P(AT _u)}dF(u)

0 =foopp(AT,u),E(AT,,,,,,)]dF(u)

= E[2F(AT)+

AT

f

1

-dF(U) I = EG(AT) 4r U

(taken from Revuz & Yor (1987); see also Burkholder (1973), p. 26).

5.11. Let H = {to, , t,n}, with 0 = to < t1 < q > p, we have v,(q)(n) < v,(P)(n)

< t. = t, be a partition of [0, t]. For

max IX,- X 1 0) > 0 and assume that v,o)(n) does not tend to oo (in probability) as 111111 -0. Then we can find ö > 0, 0 < K < co, and a sequence of partitions {TIn}n',1 such that P(A)L SP(L, > 0), where A= {L, > 0, V(q)(11)

Consequently, with n = {too,.

,

K);

n

I.

t;,t}, we have

v(P)(11.) < Km?-9(X; 1111"o

n > 1.

on A,,;

This contradicts the fact that v,(Pqn) converges (in probability) to the positive random variable L, on {L, > 0}.

5.12. Because <X> is continuous and nondecreasing, we have P[<X>T,, = 0; 0 < t < oo] = 1. An application of the optional sampling theorem to the continuous martingale M A X2 - <X> yields (Problem 3.24 (i)): 0 = EMT,,, = E[XL,EXh,, which implies P[XT A, = 0] = 1, for every 0 < t < oo. The conclusion follows now by continuity. 5.17. There are sequences {Sn}, {T "} of stopping times such that S. 00, T,,T 00, and XI") A Xo.,S, Yi(n) -4 Yr, T are {,,}-martingales. Define Rn A Sn A Tn A inf It

0: I X,I = n

or

I

= nl,

1.7. Notes

45

and set Xr = X in R., g(n) = li R, Note that R T cc a.s. Since XP) =

R.,

and likewise for Y("), these processes are also {,,}-martingales (Problem 3.24), and are in because they are bounded. For m > n, XP) = X:'",,),, and so

- t n R.

is a martingale. This implies a*>, = <X(m)>,,, R.,. We can thus decree <X>, A <X(")>, whenever t < R and be assured that <X> is well defined. The process

(X> is adapted, continuous, and nondecreasing and satisfies <X>0 = 0 a.s. Furthermore,

-

R = (X;n))2 - 0, and note that n:=1 C, = 0.

2.2. First Construction of Brownian Motion

51

From {C,},'..., we may construct another sequence {D,}MD=, which has D2 2 - , n,-=, Dm= 0, and limm, Q(Dm) = E > 0. Furthermore, each Dm has the form

the properties Di

Dm = {we 01('''); ((oft 1 ), ... , co(tm))e Am}

for some Am eM(Rm), and the finite sequence t A (t1, ..., t,)e T is an extension of the finite sequence t,,,_, A (ti,..., tm_, ) e T, m > 2. This may be accomplished as follows. Each Ck has a form Ck = lw e RE'''); (w(t, ), ... , w(t,k))e ilmk 1;

Am, e .4(Ork),

where tmk = (t1,..., t,de T. Since C, Ck, we can choose these representais an extension of t and Amk +, Am,, x 08mk+1 Mk. Define tions so that t D, = lw; w(t,)eRI,...,Dmi_, = Ico;()(t 1), ... ,co(t,,,j_1))e gr.-11 and Dm, = C1, as well as

Dm. +, = {a);(co(ti),..., w(tm,), a(t,,,i+i))e Am, x RI,..., D,2_ 1 = lw;(w(t 1), . . . , w(t ,,), w(t , +1), . . . , w(t,2_1))e A,, x Rm2-"11-11

and D,2 = C2. Continue this process, and note that by construction n m1 m =, Dm =

n:=1 Cm= 0. 2.4 Problem. Let Q be a probability measure on (R", .4(R")). We say that A e R(111") is regular if for every probability measure Q on (R", ,4(R")) and for

every C > 0, there is a closed set F and an open set G such that FgA G and Q(G\F) < E. Show that every set in M(01") is regular. (Hint: Show that the collection of regular sets is a a-field containing all closed sets.) According to Problem 2.4, there exists for each in a closed set Fm Am such that Q, (Am \Fm) < e/2m. By intersecting Fm with a sufficiently large closed sphere centered at the origin, we obtain a compact set Km such that, with E,,, A {we 0:8[°');(to(t 1), we have Em

'Mt.)) G Km),

D,,, and

Q(Dm \Em) = Q, (Am \Km)
0,

has stationary, independent increments. An increment B, - B3, where 0 < s < t, is normally distributed with mean zero and variance t - s.

B. The Kolmogorov-entsov Theorem Our construction of Brownian motion would now be complete, were it not for the fact that we have built the process on the sample space FEE°'"') of all real-valued functions on [0, co) rather than on the space C[0, co) of continuous

functions on this half-line. One might hope to overcome this difficulty by showing that the probability measure P in Corollary 2.6 assigns measure one to C[0, co). However, as the next problem shows, C[0, co) is not in the a-field

,4(0110'')), so P(C[0, co)) is not defined. This failure is a manifestation of the fact that the a-field gi(011°.')) is, quite uncomfortably, "too small" for a space as big as 01[0'`°); no set in .4(0110'"')) can have restrictions on uncountably

many coordinates. In contrast to the space C[0, co), it is not possible to determine a function in 0:11°'') by specifying its values at only countably many coordinates. Consequently, the next theorem takes a different approach, which is to construct a continuous modification of the coordinate mapping process in Corollary 2.6. 2.7 Exercise. Show that the only M(RE°''))-measurable set contained in C[0, co)

is the empty set. (Hint: A typical set in a(fle'')) has the form E = {co e RE°''); (co(t ,), co(t 2), . . .) e AI,

where A e a(fps x 01 x

)).

2.8 Theorem (Kolmogorov,

entsov (1956a)). Suppose that a process X =

{X1; 0 < t < T} on a probability space (LI, 31%, P) satisfies the condition (2.7)

EIX, - XX' < CI t - sl",

0

s, r 5 T,

for some positive constants a, fi, and C. Then there exists a continuous modification

1 = {gt; 0 5 t < T} of X, which is locally Holder-continuous with exponent y

2. Brownian Motion

54 for every ye (0, )3/ cc), i.e., (2.8)

P[co;

It(w)

sup

s(w)i

= 1,

It - slY

o 0, we have

P[IX, - XsI

E IX,

c]

-XI < Cc'lt - " s

Ex

and so Xs -> X, in probability ass t. Second, setting t = kr,s= (k - 1)/2", and e = 2-Yn (where 0 < y < 13/a) in the preceding inequality, we obtain

P[142. - 4-1)/2.1 > 2-Y1 < C 2-n( 1 +fl-", and consequently, P [ max 142. - X(k_i On I

(2.9)

2-Yn

1 Sk n*(w), and show that for every m > n, we have

1,

(2.11)

I Xt(w) - Xs(w) I

2

E j=n+1

v t,

se D 0 < t - s < 2

For m = n + 1, we can only have t = (k/r), s = ((k - 1)/2'), and (2.11) follows from (2.10). Suppose (2.11) is valid for m = n + 1, M - 1. Take s < t, s, te Dm, consider the numbers t1 = max{u eDm_i; u 5 t} and s1 = min fu e Dm_i; u > s }, and notice the relationships s 5 s1 < t1 < t, s1 - s 2-14, t - t1 5 2'. From (2.10) we have I Xs, (w) - Xs(c0)1 < 2 Ym, Xt(w) (w) I < 2-7m, and from (2.11) with m = M - 1, M -1

I Xit(w) - Xsi(w)I < 2 E

j=n+1

We obtain (2.11) for m = M.

2.2. First Construction of Brownian Motion

55

We can show now that { X,(w); t E DI is uniformly continuous in t for every

to en*. For any numbers s, t e D with 0 < t - s < h(w) g 2-"a", we select n > n*(w) such that 2-("+1) < t -s < 2-". We have from (2.11) (2.12)

I Xi(w) - Xs(w)I

2

bit - slY,

2-Yi

0 < t - s < h(C0),

j=n+1

where .5 = 2/(1 - 2-Y). This proves the desired uniform continuity.

We define 2 as follows. For w n*, set 21(w) = 0, 0 < t < 1. For w e n* and t e D, set 2,(co) = X,(w). For w E fr and t E [0, 1] n DC, choose a sequence

D with s t; uniform continuity and the Cauchy criterion imply that {X,(w)}T=1 has a limit which depends on t but not on the particular sequence {s},T=1 g D chosen to converge to t, and we set 2,(w) = Xs.(co). The resulting process X is thereby continuous; indeed, 2 satisfies (2.12), so {s},,`°_,

(2.8) is established.

To see that 2 is a modification of X, observe that 2, = X, a.s. for teD; for t e [0, 1] n DC and {s}'_,

and X, -51, a.s., so

=

t, we have

D with sn a.s.

X, in probability

2.9 Problem. A random field is a collection of random variables {X1; t Ed}, where d is a partially ordered set. Suppose {X,; t e [0, T]d }, d > 2, is a random field satisfying El X, - X512 < C Ilt

(2.13)

-

for some positive constants ; f3, and C. Show that the conclusion of Theorem 2.8 holds, with (2.8) replaced by (2.14)

+;

sup s,re[0,T[a

I ii(w) - XS(w)1

Ilt - sll.

< .5] = 1.

2.10 Problem. Show that if B, - Bs, 0 < s < t, is normally distributed with mean zero and variance t - s, then for each positive integer n, there is a positive constant C for which E IB, -

clt - sr.

2.11 Corollary to Theorem 2.8. There is a probability measure P on (41[0,-), ; t > 0} on the same space, a(11V0.))), and a stochastic process W = {W such that under P, W is a Brownian motion. PROOF. According to Theorem 2.8 and Problem 2.10, there is for each T > 0 a modification WT of the process B in Corollary 2.6 such that WT is continuous on [0, T]. Let = {w; WT (w) = B,(w) for every rational t e [0, T] },

so P(S1T) = 1. On n A WT (w)

n T =1 nT, we have for positive integers T1 and "2, wt T2((0), for every rational t e [0, T1 A T2].

2. Brownian Motion

56

Since both processes are continuous on [0, T, A T2], we must have W,Ti (co) --

W,T2(co) for every t e[0, T, A T2], to e n. Define W,(w) to be this common value. For co 0 n, set W,(w) = 0 for all t > 0.

2.12 Remark. Actually, for P-a.e. to e FRE'"), the Brownian sample path { W,(w); 0 < t < col is locally Holder-continuous with exponent y, for every y e (0, 1/2). This is a consequence of Theorem 2.8 and Problem 2.10.

2.3. Second Construction of Brownian Motion This section provides a succinct, self-contained construction of Brownian motion. It may be omitted without loss of continuity. Let us suppose that {B ..F,; t > 0} is a Brownian motion, fix 0 < s < t < co, and set 0 A (t + s)/2; then, conditioned on Bs = x and B, = z, the random variable Be is normal with mean it A (x + z)/2 and variance a2 A (t - s)/4. To verify this, observe that the known distribution and independence of the increments B Be - Bs, and B, -Bo lead to the joint density ;x

P[Bse dx, Bo e dy, B, e dz] = p(s;0, x)p

= p(s; 0, x)p(t - s; x, z)

1

a .,/ 2it

exp

{ (Y

y) p

12)2 }

2a2

(t

S

; y,

z) dx dy dz

dx dy dz

in the notation of (2.6), after a bit of algebra. Dividing by P[Bse dx, B, e dz] = p(s; 0, x)p(t - s; x, z) dx dz,

we obtain (3.1)

P[B0+.0/2 E dy IA = x, B, = z] =

1

e-(Y-P)212a2 dy.

a .,/ 2it

The simple form of this conditional distribution for B(, +S) /2 suggests that we can construct Brownian motion on some finite time-interval, say [0, 1], by interpolation. Once we have completed the construction on [0, 1], a simple "patching together" of a sequence of such Brownian motions will result in a Brownian motion defined for all t > 0. To carry out this program, we begin with a countable collection IV,"); k e 1(n), n = 0, 1, ... } of independent, standard (zero mean and unit variance) normal random variables on a probability space (Q,,5--., P). Here 1(n) is the set of odd integers between 0 and 2"; i.e., /(0) = {1}, /(1) = {1}, /(2) = 11, 31, etc. For each n > 0, we define a process B(n) = IBI"); 0 < t 5 1} by recursion and

linear interpolation, as follows. For n > 1, /3,172,, , will agree with B172-.91, k = 0, 1, ..., 2"-`. Thus, for each value of n, we need only specify B,13., for k e 1(n). We set

2.3. Second Construction of Brownian Motion

57

/V) =

Bo) = 0,

If the values of 13172-.1)i , k = 0, 1, ..., 2"-1 have been specified (so $"-1) is defined for 0 < t < 1 by piecewise-linear interpolation) and k E 1(n), we deand o-2 = (t - s)/4 = = (k + 1)/2", p = Rjr-1) note s = (k 1/2'1 and set, in accordance with (3.1), /1 + 6i;k").

B(7+ s)/ 2

We shall show that, almost surely, $10 converges uniformly in t to a continuous

function B and {B ,,11; 0 < t < 1) is a Brownian motion. Our first step is to give a more convenient representation for the processes Boo, n = 0, 1, .... We define the Haar functions by I-11°)(t) = 1, 0 < t < 1, and 1, k e I(n), for n

k-1

2(n-1)/2,

Hr)(t) =

0 (3.3)

PNri > x]

e-"2/2 du

-2

2 7i

-u C"2/2 du

xX

=

- e--x2/2 TC X

which gives

P[bn >

=

Pr keI

>

2" P[VI

n]

, n > 1.

2. Brownian Motion

58

Now E'_, 2"C"212/n < co, so the Borel-Cantelli lemma implies that there is a set n with P(C2) = 1 such that for each w e n there is an integer n(co) satisfying b(co) < n for all n > n(w). But then

E idn'snoi < E n2-("+"2 < co;

E

n=n(w) k E 1(n)

n=n(w)

so for wen, Bp)(0 converges uniformly in t to a limit Bt(co). Continuity of {B,(co); 0 < t < 1} follows from the uniformity of the convergence.

Under the inner product = 14 f(t)g(t)dt, L2[0,1] is a Hilbert space, and the Haar functions IM"); ke 1(n), n > 0} form a complete, orthonormal system (see, e.g., Kaczmarz & Steinhaus (1951), but also Exercise 3.3 later). The Parseval equality CO

= applied to f =

E 0, there exists SW > 0 such that a whenever 0 < s, t < T and Is - tI < (5(a). The same in16.(s) equality, therefore, holds for w when we impose the additional condition s, t e Q+ . It follows that w is uniformly continuous on [0, T] n 12÷ and so has an extension to a continuous function, also called w, on [0, T]; furthermore,

lw(s) - w(t)I < a whenever 0 < s, t < T and Is - tI < Ma). For n sufficiently large, we have that whenever t e [0, T], there is some rk e Q+ with k < n and It -rk S(a). For sufficiently large M > n, we have Ith,,,(rJ) - co(ri)1 5 a for all j = 0, 1, , n and m > M. Consequently,

Ith.(t) -

- 65.0.01 + Idim(rk) - co(rk)1 + ko(rk) - w(t)I

< 3E, Vm> M,0 0, so {(7)1,7_, converges uniformly on bounded intervals to the function we C[0, co). I=1

4.10 Theorem. A sequence IP1,1'._, of probability measures on (C[0, x), a(c[o, co))) is tight if and only if (4.6)

lim sup P,,[(0; At

(4.7)

> A] = 0,

n> 1

lim sup P[w; mT(w,(5) > a] =0; V T > 0, a > 0. blo n>

PROOF. Suppose first that {P.},-1 is tight. Given ri > 0, there is a compact set K with PP(K) > I - ry, for every n > 1. According to Theorem 4.9, for sufficiently large 1. > 0, we have 103(0)1 < 2 for all we K; this proves (4.6). According to the same theorem, if T and a are also given, then there exists So such that mT(w, (5) < a for 0 < S < (50 and we K. This gives us (4.7). Let us now assume (4.6) and (4.7). Given a positive integer T and n > 0, We choose

> 0 so that

64

2. Brownian Motion

1/2'.

sup P[(o; 10)(0)1 > n >1

We choose (5k > 0, k = 1, 2, ... such that sup P,,[(2.); mT (co, bk) >

1

qi2T+k+1.

r1.1

Define the closed sets 1

AT = {(0,10)(0)1

A, mT (co, Sk)

k = 1, 2, ...},

A= n

AT,

T=1

so Pn(AT) > 1 - E,T=o 17/27--f-k+i = 1 - 17/2T and PP(A) > 1 - n, for every n By Theorem 4.9, A is compact, so {P},°,p_1 is tight.

1. 1=1

4.11 Problem. Let {X(m)}:=1 be a sequence of continuous stochastic processes X(m) = IX:m); 0 < t < co} on (SI, F, P), satisfying the following conditions:

(i) sup,i EIXr1" A M < oo, (ii) supm,i El X; m) - X.1m)12 < CT1t

V T > 0 and 0 < s, t < T

s11+13;

for some positive constants a, /3, v (universal) and CT (depending on T > 0). Show that the probability measures Pm P(X(m))-1; m > 1 induced by these processes on (C[0, oo), .4(C [0, co))) form a tight sequence. (Hint: Follow the technique of proof in the Kolmogorov-Centsov Theorem 2.8, to verify the conditions (4.6), (4.7) of Theorem 4.10).

4.12 Problem. Suppose {P}n'=, is a sequence of probability measures on (C[0, oo),M(C[0, co))) which converges weakly to a probability measure P. Suppose, in addition, that If1,;°_1 is a uniformly bounded sequence of realvalued, continuous functions on C[0, co) converging to a continuous function f, the convergence being uniform on compact subsets of C[0, co). Then (4.8)

lim n

f CIO ,

f(co)dP,,(w) = )

f

f(w)dP(w).

Cio . 00)

4.13 Remark. Theorems 4.9, 4.10 and Problems 4.11, 4.12 have natural extensions to C[0, oor, the space of continuous, Rd-valued functions on [0, co). The proofs of these extensions are the same as for the one-dimensional case.

C. Convergence of Finite-Dimensional Distributions Suppose that X is a continuous process on some (n,

P). For each co, X,(w) is a member of C[0, co), which we denote by X(w). Since Al(C[0, co)) is generated by the one-dimensional cylinder sets and X1() is .F-measurable for each fixed t, the random function X: S2 -> C[0, oo) is .97,4(C[0, co))-measurable. Thus, if {X(")}'_1 is a sequence of continuous processes (with each X(") defined on a perhaps distinct probability space the function t

2.4. The Space C[0, co], Weak Convergence, and Wiener Measure

65

(0.,:y7, Pa)), we can ask whether V") X in the sense of Definition 4.4. We can also ask whether the finite-dimensional distributions of {X(")}x_i converge to those of X, i.e., whether (X7), , )0")) 4 (x,,, xt2,..., xtd).

The latter question is considerably easier to answer than the former, since the convergence in distribution of finite-dimensional random vectors can be resolved by studying characteristic functions. For any finite subset { t1, Rd as td: C[0, CO)

,

td} of [0, co), let us define the projection mapping = (co(t

),

co(td)).

If the function f: Rd IR is bounded and continuous, then the composite mapC[0, co) -.CI enjoys the same properties; thus, X(n) 4: X ping f 0 implies lim En fM7),

,

n-,co

lim En(f 0 n

= E(f 0

= Ef(X,,,

, Xid).

In other words, if the sequence of processes {X(")}`°_1 converges in distribution

to the process X, then all finite-dimensional distributions converge as well. The converse holds in the presence of tightness (Theorem 4.15), but not in general; this failure is illustrated by the following exercise. 4.14 Exercise. Consider the sequence of (nonrandom) processes 1q") = nt

(t1+ (1 - nt)

1[0, 1/2n]. ,

1,112n, lin](t);

0 < t < oo, n > 1 and let X, = 0, t > 0. Show that all finite-dimensional distributions of X(n) converge weakly to the corresponding finite-dimensional distributions of X, but the sequence of processes {X(")}'_, does not converge in distribution to the process X. 4.15 Theorem. Let {X(")}°°_, be a tight sequence of continuous processes with < td < co, then the sequence of random the property that, whenever 0 < t, < vectors X1,1,))1°11 converges in distribution. Let Pn be the measure induced on (C[0, co), Ac[o, co))) by V"). Then {Pn},T=1 converges weakly to a measure P, under which the coordinate mapping process W,(w) w(t) on C[0, co) satisfies (X:7),

,

X:d")) 4 (w,,,..., w,), 0 < ti
1, X(n) and Y(") are defined on the same probability space. If X(") - X and p(X("), Y(")) --> 0 in

probability, as n - co, then Y(n) 4 x as n --> co.

D. The Invariance Principle and the Wiener Measure Let us consider now a sequence gilT,i of independent, identically distributed random variables with mean zero and variance a2, 0 < o-2 < co, as well as the sequence of partial sums So = 0, Sk = El=i , k > 1. A continuous-time process Y = { Y; t 0} can be obtained from the sequence {Sk},T=0 by linear interpolation; i.e., (4.9)

Y, = ST,1 + (t - TtKp.,,

t

0,

where N] denotes the greatest integer less than or equal to t. Scaling appropriately both time and space, we obtain from Y a sequence of processes pOnl: (4.10)

X: n) =

aJ Y. 1

t>

0.

2.4. The Space C[0, co], Weal. Convergence, and Wiener Measure

67

Note that with s = k/n and t = (k + 1)/n, the increment Xr -Xr = (1/o-ji)k+i is independent of

=

1,

Furthermore, Xr -

,

has zero mean and variance t - s. This suggests that {X:"); t > 01 is approximately a Brownian motion. We now show that, even though the random variables are not necessarily normal, the central limit theorem dictates that the limiting distributions of the increments of X(n) are normal.

4.17 Theorem. With {VI defined by (4.10) and 0 < t1

o-,/n

0} by (4.10), and let Pn be the measure induced by X(n) on (C[0, co), .4(C[0, co))). Then {Pn}n'=1 converges weakly to a measure P*,

2.5. The Markov Property

71

under which the coordinate mapping process W(co) standard, one-dimensional Brownian motion.

w(t) on C[0, co) is a

PROOF. In light of Theorems 4.15 and 4.17, it remains to show that {X(")}x_1 is tight. For this we use Theorem 4.10, and since X' = 0 a.s. for every n, we need only establish, for arbitrary s > 0 and T > 0, the convergence

lim sup

(4.14)

64.0 n> 1

P[

max PC.!") - XP) I > El = 0.

Is-tl1 in this expression by lim, since for a finite number of integers n we can make the probability appearing in (4.14) as small as we choose, by reducing 6. But

P[

max

1X1n)

- x:")I > 61= fi[ max

I Ys -

> so- ,A11,

rib

0 <s,i< T

0 <s,t , = 654.; 1 < , M(d)) is independent of .yo.

j < d. Furthermore, the vector of martingales M = (M(1),

5.6 Definition. Given a metric space (S, p), we denote by AS)" the completion

of the Borel o-field .4(S) (generated by the open sets) with respect to the finite measure ji on (S, .4(S)). The universal o--field is ail(S) A 11,,,R(S)P, where

the intersection is over all finite measures (or, equivalently, all probability measures) u. A ail(S)/.4(R)-measurable, real-valued function is said to be universally measurable.

5.7 Problem. Let (S, p) be a metric space and let f be a real-valued function defined on S. Show that f is universally measurable if and only if for every finite measure g on (S, a(S)), there is a Borel-measurable function g: S such that pfx ES; f(x) g (x)} = 0. 5.8 Definition. A d-dimensional Brownian family is an adapted, d-dimensional

process B = {B

t > 0} on a measurable space (S/,.F), and a family of

probability measures {Px Ix c Re, such that

(i) for each F e Px(F) is universally measurable; the mapping (ii) for each x e Rd, /3' [Bo = x] = 1;

(iii) under each Px, the process B is a d-dimensional Brownian motion starting at x.

We have already seen how to construct a family of probability measures {Px} on the canonical space (C[0, oor,.4(C[0, °o)")) so that the coordinate mapping process, relative to the filtration it generates, is a Brownian motion

starting at x under any Px. With

= .4(C[0, cor), Problem 5.2 shows

that the universal measurability requirement (i) of Definition 5.8 is satisfied. Indeed, for this canonical example of a d-dimensional Brownian family, the mapping xi- Px(F) is actually Borel-measurable for each F e ,F. The reason we formulate Definition 5.8 with the weaker measurability condition is to allow expansion of .F to a larger o--field (see Remark 7.16).

2. Brownian Motion

74

B. Markov Processes and Markov Families Let us suppose now that we observe a Brownian motion with initial distribution y up to time s, 0 < s < t. In particular, we see the value of Bs, which we call y.

Conditioned on these observations, what is the probability that B, is in some set f e MO:id)? Now B, = (B, - Bs) + Bs, and the increment B, -Bs is independent of the observations up to time s and is distributed just as B,_, is under P°. On the other hand, Bs does depend on the observations; indeed,

we are conditioning on B, = y. It follows that the sum (B, - Bs) + Bs is distributed as B,_, is under P. Two points then become clear. First, knowledge of the whole past up to time s provides no more useful information about B, than knowing the value of Bs; in other words, (5.3)

P"[B, e F I.Fs] = P"[B, e Fl Bs],

0 < s < t, F e a(Rd).

Second, conditioned on Bs = y, B, is distributed as B,_, is under PY; i.e., (5.4)

P"[B,e rIB, = y] = PY [B,_, e fa o

s < t, re.4(01d).

5.9 Problem. Make the preceding discussion rigorous by proving the following. If X and Y are d-dimensional random vectors on (52,.f, P), W' is a sub-a-field of F, X is independent of W and Y is p-measurable, then for every rel/(01°): (5.5)

P[X + Ye rm = P[X + Ye TI Y], a.s. P;

(5.6)

P[X + Y e ri Y = y] = P[X + y e r], for PY-1-a.e. y e Old

in the notation of (4.2).

5.10 Definition. Let d be a positive integer and it a probability measure on (Rd, Al(Rd)). An adapted, d-dimensional process X = {X .f;; t > 0} on some probability space (52,.x, P ") is said to be a Markov process with initial distribution it if

(i) P"[X0 e r] = p(r), vre M(Rd); (ii) for s, t _. 0 and re A(V),

P"[X,e ri,Fs] = PP[X,e rixj,

P"-a.s.

Our experience with Brownian motion indicates that it is notationally and conceptually helpful to have a whole family of probability measures, rather than just one. Toward this end, we define the concept of a Markov family. 5.11 Definition. Let d be a positive integer. A d-dimensional Markov family

is an adapted process X = {X Ft; t > 0} on some (S2,,), together with a family of probability measures {Px} x, Re on (SI, F), such that (a) for each F e .9 ", the mapping x i-3 Px(F) is universally measurable; (b) Px[X0 = x] = I, V x e Rd;

2.5. The Markov Property

75

(c) for x E Rd, s, t> 0 and T e .4(Rd), Px

(d) for x e Rd, s, t

e F .Fs] = Px [XH, e r I xs],

Px-a.s;

0 and F e *Rd),

Px[X,+se F I X, = y] = PY [Xi e

y

in the notation of (4.2). The following statement is a consequence of Problem 5.9 and the discussion preceding it.

5.12 Theorem. A d-dimensional Brownian motion is a Markov process. A d-dimensional Brownian family is a Markov family.

C. Equivalent Formulations of the Markov Property The Markov property, encapsulated by conditions (c) and (d) of Definition 5.11,

can be reformulated in several equivalent ways. Some of these formulations amount to incorporating (c) and (d) into a single condition; others replace the evaluation of X at the single time s + t by its evaluation at multiple times after s. The bulk of this subsection presents those formulations of the Markov property which we shall find most convenient in the sequel. Given an adapted process X = {X t 0} and a family of probability measures {Px gld on (S/, g;), such that condition (a) of Definition 5.11 is satisfied, we can define a collection of operators { Ut}t, 0 which map bounded, Borel-measurable, real-valued functions on R' into bounded, universally measurable, real-valued functions on the same space. These are given by (5.7)

(Utf )(x) A Ex f (X

In the case where f is the indicator of F E M(Rd), we have Exf(X,) = Px [X, e F], and the universal measurability of U, f follows directly from Definition 5.11 (a);

for an arbitrary, Borel-measurable function f, the universal measurability of EU is then a consequence of the bounded convergence theorem. 5.13 Proposition. Conditions (c) and (d) of Definition 5.11 can be replaced by:

(e) For x e

s, t > 0 and T e M(Rd),

Px[X,,,e1-1,s] =

Px-a.s.

PROOF. First, let us assume that (c), (d) hold. We have from the latter:

Px[X,,serix, = y] = (U, 1 )(y) for PxXs-l-a.e. y

Rd.

If the function a(y) g (U,1,-)(y): Rd [0, 1] were AR ")- measurable, as is the case for Brownian motion, we would then be able to conclude that, for all

76

2. Brownian Motion

x e Rd, s 0: Px[Xt+serixs] = a(Xs), a.s. Px, and from condition (c): Px[X,e I" L. s] = a(Xs), a.s. Px, which would then establish (e). However, we only know that Utl r( ) is universally measurable. This means (from Problem 5.7) that, for given s, t > 0, x e Rd, there exists a Borel-measurable function g: Rd --, [0, 1] such that (5.8)

(u, 1r) (y) = g(y),

for PxXs 1-a.e. y e Rd,

whence

(Utir)(Xs) = g(Xs),

(5.9)

a.s. Px.

One can then repeat the preceding argument with g replacing the function a. Second, let us assume that (e) holds; then for any given s, t > 0 and x e Rd, (5.9) gives (5.10)

Px[X,e F I's] = g(X,),

a.s. Px.

It follows that Px [X,,, e I- 1,:j has a o-(Xs)-measurable version, and this establishes (c). From the latter and (5.10) we conclude

Px[X,,se rix, = y] = g(y) for PxX;-1-a.e. ye Rd, and this in turn yields (d), thanks to (5.8). 5.14 Remark on Notation. For given w e SI, we denote by X.(w) the function

t 1- X,(w). Thus, X. is a measurable mapping from 0,39 into ((Rd)1°'), ge(oRdyo,c.),)( the space of all Rd-valued functions on [0, co) equipped with

the smallest a-field containing all finite-dimensional cylinder sets. 5.15 Proposition. For a Markov family X, (II, .x), {Px}xe [Iv, we have:

(c') For x e Rd, s> 0 and F e A(Rdy°,03)), Px [Xs+. e Figs] = Px[Xs+. e FI Xs], Px-a.s; (d') For x e Rd, s > 0 and F e.4((Rd)1° "))),

Px[Xs+.e FIX, = y] = PY[X.e F], PxXs-l-a.e. y. (Note: If f E .4(Rd) and F = {w e(Rd)1°''); w(t)ET }, for fixed t and (d') reduce to (c) and (d), respectively, of Definition 5.11.)

0, then (c')

The collection of all sets F e .4((Rd)""3)) for which (c') and (d') hold forms a Dynkin system; so by Theorem 1.3, it suffices to prove (c') and (d') for finite-dimensional cylinder sets of the form PROOF.

F = {0.) e(Old)(°'x'); w(to) e ro,

.

., co(tn_1)er,,_1,

w(tn)er},

where 0 = to < ti < - < t, Tie mold), i = o, 1, ..., n, and n _. O. For such an F, condition (c') becomes

2.5. The Markov Property (5.11)

77

Px[Xs efo, ... , Xs+,_, e rn_ Xs+t.E rn IA]

= P[Xsero, ..., Xs+,._,ern_i, Xs+,. 6 rnIxs], P-a.s. We prove this statement by induction on n. For n = 0, it is obvious. Assume it true for n - 1. A consequence of this assumption is that for any bounded, Borel-measurable cp: IRd" -, 01,

(5.12) .Ex[cp(Xs, ..., Xs+,._,)Igs] = Ex[p(Xs, ..., Xs+,._,)1X,],

P-a.s.

Now (c) implies that (5.13)

P[Xsero, ..., Xs+tn_ie rn_i, Xs+,. e rn I gs] = Ex[1{x.e 1-0,...,xs*,._1 er_,}Px[Xs+t e rnigs+tn_jigs] = Ex [1(x. e ro,...,x.._, er_,}Px[Xs+t,, 6 TnIxs+t,,_,] Igs].

Any gx.+, -measurable random variable can be written as a Borel-measurable function of Xs+,._i (Chung (1974), p. 299), and so there exists a Borel-measurable

function g: Rd -- [0, 1], such that Px[Xs+,. e 1-.1Xs+/_, ] = g(Xs+in_i), a.s. P.

Setting cp(x0, ... , x_1) g'-- 100... 1,-;_i(x_,)g(xn_i), we can use (5.12) to replace gs by Q(Xs) in (5.13) and then, reversing the previous steps, to obtain (5.11). The proof of (d') is similar, although notationally more complex.

It happens sometimes, for a given process X = {X gt; t _. 0} on a measurable space (SI, g), that one can construct a family of so-called shift operators Os: SI - SI, s > 0, such that each Os is g/g--measurable and (5.14)

Xs,(co) = X, (Cisco);

V co e SI,

s, t > 0.

The most obvious examples occur when SI is either (W)1°.') of Remark 5.14 or C[0, co)" of Remark 4.13, .97 is the smallest o--field containing all finitedimensional cylinder sets, and X is the coordinate mapping process Xt(co) = (0(4 We can then define 00a) = co(s + ), i.e., (5.15)

(Osco)(t) = co(s + t),

t > 0.

78

2. Brownian Motion

When the shift operators exist, then the function Xs+.(w) of Remark 5.14

"

is none other than X.(0sw), so {X.eF} = BS {X. e F}. As F ranges over {X. a F} ranges over 'cx. Thus, (c') and (d') can be reformulated as follows: for every F and s > 0, Px[tV1FI.Fs] = Px[tV1FIXs],

(c")

(d")

Px[0;1 FIX, =

= PY [F],

PXXs I a.e. y.

In a manner analogous to what was achieved in Proposition 5.13, we can capture both (c") and (d") in the requirement that for every F e F cox and s > 0,

Px[VFIF] = Px.(F),

(e")

Px-a.s.

Since (e") is often given as the primary defining property for a Markov family, we state a result about its equivalence to our definition.

5.16 Theorem. Let X = {X

t > 0} be an adapted process on a measurable space (52,,), let it-Px xe Rd be a family of probability measures on (SI, .F), and

let {0,},0 be a family of .97/,-measurable shift-operators satisfying (5.14). Then X, (1/,.F), {Px }Re is a Markov family if and only if (a), (b), and (e") hold.

5.17 Exercise. Suppose that X, (11,,), {Px}xE Rd is a Markov family with shift-operators {Os}s,o. Use (c") to show that for every x e Rd, s > 0, GE",

and FED, px[G

osiFixs]

px[Gixipx[os-iFixj,

px_a.s.

We may interpret this equation as saying the "past" G and the "future" 0;1F are conditionally independent, given the "present" X. Conversely, show that (c'") implies (c").

We close this section with additional examples of Markov families.

5.18 Problem. Suppose X = {X

t > 0} is a Markov process on (CI, P) Rd and T: [0, co) L(l V,OV), the space of linear transformations from Rd to Rd, are given (nonrandom) functions with (p(0) = 0 and T(t) and gyp: [0, co)

nonsingular for every t > 0. Set Y, = (p(t) + T(t)X Then Y = is also a Markov process.

Yi,

t

0}

5.19 Definition. Let B = {13,, t > 0 }, (S2,F), {Px}xE Rd be a d-dimensional Brownian family. If /I E Rd and a n L(Rd, Rd) are constant and a is nonsingular, then with Y, + o-B we say Y = Y t 0), (C/,37), {P° lx}xRd is a d-dimensional Brownian family with drift pi and dispersion coefficient a.

This family is Markov. We may weaken the assumptions on the drift and diffusion coefficients considerably, allowing them both to depend on the location of the transformed process, and still obtain a Markov family. This is the subject of Chapter 5 on stochastic differential equations; see, in particular, Theorem 5.4.20 and Remark 5.4.21.

2.6. The Strong Markov Property and the Reflection Principle

79

5.20 Definition. A Poisson family with intensity A > 0 is a process N = IN ,Ft;

t > 01 on a measurable space (Si, .F) and a family of probability measures { px}xER, such that

(i) for each E e F, the mapping x 1-3 Px(E) is universally measurable; (ii) for each x e gri, Px [No = x] = 1; (iii) under each Px, the process IN, = N, - No, ,Ft: t > 01 is a Poisson process with intensity A.

5.21 Exercise. Show that a Poisson family with intensity A > 0 is a Markov family. Show furthermore that, in the notation of Definition 5.20 and under any I', the 6- fields Fco and Fo are independent.

Standard, one-dimensional Brownian motion is both a martingale and a Markov process. There are many Markov processes, such as Brownian motion with nonzero drift and the Poisson process, which are not martingales. There are also martingales which do not enjoy the Markov property.

5.22 Exercise. Construct a martingale which is not a Markov process.

2.6. The Strong Markov Property and the Reflection Principle Part of the appeal of Brownian motion lies in the fact that the distribution of certain of its functionals can be obtained in closed form. Perhaps the most fundamental of these functionals is the passage time 'I', to a level b e H, defined by (6.1)

Tb(co) = inf { t > 0; 13,(w) = b }.

We recall that a passage time for a continuous process is a stopping time (Problem 1.2.7).

We shall first obtain the probability density function of Tb by a heuristic argument, based on the so-called reflection principle of Desire Andre (Levy

(1948), p. 293). A rigorous presentation of this argument requires use of the strong Markov property for Brownian motion. Accordingly, after some motivational discussion, we define the concept of a strong Markov family and

prove that any Brownian family is strongly Markovian. This will allow us to place the heuristic argument on firm mathematical ground.

A. The Reflection Principle Here is the argument of Desire Andre. Let IB Ft; 0 5 t < col be a standard, one-dimensional Brownian motion on (0, ,F, P°). For b > 0, we have

2. Brownian Motion

80

p0 [Tb < t]

pOrL 1 b < t, B, > b] + P°[Tb < t, B, < b].

Now P° [Tb < t, A > b] = P° [B, > b]. On the other hand, if Tb < t and B, < b, then sometime before time t the Brownian path reached level b, and then in the remaining time it traveled from b to a point c less than b. Because of the symmetry with respect to b of a Brownian motion starting at b, the "probability" of doing this is the same as the "probability" of traveling from

b to the point 2b - c. The heuristic rationale here is that, for every path which crosses level b and is found at time t at a point below b, there is a "shadow path" (see figure) obtained from reflection about the level b which exceeds this level at time t, and these two paths have the same "probability." Of course, the actual probability for the occurrence of any particular path is zero, so this argument is only heuristic; even if the probability in question were positive, it would not be entirely obvious how to derive the type of "symmetry" claimed here from the definition of Brownian motion. Nevertheless, this argument leads us to the correct equation P ° [ Tb < t, B, < b] = P°[Tb < t, B, > b] = P°[B, > IA,

2b -

which then yields 0,

(6.2)

P° [Tb < t] = 2P° [B, > b] =

7t1bt -1/2

e-x212 dx.

Differentiating with respect to t, we obtain the density of the passage time (6.3)

P°[Tbe d[] =

,J2irt3 /27rt3

e-b2/2`dt;

t > 0.

.,

The preceding reasoning is based on the assumption that Brownian motion "starts afresh" (in the terminology of Ito & McKean (1974)) at the stopping


81

time Tb, i.e., that the process {Bt +Tb - BTh; 0 < t < co} is Brownian motion, independent of the cr-field ,Tb. If T6 were replaced by a nonnegative constant, it would not be hard to show this; if Tb were replaced by an arbitrary random time, the statement would be false (cf. Exercise 6.1). The fact that this "starting afresh" actually takes place at stopping times such as '1,, is a consequence of the strong Markov property for Brownian motion.

6.1 Exercise. Let {B ",; t > 0} be a standard, one-dimensional Brownian motion. Give an example of a random time S with P[0 < S < co] = 1, such that with W, A Bs+, - Bs, the process W = {Wt, Fr; t > 0} is not a Brownian motion.

B. Strong Markov Processes and Families 6.2 Definition. Let d be a positive integer and ti a probability measure on (Rd, .4(Rd)). A progressively measurable, d-dimensional process X = {X .F,; t > 0} on some (S2, F, P") is said to be a strong Markov process with initial distribution t if

(i) Pu[Xoe 1] =µ(r), V I" e .4(01°); (ii) for any optional time S of {Ft}, t

0 and I" eR(Rd),

r[Xs+, erls+] = PP[xs÷terixs],

P" -a.s. on {S < co}.

6.3 Definition. Let d be a positive integer. A d-dimensional strong Markov family is a progressively measurable process X = {X ,F,; t > 0} on some (C/, F), together with a family of probability measure {Px}xeRa on (SZ, g"), such that: (a) for each F e.F, the mapping x 1-* Px(F) is universally measurable; (b) Px[Xo = x] = 1, V x elle; (c) for x e Rd, t > 0, r eM(Rd), and any optional time S of {Ft},

Px[Xs+teri,s+] = Px[xs+ieriXs], Px-a.s. on {S < co}; (d) for x e Rd, t > 0, F e .4(Rd), and any optional time S of {A},

Px[xs+,e I-I Xs = y] = PY [X, e 11 PxXV -a.e. y.

6.4 Remark. In Definitions 6.2, 6.3, {Xs+, e I"} A {S < co, Xs,,e1"} and PxXV (Rd) = Px(S < co). The probability appearing on the right-hand side of Definition 6.2(ii) and Definition 6.3(c) is conditioned on the cr-field generated by Xs as defined in Problem 1.1.17. The reader may wish to verify in this connection that for any progressively measurable process X,

PPLXs+terls+i = PilXs+teriXs] = 0, P" -a.s. on {S = co}, and so the restriction S < co in these conditions is unnecessary.

2. Brownian Motion

82

6.5 Remark. An optional time of I.F,1 is a stopping time of {,t+} (Corollary 1.2.4). Because of the assumption of progressive measurability, the random

variable Xs appearing in Definitions 6.2 and 6.3 is A,-measurable (Proposition 1.2.18). Moreover, if S is a stopping time of {F,}, then Xs is Fsmeasurable. In this case, we can take conditional expectations with respect to A on both sides of (c) in Definition 6.3, to obtain

Px[Xs+i E FIFO = Px[Xs+IEFIXA,

Px-a.s. on IS
0, we obtain condition (c) of Definition 5.11. Thus, every strong Markov family is a Markov family. Likewise, every strong

Markov process is a Markov process. However, not every Markov family enjoys the strong Markov property; a counterexample to this effect, involving a progressively measurable process X, appears in Wentzell (1981), p. 161.

Whenever S is an optional time of IA and u > 0, then S + u is a stopping time of (Problem 1.2.10). This fact can be used to replace the constant s

in the proof of Proposition 5.15 by the optional time S, thereby obtaining the following result.

6.6 Proposition. For a strong Markov family X = IX Ft; t > 0 }, (S2, F), IPx}x. Re, we have

(c') for x c Rd, F c a((R' r,-)), and any optional time S of

Px[Xs+.E FIAJ = Px[Xs+.E PIXs], Px-a.s. on {S < oo 1; (d') for x E Rd, F E a((Rdr,-)), and any optional time S of {A}, Px[xs+. E FIRS = y] = PY [X. E F],

PxXS-1-a.e. y.

Using the operators IU,1, in (5.7), conditions (c) and (d) of Definition 6.3 can be combined. 6.7 Proposition. Let X = {X ,E; t > 0} be a progressively measurable process on (52,,9), and let {Px Re be a family of probability measures satisfying (a) and (b) of Definition 6.3. Then X, (52, 3T), kPx, 1 xERd is strong Markov if and only if for any {3 }- optional time t > 0, and x e Rd, one of the following equivalent conditions holds: (e)

for any r E a(OBd),

Px[X,EFIAJ =

r)(Xs),

Px-a.s. on {S < col;

(e') for any bounded, continuous f: Rd -> Ex[f(Xs+f)IFs+] = (U1.1)(Xs),

Px-a.s. on {S
0}, (R.F),

is a strong Markov

family and tt is a probability measure on (Rd, .4(Rd)), we can define a probability measure P" by (5.2) for every F e.517, and then X on (SI, , P") is a strong Markov process with initial distribution it. Condition (ii) of Definition 6.2 can be verified upon writing condition (e) in integrated form:

(Utlr)(Xs)dPx = Px[X,s+ter, F];

FE F +,

SF

and then integrating both sides with respect to it. Similarly, if X, (52, 5), {Px}xeRd is a Markov family, then X on (S2,..97,P") is a Markov process with initial distribution it.

It is often convenient to work with bounded optional times only. The following problem shows that stating the strong Markov property in terms of such optional times entails no loss of generality. We shall use this fact in our proof that Brownian families are strongly Markovian. 6.9 Problem. Let S be an optional time of the filtration {Ft} on some (C/,..F, P).

(i) Show that if Z1 and Z2 are integrable random variables and Z1 = Z2 on some ,975.4.-measurable set A, then

E[Z,1,53-5+] = E[Z21F.j,

a.s. on A.

(ii) Show under the conditions of (i) that if s is a positive constant, then

=

-

tS 4+1 a.s. on {S < s} n A.

(Hint: Use Problem 1.2.17(i)).

(iii) Show that if (e) (or (e')) in Proposition 6.7 holds for every bounded optional time S of

then it holds for every optional time.

Conditions (e) and (e') are statements about the conditional distribution of X at a single time S + t after the optional time S. If there are shift operators Als satisfying (5.14), then for any random time S we can define the random shift Os: IS < oo} -*CI by

= 0, on {S = s}. In other words, 0 is defined so that whenever S(w) < co, then

2. Brownian Motion

84

Xs(.)-Fi(w) = Xi(Os(w))

{X. e E}, and (c') and (d') are, respecIn particular, we have {Xs+. e E} = tively, equivalent to the statements: for every x e Rd, Fegicx, and any optional time S of {A}, px[os-iFIAA

(c")

Px-a.s. on {S < col;

Px [os'Fl Xs = y] = PY (F),

(d")

Px

y.

Both (c") and (d") can be captured by the single condition: (e") for x e Rd,

F

e 'co', and any optional time S of {A}, px[vFigFs+]

=

pxs,-kr,)

Px-a.s. on {S < co}.

Since (e") is often given as the primary defining property for a strong Markov family, we summarize this discussion with a theorem.

6.10 Theorem. Let X = IX ,Ft; t > 01 be a progressively measurable process on (SI, F), let {Px}xE Rd be a family of probability measures on (e, F), and let {0,}s>o be a family of F/g-measurable shift operators satisfying (5.14). Then X , (Si,

{Px} x E Rd is a strong Markov family if and only if (a), (b), and (e") hold.

6.11 Problem. Show that (e") is equivalent to the following condition:

(e"') For all x e gl, any bounded, gc,X-measurable random variable Y, and any optional time S of {gt}, we have Ex[Y 0 Osigs+] = Exs(Y),

Px-a.s. on {S < oo}.

(Note: If we write this equation with the arguments filled in, it becomes Ex [ Y 0 OsIgs+] (w) =

Y(w')Pxs(-)(')(dw'),

Px-a.e. w e {S
[0, 1] such that (i) for each wee, Q(co; ) is a probability measure on (S,M(S)), (ii) for each E e M(S), the mapping Q(w; E) is W-measurable, and (iii) for each E e .4(S), P[X e Elfl(co) = Q(w; E), P-a.e. w.


85

Under the conditions of Definition 6.12 on X, (S2, .y, P), (S, AS)), and , a regular conditional probability for X given W exists (Ash (1972), pp. 264-265, or Parthasarathy (1967), pp. 146-150). One consequence of this fact is that

the conditional characteristic function of a random vector can be used to determine its conditional distribution, in the manner outlined by the next lemma.

6.13 Lemma. Let X be a d-dimensional random vector on (CI, 37, P). Suppose is a sub-a-field of F and suppose that for each w e SI, there is a function p(w; -): Rd C such that for each u e Rd, cp(w; u) = E[ei(".x)ifl(w),

P-a.e. w.

If, for each w, cp(co; ) is the characteristic function of some probability measure P`d on (Rd ,.4(Rd)), i.e.,

(p(p; u) =

f

el("P'(dx),

Rd

where i = .J -1, then for each T a (Rd), we have

P[X el- l5](w) = P'(F), P-a.e. w. PROOF. Let Q be a regular conditional probability for X given fixed u a Rd we can build up from indicators to show that (6.4)

cp(w; u) = E[ei("m151(w) = I

ei(u.x)Q(a); dx),

, so for each

P-a.e. w.

Rd

The set of w for which (6.4) fails may depend on u, but we can choose a countable, dense subset D of Rd and an event n EY; with p(n) = 1, so that (6.4) holds for every w e n and u e D. Continuity in u of both sides of (6.4) allows us to conclude its validity for every wen and ue Rd. Since a measure is uniquely determined by its characteristic function, we must have P' = Q(w; ) for P-a.e. w, and the result follows.

Recall that a d-dimensional random vector N has a d-variate normal distribution with mean p e Rd and (d x d) covariance matrix / if and only if it has characteristic function (6.5)

Eei(u.N) = e1(")-(1"uo;

u e Rd.

Suppose B = IB A; t > 0 }, (52, 37), {Px}Egid is a d-dimensional Brownian family. Choose u a Rd and define the complex-valued process M, A exp [i(u, B,) + -1102 ], 2

t > 0.

We denote the real and imaginary parts of this process by R, and l respectively.

2. Brownian Motion

86

6.14 Lemma. For each x e Rd, the processes {R A; t > 0} and {I ;; t

0}

are martingales on 4,, Pl. PROOF. For 0 < s < t, we have PEA '1,1.9';] = Ex[Ms exp (i(u,

A -A ) + t

= Ms Ex [exp (i(u, A

- A) + t

S

-

II

u112) F1

S 11U112)]

= Ms,

where we have used the independence of B, -Bs and Fs, as well as (6.5). Taking real and imaginary parts, we obtain the martingale property for IR .; t ..._. 01

0

and II A; t > 01.

6.15 Theorem. A d-dimensional Brownian family is a strong Markov family. A d-dimensional Brownian motion is a strong Markov process. PROOF. We verify that a Brownian family B = {B ..Ft; t .. 0 }, (SI, ..F), NP{x1,.ER.

satisfies condition (e) of Proposition 6.7. Thus, let S be an optional time of {F,}. In light of Problem 6.9, we may assume that S is bounded. Fix x e Rd. The optional sampling theorem (Theorem 1.3.22 and Problem 1.3.23 (i)) applied to the martingales of Lemma 6.14 yields, for Px-a.e. w e a Ex [exp(i(u, Bs+,))1..Fs,](w) = exp [i(u, Bs,,s (w)) )

2

II

ul12] .

Comparing this to (6.5), we see that the conditional distribution of Bs, given Fs+, is normal with mean Bsos)(w) and covariance matrix t/d. This proves (e).

0 We can carry this line of argument a bit further to obtain a related result. 6.16 Theorem. Let S be an a.s. finite optional time of the filtration {. } for the d-dimensional Brownian motion B = {B A; t > 0}. Then with Wt -'4- Bs+t - Bs, the process W = {147i, Fri; t > 0} is a d-dimensional Brownian motion, independent of Fs+.

PROOF. We show that for every n u e Rd, we have a.s. P: (6.6) E [exp (i kEi (uk, "tk - w,,,_,))

1, 0 < to
0} be a standard, one-dimensional Brownian motion, and for b 0, let Tb be the first passage time to b as in (6.1). Then Tb has the density given by (6.3). PROOF. Because { -B ,57;,; t > 0} is also a standard, one-dimensional Brownian

motion, it suffices to consider the case b > 0. In Corollary 6.18 set S = Tb, ft

T=

oo

if S < t, if S > t,

and F = ( - oo, 6). On the set {T < co} -- {S < t}, we have Bs(w)(a)) = b and (Urm_s(a,)1,-)(Bs(w)(w)) = 1. Therefore, P °[Tb

< t, B, < b] =

f

P°[137- e I- 1 Fs+] dP"

1

= -2 P° [Tt, < t].

(rbo

7.6 Problem. If the process X has left-continuous paths, then the filtration 1,11 is left-continuous.

A.

Right-Continuity of the Augmented Filtration for a Strong Markov Process

We are ready now for the key result of this section.

7.7 Proposition. For a d-dimensional strong Markov process X = {Xt, Fir; t > 0} with initial distribution it, the augmented filtration 1,971 is rightcontinuous.

Pg) be the probability space on which X is defined. Fix s > 0 and consider the degenerate, {,F,,x}-optional time S = s. With 0 < to < t, < < tn < s < t,,,, < < t. and ro, F, in *Rd), the PROOF. Let (SI,

strong Markov property gives [X,0 E ro, = 1{Xt0Ero,

Xt,,,E F, fix], Pg [X tn Ern,

G rdxs],

P" -a.s. It is now evident that Pg[X toc To, ..., X tn,E1-.1,97,x+] has an ,Fsx-

2.7. Brownian Filtrations

91

measurable version. The collection of all sets Fegl for which PP[FI9,x+] has an ,sx-measurable version is a Dynkin system. We conclude from the Dynkin System Theorem 1.3 that, for every FE the conditional probability PP[FI,Fsx,] has an .Fsx-measurable version.

Let us take now FEX _c .91; we have P[F137,x+] = IF, a.s. I", so IF has an 37sx-measurable version which we denote by Y. Because G A { Y = 1} e

.f and F AG g_ {1, Y} we have Feg: and consequently ..sf, g:; s > 0. Now let us suppose that FE. for every integer n > 1 we have Fes1,11,,,), as well as a set Gne.Fx, - ,-(1/n) such that F A G EXP. We define G A n:=1U c°=m G, and since G= n := m U ,T= m G,, for any positive integer M, ,°"Fr. To prove that Fe,f, it suffices to show F A GeXP. we have GeFs'f,_ Now

G\F g.

03

00

n =1

n=1

(U G)\F = U (G\F) e X"-

On the other hand,

F\G = F n (() 0 G,,)c = F n (0 () G,`,) m=1 n=m

co r

co

= m=i U [ F n ( n=m n

m=1 n=m CO

CO

Gf)] s U (F n G,`) = U (F\Gm)e .A7. m=1

m=1

It follows that F e 0} be a d-dimensional Brownian motion with initial distribution p on (f2, .9";c0B, Pl. Relative to the filtration {,F,"}, {B t > 0} is still a d-dimensional Brownian motion. PROOF. Augmentation of a-fields does not disturb the assumptions of Definition 5.1.

7.10 Remark. Consider a Poisson process IN .F11; 0 < t < ool as in Definition 1.3.3 and denote by {,F,} the augmentation of {..Fin. In conjunction with Problems 6.21 and 7.3, Proposition 7.7 shows that { F,} satisfies the usual conditions; furthermore, {N, ,Ft; 0 < t < oo} is a Poisson process.

Since any d-dimensional Brownian motion is strongly Markov (Theorem 6.15), the augmentation of the filtration in Theorem 7.9 does not affect the strong Markov property. This raises the following general question. Suppose {X ,,x; t > 0} is a d-dimensional, strong Markov process with initial distri-

2. Brownian Motion

92

bution u on (SI, Flo`, P4). Is the process {X Ft"; t > 0} also strongly Markov?

In other words, is it true, for every optional time S of {Fil}, t > 0 and F e R(W), that (7.1)

1"[XS+t E r 1 -91+ ] = 1"[Xs+t E r 1 Xs],

PP-a.s. on {S < co }?

Although the answer to this question is affirmative, phrased in this generality the question is not as important as it might appear. In each particular case, some technique must be used to prove that {X ,97,x; t > 0} is strongly Markov in the first place, and this technique can usually be employed to establish the strong Markov property for {X F,P; t > 0) as well. Theorems 7.9 and 6.15 exemplify this kind of argument for d-dimensional Brownian motion. Nonetheless, the interested reader can work through the following series of exercises to verify that (7.1) is valid in the generality claimed.

In Exercises 7.11-7.13, X = {X ..f.fx ; 0 < t < co} is a strong Markov process with initial distribution it on (n, 'cox, P4).

7.11 Exercise. Show that any optional time S of {F,P} is also a stopping time of this filtration, and for each such S there exists an optional time T of {ix } with {S 0 TI e .AfP. Conclude that ..fli., = ,fl = .FT, where Ff is defined to be the collection of sets A e FP satisfying A nIT < tl e F,P, V 0 < t < co. 7.12 Exercise. Suppose that T is an optional time of {Fix }. For fixed positive integer n, define

on IT = col T. = 2 "'

on

{ic

- 1 < T < k} 2"

2"

-

Show that 1,, is a stopping time of {Fix}, and Ff- g o-(gq. u Al"). Conclude that 9-74! g o-(FIL u .A14). (Hint: Use Problems 1.2.23 and 1.2.24.)

7.13 Exercise. Establish the following proposition: if for each t > 0, f ea(ild), and optional time T of {Fix}, we have the strong Markov property (7.2)

PP[X,.. e FIFIQ = 134[XT.eFIXT], PP-a.s. on {T < co I,

then (7.1) holds for every optional time S of {Fr }. This completes our discussion of the augmentation of the filtration generated

by a strong Markov process. At first glance, augmentation appears to be a rather artificial device, but in retrospect it can be seen to be more useful and natural than merely completing each a -field Fix with respect to PTM. It is more

natural because it involves only one collection of PP-null sets, the collection we called ,A14, rather than a separate collection for each t > O. It is more useful because completing each a-field Fix need not result in a right-continuous filtration, as the next problem demonstrates.

93

2.7. Brownian Filtrations

7.14 Problem. Let {Bt; t > 01 be the coordinate mapping process on (C [0, 04

4(C[0, oo))), P° be Wiener measure, and Fi denote the completion of under P°. Consider the set F = lw e C[0, co); w is constant on [0, s] for some e > 01.

Show that: (i) P °(F) = 0, (ii) F e F:+, and (iii) F A.

B. A "Universal" Filtration The difficulty with the filtration Ign, obtained for a strong Markov process with initial distribution ft, is its dependence on it. In particular, such a filtration

is inappropriate for a strong Markov family, where there is a continuum of initial conditions. We now construct a filtration which is well suited for this case.

Let {X

0}5 (S2, 2,1), f DX

JxeOld be a d-dimensional, strong Markov family. For each probability measure on (Rd, ARd)), we define PP as in (5.2):

PP(F)=

f

Px(F)p(dx),

VFe

,

and we construct the augmented filtration {Fr } as before. We define (7.3)

A A n FtP, 0 < t < 00,

where the intersection is over all probability measures it on (Rd, M(Rd)). Note that Fix 0 < t < CO for any probability measure jt on (Rd, gi(Rd)); therefore, if {X t > 0} and {X t z 0} are both strongly Markovian

under Po, then so is {X gt; t > 0}. Because the order of intersection is interchangeable and {Ft"} is right-continuous, we have

= s>tn n p

Thus

=n ns>tgp= n

=

} is also right-continuous.

7.15 Theorem. Let B = IA, FtB; t > ol {Px}xco, be a d-dimensional Brownian family. Then {B gi; t > 0}, (S/, g), IPx}.E gad is also a Brownian family.

PROOF. It is easily verifed that, under each Px,{B

,; t

0} is a d-dimensional

Brownian motion starting at x. It remains only to establish the universal measurability condition (i) of Definition 5.8. Fix F e Ao. For each probability measure it on (Rd,M(Rd)), we have F e..F", so there is some G e.: F! with F AG eAr". Let N e satisfy FAG N and Pv(N)= 0. The functions g(x) g PX(G) and n(x) g PX(N) are universally measurable by assumption. Furthermore,

2. Brownian Motion

94

n(x)ii(dx) = r(N)= 0, The nonnegative functions hi (x) Px(F\G) and h2(x) so n = 0, Px(G\F) are dominated by n, so h, and h2 are zero ft-a.e., and hence h, and h2 are measurable with respect to .4(Rd)", the completion of AR") under /.1. Set f(x) Px(F). We have f(x) = g(x) + hi(x) - h2(x), so f is also .4(Rd)"measurable. This is true for every it; thus, f is universally measurable.

7.16 Remark. In Theorem 7.15, even if the mapping x 1--.Px(F) is Borel(c.f. Problem 5.2), we can conclude only its measurable for each F e universal measurability for each F e yam. This explains why Definition 5.8 was designed with a condition of universal rather than Borel-measurability.

C. The Blumenthal Zero-One Law We close this section with a useful consequence of the results concerning augmentation.

,),

7.17 Theorem (Blumenthal (1957) Zero-One Law). Let IB A; t is given by (7.3). If {Px })C Rd be a d-dimensional Brownian family, where F e go, then for each x e Rd we have either Px(F) = 0 or Px(F) = 1. PROOF. For F ego and each x e Rd, there exists G E .a)7011 such that Px(F A G) = 0.

But G must have the form G = {Boer} for some T eM(Rd), so Px(F) = r(G) = Px {Bo e F} = 1,-(x).

7.18 Problem. Show that, with probability one, a standard, one-dimensional Brownian motion changes sign infinitely many times in any time-interval [0, s], e > 0.

7.19 Problem. Let

Wt, .Ft; 0 < t < co} be a standard, one-dimensional

Brownian motion on (S1,

P), and define

Sb = inf It

0; Wt > bl;

b

O.

(i) Show that for each b > 0, P['T,, 0 Sb] = 0.

(ii) Show that if L is a finite, nonnegative random variable on (SI, which is independent of Fm, then {TL 0 SL} E.9 and P[7i,

P)

SL] = 0.

2.8. Computations Based on Passage Times In order to motivate the strong Markov property in Section 2.6, we derived the density for the first passage time of a one-dimensional Brownian motion from the origin to b 0 0. In this section we obtain a number of distributions

2.8. Computations Based on Passage Times

95

related to this one, including the distribution of reflected Brownian motion, Brownian motion on [0, a] absorbed at the endpoints, the time and value of the maximum of Brownian motion on a fixed time interval, and the time of the last exit of Brownian motion from the origin before a fixed time. Although derivations of all of these distributions can be based on the strong Markov property and the reflection principle, we shall occasionally provide arguments based on the optional sampling theorem for martingales. The former method yields densities, whereas the latter yields Laplace transforms of densities (moment generating functions). The reader should be acquainted with both methods.

A. Brownian Motion and Its Running Maximum Throughout this section, { W .Ft; 0 < t < (S/, 37;), {Px}xcgg will be a onedimensional Brownian family. We recall from (6.1) the passage times 0; W, = b};

Tb = inf {t

be

and define the running maximum (or maximum-to-date)

M, = max Ws.

(8.1)

8.1 Proposition. We have for t > 0 and a < b, b > 0: (8.2)

P°[W, e da, M, e db] =

2(2b - a)

2irt3

exp

{

(2b -

2)

da db.

2t

PROOF. For a < b, b > 0, the symmetry of Brownian motion implies that

(U,,1(_,1)(b) A Pb[W,,

2b - a]

a] =

A (Ut-s1[2b-a,0)(b);

0 < s < t.

Corollary 6.18 then yields P°[Wt < al b}, we obtain P° [ W,

a, M,

b] = P°[W, = P° [W,

Differentiation leads to (8.2).

2b - a,

26 - a] =

b] 1

2rct

e-x2/2t dx. 26-a

2. Brownian Motion

96

8.2 Problem. Show that for t > 0, (8.3)

P° [Mt e db] = P° [IWt 1 e db] = P° [Mt -W, e db] 2

/2nt

b > 0.

e-b212' db;

8.3 Remark. From (8.3) we see that (8.4)

t] = P° [Mt > 1)] =

P°

2

n'

/2,7c J

Cx2/2 dx;

b > 0.

By differentiation, we recover the passage time density (6.3):

P°[Tbe dt] =

(8.5)

,

b

e- 1'212' dt;

b > 0, t > 0.

2nt3

For future reference, we note that this density has Laplace transform Eoe-arb

(8.6)

e-b,f2;

b > 0, a > 0.

By letting t r co in (8.4) or a 10 in (8.6), we see that P° [T1' < co] = 1. It is clear from (8.5), however, that E° 'Tb = co.

8.4 Exercise. Derive (8.6) (and consequently (8.5)) by applying the optional sampling theorem to the {. }-martingale X, = exp{.1W, - IA2t}; 0

(8.7)

t < co,

> 0.

with A =

The following simple proposition will be extremely helpful in our study of local time in Section 6.2. 8.5 Proposition. The process of passage times T = {T, .97,-.4.; 0 < a < co} has

the property that, under P° and for 0 < a < b, the increment Tb -T, is independent of FT.., and has the density P° [Tb - 'Tae dt] =

b -a e-(6-0212t dt; ,

/ 2nt3

0 < t < co.

In particular, (8.8)

Eo[e-ct(Tb-T.)1,T.+]

e-(b-a) .12a;

0.

PROOF. This is a direct consequence of Theorem 6.16 and the fact that Tb -7; = inf ft 0; WT.+t WT. = b al.


97

B. Brownian Motion on a Half-Line When Brownian motion is constrained to have state space [0, oo), one must specify what happens when the origin is reached. The following problems explore the simplest cases of absorption and (instantaneous) reflection. 8.6 Problem. Derive the transition density for Brownian motion absorbed at the origin {W, To, ,Ft; 0 < t < 00 }, by verifying that (8.9)

I' [W, e dy, T, > t] = p _(t; x, y)dy -41- [p(t; x, y) - p(t; x, -y)] dy;

t > 0, x, y > O.

8.7 Problem. Show that under P°, reflected Brownian motion 'WI A {114I, A; 0 < t < co} is a Markov process with transition density (8.10)

P° El wt+sl e 411 Wt I = x] = p +(s; x, y) dy

A [p(s; x, y) + p(s; x, - y)] dy;

s > 0, t _.. 0 and x, y _. O.

8.8 Problem. Define Y, A M, - KO < t < co. Show that under P°, the process Y = {Y,, .9,; 0 < t < co} is Markov and has transition density (8.11)

P° [Y,+, e dy I Y, = x] = p +(s; x, y) dy;

s > 0, t > 0 and x, y .. 0.

Conclude that under P° the processes IWI and Y have the same finitedimensional distributions.

The surprising equivalence in law of the processes Y and I WI was observed by P. Levy (1948), who employed it in his deep study of Brownian local time (cf. Chapter 6). The third process M appearing in (8.3) cannot be equivalent in law to Y and I WI, since the paths of M are nondecreasing, whereas those

of Y and I WI are not. Nonetheless, M will turn out to be of considerable interest in Section 6.2, where we develop a number of deep properties of Brownian local time, using M as the object of study.

C. Brownian Motion on a Finite Interval In this subsection we consider Brownian motion with state space [0, a], where

a is positive and finite. In order to study the case of reflection at both endpoints, consider the function co: IR -+ [0, a] which satisfies co(2na) = 0, cp((2n + 1)a) = a; n = 0, ± 1, ±2, ..., and is linear between these points.

8.9 Exercise. Show that the doubly reflected Brownian motion {9(W), A; 0 < t < col satisfies X

Pq9(W,) e dy] = E p +(t; x, y + 2na) dy; 0 < y < a, 0 < x < a, t > O. n = -x

2. Brownian Motion

98

The derivation of the transition density for Brownian motion absorbed at 0 and a i.e., { WtA To A Ta, d't; 0 < t < co}, is the subject of the next proposition.

8.10 Proposition. Choose 0 < x < a. Then for t > 0, 0 < y < a: (8.12)

> t] = j p_(t; x, y + 2na) dy.

Px[W, a dy, To A

n=-3c

PROOF. We follow Dynkin & Yushkevich (1969). Set a0 A 0, to -4 To, and define recursively an -A inf{t > Tn_1; 147, = in = inf It > an; W, = 0 }; n = 1, 2, .... We know that Px[to < co] = 1, and using Theorem 6.16 we can show by induction on n that an - T_1 is the passage time of the standard Brownian to a, T -an is the passage time of the standard motion -a, and the sequence of differences a, - To, Brownian moton W.". -W ri - at, a2 - t1, t2 - a21 consists of independent and identically distributed random variables with moment generating function e-°/' (c.f. (8.8)). It follows that to - To, being the sum of 2n such differences, has moment generating function e-2na,hat, and so

-

Px[T - To

t] = P °[T2n° < t].

We have then

lim Px[T < t] = 0; 0 < t < GO.

(8.13)

n-oo

For any y e (0, co), we have from Corollary 6.18 and the symmetry of Brownian motion that = Px[W,

Px[W,

-yl",.+]

on {T < t },

and so for any integer n > 0, (8.14)

13' [W,

Similarly, for y

y, T

t] = Px[W,

- y, T.

t] = Px[W,

-y, an

t].

- 09, a), we have

= Px[W, > 2a -y1,97,+]

Px[W,

on {an 5_ t },

whence (8.15)

Px[W,

y, an

t] = Px[W,

2a - y, an 5_ t]

= Px[W, > 2a - y, tn_i < t];

n > 1.

We may apply (8.14) and (8.15) alternately and repeatedly to conclude, for 0 < y < a, n > 0: Px CW1

to

Px[Wi

)7, an

= Px[W, < -y - 2na], = Px[Wr y- 2na],

and by differentiating with respect to y, we see that


Px[Hie dy,

(8.16)

to

99

< t] = p(t; x, -y - 2na)dy,

[W, e dy, on 5 t] = p(t; x, y - 2na)dy.

(8.17)

Now set no = 0, Po = T., and define recursively nn

= inf {t > p,,_1; 14/, = 0}, p = inf { t > re.; W = a}; n = 1, 2,

....

We may proceed as previously to obtain the formulas

t] = 0; 0 < t
0.

It is easily verified by considering the cases To < Ta and To > T. that r_1 V p_, = cr. A n and on v 7r,, = to A p,,; n 1. Consequently, Px[Wte dy, r_1 ^ Pn -i < t] = 13' [W, e dy, -c_, < t] + Px[W, e dy, Pn -i < t] - Px[W, e dy,

(8.21)

on A nn

< t],

and

t] = Px[W, e dy, a

Px[Wtedy, a A 7c.

(8.22)

- Px [W te dy,

t] + Px[Wte dy, 7t

Ap

t]

t].

Successive application of (8.21) and (8.22) yields for every integer k

1:

(8.23)

Px[141,e dy, To A Po < t] = E IPx[Wte dy, t_1

t] + Px[Wte dY, Pa-i < t]

n=1

-

[147, e dy, on < t] - Px[Wie dy,

nn

< t]l

+ Px[W, e dy, t k n Pk < t].

Now we let k tend to infinity in (8.23); because of (8.13), (8.18) the last term converges to zero, whereas using (8.16), (8.17) and (8.19), (8.20), we obtain from

the remaining terms: Px[Wie dy, To A

Ta

> t] = Px[Wie dy] - Px[Wie dy, To A Po < t] CO

= E p_(t; x, y + 2na)dy; 0 < y < a, t > O. 8.11 Exercise. Show that for t > 0, 0 < x < a: (8.24)

Px[ To A

e dt] =

,1

E

+ x) exp

(2na

x)2 1

\/2/rt3 n=

+ (2na + a - x) ex p jl

(2na + a - x)2 1 ] 2t

dt.

2. Brownian Motion

100

It is now tempting to guess the decomposition of (8.24): (8.25)

Px [To E dt, To < To] =

1

(2na + x)2 dt, 2t

(2na + x) exp 1

\/27rt3 n=-co

(8.26)

e dt, 7; < To] =

1

2nt3

e°

E (2na + a

(2na + a - x)2

x) exp

2t

dt.

Indeed, one can use the identity (8.6) to compute the Laplace transforms of the right-hand sides; then (8.25), (8.26) are seen to be equivalent to (8.27)

Ex [e'T°1(ro 0.

8.18 Problem. Define the time of last exit from the origin before t by (8.35)

y,

sup{0

s

t; W = 0}.

Show that y, obeys the arc-sine law; i.e.,

P°[y, e ds] -

ds

/ s(t - s)

;

0 < s < t.

(Hint: Use Problem 8.8.)

8.19 Exercise. With y, defined as in (8.35), derive the quadrivariate density

2.9. The Brownian Sample Paths

103

PqW,e da, M,Edb, y,e ds, 0,e du]

-2ab 2 (2nu(s - u)(t - s))312

a

exp - 2u(sub2- u)

2(t-2

da db ds du;

0 < u < s < t, a < 0 < b.

2.9. The Brownian Sample Paths We present in this section a detailed discussion of the basic absolute properties

of Brownian motion, i.e., those properties which hold with probability one (also called sample path properties). These include characterizations of "bad" behavior (nondifferentiability and lack of points of increase) as well as "good" behavior (law of the iterated logarithm and Levy modulus of continuity) of the

Brownian paths. We also study the local maxima and the zero sets of these paths. We shall see in Section 3.4 that the sample paths of any continuous martingale can be obtained by running those of a Brownian motion according to a different, path-dependent clock. Thus, this study of Brownian motion has much to say about the sample path properties of much more general classes of processes, including continuous martingales and diffusions.

A. Elementary Properties We start by collecting together, in Lemma 9.4, the fundamental equivalence transformations of Brownian motion. These will prove handy, both in this section and throughout the book; indeed, we made frequent use of symmetry in the previous section. 9.1 Definition. An Rd-valued stochastic process X = {X,; 0 < t < co} is called < t, < co, Gaussian if, for any integer k > 1 and real numbers 0 < t, < t2
0, and its covariance matrix

p(s, t) A E[(X, - m(s))(X, - m(t))T]; s, t

0,

where the superscript T indicates transposition. If m(t) = 0; t > 0, we say that X is a zero-mean Gaussian process.

9.2 Remark. One-dimensional Brownian motion is a zero-mean Gaussian process with covariance function (9.1)

p(s,t) = s

t;

s, t

0.

2. Brownian Motion

104

Conversely, any zero-mean Gaussian process X = {X ,x; 0 S t < col with a.s. continuous paths and covariance function given by (9.1) is a onedimensional Brownian motion. See Definition 1.1.

Throughout this section, W = {W, gt; 0 5 t < co } is a standard, onedimensional Brownian motion on (SI, Sri, P). In particular Wo = 0, a.s. P. For fixed w e SI, we denote by W(w) the sample path t W(co).

93 Problem (Strong Law of Large Numbers). Show that (9.2)

lim

-Wt

t-,a)

t

= 0,

a.s.

(Hint: Recall the analogous property for the Poisson process, Remark 1.3.10.)

9.4 Lemma. When W = {W,

0 < t < co} is a standard Brownian motion,

so are the processes obtained from the following "equivalence transformations":

(i) Scaling: X = {X Fet; 0 < t < co} defined for c > 0 by

=

(9.3)

(ii) Time-inversion: Y = { (9.4)

1

Wc,

0 1, k > 1, we define the set (9.16)

Aik

= U

to.0,11

n

{wen; iw,,,,(w) - wt(coi

he[0.11k1

Certainly we have

{we); -co < D+147,(w)

147,(w) < co, for some t e [0, 1]} = U U 1=1 k=1

and the proof of the theorem will be complete if we find, for each fixed j, k, an event C e ,F with P(C) = 0 and Aik C.

Let us fix a sample path co e Aik, i.e., suppose there exists a number t e [0,1] with 1147,(w) - 14/,(w)1 < jh for every 0 < h < 1/k. Take an integer n > 4k. Then there exists an integer i, 1 < i < n, such that (i - 1)/n < t < i/n, and it is easily verified that we also have ((i + v)/n)) - t < (v + 1) /n 1/k, for v = 1, 2, 3. It follows that I

W1 +11/n(w) - Wi/n(a))1

W, +11 /l(w) - Wt(0)1 + Wirn(w) - W(w)1

< 2jn- -nj

3j n

The crucial observation here is that the assumption w e Aik provides information about the size of the Brownian increment, not only over the interval [i /n, (i + 1)/n], but also over the neighboring intervals [(i + 1) /n, (i + 2)/n] and

2.9. The Brownian Sample Paths

111

[(i + 2)/n, (i + 3)/n]. Indeed, Wo+2)/n(0) - W(i+l)in(W)1

WII+1K+1)In

1W(i+3)In(0) - WO-F 2)1n(a*

WtI +IWO+ 2)In

3j

2j

n

n

Hi d 4k. But now j1(Wy+v)In

Zv;

W(t+v-1)1n)

V = 1, 2, 3

are independent, standard normal random variables, and one can easily verify

the bound P[IZI < a] < a. It develops that

P(Cr))
4k P(U7=1 Cin)) = 0.

0

9.20 Remark. An alternative approach to Theorem 9.18, based on local time, is indicated in Exercise 3.6.6.

9.21 Exercise. By modifying the preceding proof, establish the following stronger result: for almost every cu e S2, the Brownian path 14/.(u)) is nowhere Holder-continuous with exponent y > I. (Hint: By analogy with (9.16), consider the sets (9.19) AJk A {w e Lr2;

I Wt,(w) - W(w)1 < jhY for some t e [0,1] and all h e [0,1 /1c]l

and show that each

is included in a P-null event.)

E. Law of the Iterated Logarithm Our next result is the celebrated law of the iterated logarithm, which describes the oscillations of Brownian motion near t = 0 and as t o o. In preparation for the theorem, we recall the following upper and lower bounds on the tail of the normal distribution.

2. Brownian Motion

112

9.22 Problem. For every x > 0, we have x

(9.20)

1

-x2/2 < j.co e -"2

1+x

< -e-x212. x

x

9.23 Theorem. (Law of the Iterated Logarithm (A. Hin6in (1933))). For almost every w e O., we have Wt((0)

(i) lim .%/ 2t

log log(1 /t)

Wi(w)

(iii) lim

t-.

(ii) lim

= 1,

Wt(w)

Wi(w)

(iv) lira

= 1,

1,

.,./2t log log(l/t)

=

t-. .\/2t log log t

2t log log t

1.

9.24 Remark. By symmetry, property (ii) follows from (i), and by time-inversion,

properties (iii) and (iv) follow from (i) and (ii), respectively (cf. Lemma 9.4). Thus it suffices to establish (i).

PROOF. The submartingale inequality (Theorem 1.3.8 (i)) applied to the exponential martingale {X 5c; 0 t < oo} of (8.7) gives for A > 0,11 > 0: (9.21)

P max Ws - -sA2 > / 3 = P max Xs > elfl 0.,5..t

< e-lfl.

ri._.5.i

With the notation h(t)-

72t log log(l/t) and fixed numbers 0, .5 in (0, 1), we choose A --= (1 + .5)0-"h(0"), 13 = lh(0"), and t = 0" in (9.21), which becomes

P[

max (Ws o<s<en

-

>

e- x212

>

const.

n>

-

.,./27c(x + 1/x)

n

1

log 0

Now the last expression is the general term of a divergent series, and the second half of the Borel-Cantelli lemma (Chung (1974), p. 76, or Ash (1972), p. 272)

guarantees the existence of an event n,e.F with Pod = 1 such that, for every co e 00 and k 1, there exists an integer m = m(k, co) ._ k with (9.23)

14/0-((o) - Wo-,,(0)

-11 - 011(0m).

On the other hand, (9.22) applied to the Brownian motion -W shows that there exist an event SI* e g" of probability one and an integer-valued random variable N*, so that for every w e SP (9.24)

- W,..i(w)
(1 +

The probability in the last summand of (9.28) is bounded above, thanks to (9.20), by a constant multiple of n-112(k2')" "2, and 2,0+1

x(1"2 dx = k=1

(2" + 1)v V

0

where v = 1 + (1 + 6)2. Therefore,

P[

max 0 < j< 2. k=j-i52.°

1W.i/2"

Wi/2.I

9(k /2")

> 1 + 61
0, we have a closed set F

and an open set G such that F g A g_ G and Q(G\F) < E. But then Fc is open, G` is closed, Gr c A' S F`, and Q(P\G`) = Q(G\F) < E; therefore, ACE F. To show ,F is closed under countable unions, let A = (.);,°=1 Ak, where Ak E3-4.- for each k. For each E > 0, there is a sequence of closed sets {Fk} and a sequence of k = 1, 2, .... Let open sets {Gk} such that Fk g Ak g Gk and Q(Gk \Fk)
n*(w), and show that for every m > n, we have 171

(10.2)

IX,(co) - Xs(co)I 5 2d E 2-7i; V t, s E L s < t, lilt - sill < j=n+1

For in = n + 1, we can only have tE R,(s), and (10.2) follows from (10.1). Suppose (10.2) is valid for m = n + 1, , M - 1. Take t, s E Lm, s t. There is a vector t1 t. s' E L,_, n R m(s) and a vector ti E Lf_, with t E R f(t') such that s From (10.1) we have,

IX,,(co) - Xs(co)1 _5 dr'', PC, (co) - X,i(o.))1 5_ d2"m, and from (10.2) with in = M - 1, we have m-i

PC,i(a)) - Xsi(co)1 5_ 2d E 2i=n+1

We obtain (10.2) for in = M. For any vectors s, t E L with s t and 0 < lilt - sill < h(co) A 2 ""), we select n > n*(w) such that 2-("+" 5_ lilt - sill < 2-". We have from (10.2) 2 YJ 5_ Slllt -

I Xi(co) - Xs(co)I 5_ 2d j=n+1

where (5 = 2d/(1 - 2-Y). We may now conclude as in the proof of Theorem 2.8.

4.2. The n-dimensional cylinder sets are generated by those among them which are n-fold intersections of one-dimensional cylinder sets; the latter are generated by sets of the form H = {we C[0, co); w(t,) G}, where G is open in R. But H is open in C[0, co), because for each coc,eli, this set contains a ball B(coo, 8) A {we C[0, oo); p(co,coo) < El, for suitably small E > 0. It follows that W .4(C[0, co)). Because C[0, co) is separable, the open sets are countable unions of open balls of the form B(coo,E) as previously. Let Q be the set of rationals in [0, co). We have {(.0:

E - sup (lco(t) - wo(01 A 1) < El. EB(0)0,0=

n=1 2" 0 0 there exists a compact set K c S, such that P [X(M)E K] > 1 - e/6M, V n > 1. Choose 0 < S < 1 so I f(x) - f(y)I < c/3 whenever XE K and p(x, y) < S.

Finally, choose a positive integer N such that P[p(X("),Y(")) > (5] < e/6M, Vn N We have

< - P [X(") e K, p(X("), V")) < .5]

-3

n

+ 2M P[X(")

IC]

+ 2M P[p(X("), r")) > (5] < E. 5.2. The collection of sets F E a(C. [0, 00 )d ) for which x 1- P'(F) is .4(08°)/,4([0, 1])measurable forms a Dynkin system, so it suffices to prove this measurability for


121

all finite-dimensional cylinder sets F of the form F = {aye CTO, oo)d; co(t 0) E FP,

where 0 = t, < t, < P'(F) = lro(x)

< t, re.4)(Rd), i = 0, 1,

Jr,r,

I

Jr

Pe(t 1; x, Yi)

, co(t.)e F} , , n. But

Pa(t. -4 -1; Y.-1, Y,OdY.

dYi,

where pd(t; x, y) A (2nt)-da exp{ -(II x - y112 /2t) }. This is a Borel-measurable function of x. 5.7. If for each finite measure p on (S, AS)), there exists g,, as described, then for each cc e R, {x ES;f (x) < a} 0 {x E S; g ,,(x) < a} has p-measure zero. But {go e .4(S), so {f e At(S)". Since this is true for every p, we have If eaW(S). For the converse, suppose f is universally measurable and let a finite measure p be given. For r e Q, the set of rationals, let U(r) = {x e S; f(x) Then f(x) = inf {r e Q; x e U (r)} . Since U(r) E AST', there exists B(r)E R(S) with p[B(r),a,U(r)] = 0, r e Q. Define g m(x) A inf {r e Q; x e B(r)} = inf (1),(4 reQ

where (p,.(x) = r if X E B(r) and (p,(x) = oo otherwise. Then gu: S

measurable, and {x ES; f(x) # g,(x)}

IR is Borel-

U,e Q[B(r) 0 U(r)], which has p-

measure zero.

5.9. We prove (5.5). Let us first show that for D E *Rd x Rd), we have (10.3)

P[(X, 11E1)0] = PUX, Y)EDIn

If D = B x C, where B, C E R(M), then

E[1{xee}1{YeC" = 1{yec}E{1{XeB" = i{yc}P[XE B].

= 1{yc}P[X en so (10.3) holds for this special case. The sets D for which (10.3) holds form a Dynkin system For the same reasons, E[1{x E,3}1{y Ec} I

containing all measurable rectangles, so (10.3) holds for every D E .41(Rd x Rd). To prove (5.5), set D = {(x,y); x + ye r }. A similar proof for (5.6) is possible. 6.9. (ii) By Corollary 1.2.4, S is a stopping time of {#;.,.}. Problem 1.2.17 (i) implies

E[Z2IFs+] = EV21,(5,,o+], a.s. on {S This equation combined with (i) gives us the desired result.

(iii) Suppose that S is an optional time of {A}, and that (e) holds for every bounded optional time of {A}. Then for each s > 0, Px[X(ss)+,e FI.Fas)+] = (U,1,)(Xs s),

a.s.

But on {S < s}, we have X(s,,o+, = Xs+ so (ii) implies

Px[Xs+te rIFs+] = Px[x(s,,$),-,ErIF(s,,o+] =(U,i,-)(xs,$)= (U,1,)(Xs), 13' a.s. on {S Now let s T co to obtain (e) for the (possibly unbounded) optional time S. The argument for (e') is the same.

2. Brownian Motion

122

7.1.

=

= ns>tn.>03,sL=

(1)

(ii) The a-field Ax is generated by sets of the form F = {(X,... , 'COE r}, where

0 = t,
0} is a one-dimensional Brownian motion with initial distribution p. The set F = {w; w(1) = 0} has Po-measure zero, so F E .F4.. If F is also in the completion ,01' of under Pi', then there must be some G e ,013 with F s G and P"(G) = 0. Such a G must be of the form G = {w: w(0) e F} for some F E R(R), and the only way G can contain F is to have F = R. But then Pu(G) # 0. It follows that F is not in Foi`.

2.10. Solutions to Selectee Problems

123

7.5. Clearly, AP g_ Fu holds for every 0 < t < o o , so ..f-sg, g_ F. For the opposite inclusion, let us take any F ..F0; Problem 7.3 guarantees the existence of an event G54...,ox such that N = PAGE Ac". But now .Foox .97,,;

for every 0 < t < co), and thus G E

.FX, (we have .97,x g

N e Ar" c ,9 ; imply

F=G.LNE,Fog. 7.6. Repeat the argument employed in Solution 7.5, replacing ..F" by and ..FX by .5,'! and using the left-continuity of the filtration {V} (Problem 7.1).

7.18. Let {B t > 0 }, (0,1"), {P'}11 be a one-dimensional Brownian family. For r E a(Q8), define the hitting time Hr(co) = inf It 0; B,(a)) According to Problem 1.2.6, H(O,) is optional, so {H(,,,s) = 0} is in .510, = A. Likewise, li,,mE37:0. Because of the symmetry of Brownian motion starting at the origin, P°[H,0,, = 0] = = 0]. According to the Blumenthal zeroone law (Theorem 7.17), this common value is either zero or one. If it were zero, then P° [B, = 0, V 0 < t < s for some E> 0] = 1, but this contradicts Problem 7.14 (i). Therefore, P° [H(0a) = 0] = P°[H,_0,,o) = 0] = 1, and for each we {H(0.0,) = 0} n {11(_..0)= 0}, there are sequences s 10, tn 10 with Bs.(co) > 0, Bin(co) < 0 for every n

1.

7.19. For fixed wee, Tb(w) is a left-continuous function of b and Sb(w) is a rightcontinuous function of b. For fixed b E FR, "T is a stopping time and Sb is an optional time (Problem 1.2.6), so both are 37"..w-measurable. According to Remark 1.1.14, the set A = {(b, co) E [0, CC) x 0; Tb(w) Sb(w)} is in .4([0, cc)) (3) ,F07. Furthermore, A g {(0 e f2; (b, 0)) E Al is included in the set {we SI; BA(0)

wrb(o+t(w) - Wrb(w)(0) < 0 for some E> 0 and all t E

en,

which has probability zero because {B F,B; 0 < t < co} is a standard Brownian motion (Remark 6.20 and Theorem 6.16), and Problem 7.18 implies that B takes positive values in every interval of the form [0,E] with probability one. This establishes (i). For (ii), it suffices to show OD

P[wES); (L(co),co)e A] = J

P(Ab)P[L E db].

If A were a product set A = C x D; C E .4([0, co)), D e..Fw, this would follow from the independence of L and The collection of sets A e .4([0, co)) (3).F,w for which this identity holds forms a Dynkin system, so by the Dynkin system Theorem 1.3, this identity holds for every set in 98([0, op)) ®

8.8. We have for s > 0, t > 0, b > a, b > 0; P°[W,±s

a, M,s < b1.5rJ = P°[W,±s < a, M, < b, max W,, < b

ou5s

= 1 { ,,,,,}

P° [ W.s < a, max W, < b oso5s

The last expression is measurable with respect to the a-field generated by the pair of random variables (W M,), and so the process {(W,, M,); .F.,; 0 is Markov. Because V, is a function of (W Me), we have

P°[Y,A,EFIA] = P°[Yt+serlWr,

rEge(R)-

t
m > w, b > a, m > 0 we have

[Wr. da, Mr.

W, = w, M, = m]

= P°[W,,,E da, max W,. E dbl W, = w, M, =

13.5

[Ws da - w, Mse db - w]

Pw[WE da, Ms db] =

=

2(26 -a - w)

(26 -a - w)2}

exp

2s

.12rz.s3

da db,

thanks to (8.2), which also gives r[Wr+s E da, Mt+s = ml W, = w,M, = m] ll

= r[W,+se da, max W +SmIW = w,Mr =mJ = =

e da, Ms 1

t

[W da - w, Ms

m] =

-2s

exp

2its

m -w]

(2m -a -

exp

2s

Therefore, P°[Y,+se dy1W, = w, M, = m] is equal to

P°[WE b - dy, 111,+ sE dbl

1N,

Air

= m] db

f(m, co)

+ P°[W,+sE m - dy, M, +s = mIW, = w, Mr = m] = p+(s; m - w, y)dy. Since the finite-dimensional distributions of a Markov process are determined by the initial distribution and the transition density, those of the processes I WI and Y coincide. 8.12. The optional sampling theorem gives

= Ex

= Ex X, ,., T ATa =

[exp

Since W(AMA T. is bounded, we may let t = Ex [exp JI WT. A T -

W.,ATOAT-

-1,12(t A To A TO)].

co to obtain

1:1.2(To

'TX]

= Ex[1{7-0 0 such that -,-1W+h((o) - 14/11041 > c, V t e [0, op)] = 1. P[co al; lim h40

Nfil

For every w e 0, S,,, -4 It E [0, CO); liMh10 (1 Wt+17(04 - Wt(0))01) < 091 has been called by Kahane (1976) the set of slow points from the right for the path W.(w). Fubini's theorem applied to (9.25) shows that meas(S,o) = 0 for P a.e.

wee, but, for a typical path, S,,, is far from being empty; in fact, we have P1

En; inf lim cl.,; cf. Theorem 1.5.8. It is precisely this fact that allows one to proceed, in a highly nontrivial yet straightforward manner, with the construction of the stochastic integral (2.1) with respect to the continuous, square-integrable martingale M, for an appropriate class of integrands X. The construction is due to ItO (1942a),

(1944) for the special case that M is a Brownian motion and to Kunita & Watanabe (1967) for general Me di2. We shall first confine ourselves to M e."2, and denote by <M> the unique (up to indistinguishability) adapted, continuous, and increasing process, such that IM,2 - <M> A; 0 < t < ool is a martingale (cf. Definition 1.5.3 and Theorem 1.5.13). The construction will then be extended to general continuous, local martingales M. We now consider what kinds of integrands are appropriate for (2.1). We first define a measure gm on ([0, oo) x S2, 8([0, co)) ,(w).

We shall say that two measurable, adapted processes X = IX ,Fg; 0 < t < col and Y = { Y A; 0 < t < col are equivalent if X,(w) = Y,(co);

µM -a.e. (t, w).

This introduces an equivalence relation. For a measurable, 1,97,1-adapted process X, we define (2.3)

[X]T °= E I

X2 d<M>

provided that the right-hand side is finite. Then [X]j, is the L2-norm for X, regarded as a function of (t, w) restricted to the space [0, T] x CI, under the measure We have [X = 0 for all T > 0 if and only if X and Y are equivalent. The stochastic integral will be defined in such a manner that whenever X and Y are equivalent, then 1(X) and 1(Y) will be indistinguishable:

PUT(X) = /T(Y),

V0

T < oo] = 1.

2.1 Definition. Let 29 denote the set of equivalence classes of all measurable,

IFJ-adapted processes X, for which [X],- < co for all T > 0. We define a metric on 2' by [X - Y], where 2-"(1 A [X].).

[X] -4-n=i

3.2. Construction of the S:ochastic Integral

131

Let ..r* denote the set of equivalence classes of progressively measurable processes satisfying [X]T < oo for all T > 0, and define a metric on .2" in the same way. We shall follow the usual custom of not being very careful about the distinction between equivalence classes and the processes which are members of those equivalence classes. For example, we will have no qualms about saying, ".29* consists of those processes in .29 which are progressively measurable." Note that .29 (respectively, .2") contains all bounded, measurable, {,F, }adapted (respectively, bounded, progressively measurable) processes. Both 2' and 2* depend on the martingale M = {M t > 0}. When we wish to indicate this dependence explicitly, we write 2'(M) and 2' *(M). If the function ti- <M>,(co) is absolutely continuous for P-a.e. to, we shall be able to construct P., X, dM, for all X e and all T > 0. In the absence of this condition on <M>, we shall construct the stochastic integral for X in the slightly smaller class .29*. In order to define the stochastic integral with respect to general martingales in A/2 (possibly discontinuous, such as the compensated Poisson process), one has to select an even narrower class of integrands among the so-called predictable processes. This notion is a slight extension of leftcontinuity of the sample paths of the process; since we do not develop stochastic integration with respect to discontinuous martingales, we shall forego further discussion and send the interested reader to the literature: Kunita & Watanabe (1967), Meyer (1976), Liptser & Shiryaev (1977), Ikeda & Watanabe (1981), Elliott (1982), Chung & Williams (1983). Later in this section, we weaken the conditions that M e.ifz and [X]i < co,

V T > 0, replacing them by M e Jr' and T

P

d<M>,
T, co en. For T = co, ..r; is defined as the class of processes X e .2" for which E ft; X,2 d<M>, < co (a condition we already have for T < co, by virtue of membership in _2"). A process X e 2,;!' can be identified with one defined only for (t, w) e [0, T] x n, and so we can regard 4` as a subspace of the Hilbert space (2.4)

YfT -4 L2([0, T] x

a([0, T])

T Pm)

More precisely, we regard an equivalence class in A9T as a member of .27 if

it contains a progressively measurable representative. Here and later we replace [0, T] by [0, co) when T = co. 2.2 Lemma. For 0 < T < co, .27 is a closed subspace of .09T. In particular, .,27 is complete under the norm [X]i, of (2.3).

3. Stochastic Integration

132

PROOF. Let {X(")},111 be a convergent sequence in .27 with limit X EST. We may extract a subsequence, also denoted by {X(")},,m11, for which { (t, co) e [0, T] x

lim X: "°(co) 5 Xt(co)} = O. n-,co

By virtue of its membership in Yer, X is ,4([0, T]) not be progressively measurable. However, with

.F-measurable, but may

A -A {(t,co)e [0, T] x i2; lim X;"°(w) exists in 51),

the process )7, (w) _4_

lim XP)(co);

(t, w) e A

0;

(t, CO) A

{n-co

inherits progressive measurability from {X(")}c°_, and is equivalent to X.

0

A. Simple Processes and Approximations 2.3 Definition. A process X is called simple if there exists a strictly increasing sequence of real numbers {tn}n°3_0 with to = 0 and lim, t = oo, as well as a sequence of random variables g1,110 with supn,oln(01 < C < co, for every we 1, such that is Ft.-measurable for every n > 0 and CO

i(co)10,,,,,i(t);

X,(co) = 0(co) 1{0} (t) +

0 < t < co, co e

i=o

The class of all simple processes will be denoted by Yo. Note that, because members of ..F0 are progressively measurable and bounded, we have .290 g ...?*(M) c .29(M).

Our program for the construction of the stochastic integral (2.1) can now be outlined as follows: the integral is defined in the obvious way for X e as a martingale transform: n-1

(2.5)

it(X) A E

- m) + um, - 4)

co

= E um, A ti+, - MIA t,), i=0

t
0 is the unique integer for which t < t < tn+1. The definition is then extended to integrands X e 21* and X e .29, thanks to the crucial results which show that elements of .21* and _V' can be approximated, in a suitable sense, by simple processes (Propositions 2.6 and 2.8). 2.4 Lemma. Let X be a bounded, measurable, {g;}-adapted process. Then there exists a sequence {X('")}:=1 of simple processes such that

3.2. Construction of the Stochastic Integral

sup lira E

(2.6)

T>0 m-.co

f

133

T

RO'n) - X,12 dt = 0.

0

PROOF. We shall show how to construct, for each fixed T > 0, a sequence {x(n,T)} flop_ of simple processes so that lim E

f

I XIn'T) - X,I2dt = 0.

o

Thus, for each positive integer m, there is another integer n,,, such that m

1,0"-m) - X,I2 dt < -1

E

m

o

and the sequence {X("-m)},',., has the desired properties. Henceforth, T is a fixed, positive number. We proceed in three steps. (a) Suppose that X is continuous; then the sequence of simple processes 2.-1

)0 n)(C0) A X0(01{0}(t) + E X0-724(0) laT/2",(k+1)T/2.1(t); n _. 1, k=0

satisfies lim, E f o IX; ") - x,12 dt = 0 by the bounded convergence theorem. (b) Now suppose that X is progressively measurable; we consider the continuous, progressively measurable processes t

(2.7) F,(w)

J.

T

XJw) ds; fes'n)(co)

m[Ft(w) - Fo_(1 /m)".(co)]; m

1,

for t > 0, w e f2 (cf. Problem 1.2.19). By virtue of step (a), there exists, for

each m > 1, a sequence of simple processes {g(m")},T=1 such that limn E - -1:"1)12 dt = 0. Let us consider the A[0, n) .FTmeasurable product set A A {(t, w) e [0, T] x f2; lim it'n(co) = Xi(w)lc.

For each w e f2, the cross section A.

{t e [0, T]; (t,

e A} is in

a([0, T]) and, according to the fundamental theorem of calculus, has Lebesgue measure zero. The bounded convergence theorem now gives lim,n_,. E fpfon)- X,I2 dt = 0, and so a sequence {g(m "-)}:=1 of bounded, simple processes can be chosen, for which T

I i{"-) - X,I2 dt = 0.

lim E m-

(c)

o

Finally, let X be measurable and adapted. We cannot guarantee immediately that the continuous process F = {Ft; 0 < t < co} in (2.7) is progressively measurable, because we do not know whether it is adapted. We do

134


know, however, that the process X has a progressively measurable modification Y (Proposition 1.1.12), and we now show that the progressively measurable process {G, g po^ T Ysds, A; 0 5 t 5 T} is a modification of F. For the measurable process ;Mu)) = 0 < t < T, w en, we have from Fubini: E (T)ri,(w)dt = 11; P[x,(0)) Y,(w)]dt = 0. Therefore, 14' ri,(w)dt = 0 for P-a.e. wen. Now {F, 0 G,1 is contained in the event Iw; f o ri,(w) dt > 0 }, G, is A- measurable, and, by assumption, A contains all subsets of P-null events. Therefore, F, is also A- measurable. Adaptivity and continuity imply progressive measurability, and we may now repeat verbatim the argument in (b). 25 Problem. This problem outlines a method by which the use of Proposition 1.1.12, a result not proved in this text, can be avoided in part (c) of the proof of Lemma 2.4. Let X be a bounded, measurable, {. }- adapted process. Let 0 < T < co be fixed. We wish to construct a sequence {X(k)},T_I of simple processes so that T

lim E

(2.8)

k

1)0k) - X,I2 dt = 0.

co

To simplify notation, we set X, = 0 for t < 0. Let cp:

Or"; j =

0, + 1, + 2, ...1 be given by (0 .(t)

for

2"

1

0 is a simple, adapted process. (b) Show that lim,40 E 14. X, - X,_I2 dt = 0. (c) Use (a) and (b) to show that T

1

lim E n-

1,O"'s) - X,I2 ds dt = 0. 0

0

(d) Show that for some choice of s > 0 and some increasing sequence Ink1,°_, of integers, (2.8) holds with X(k) = X("k's). This argument is adapted from Liptser and Shiryaev (1977).

2.6 Proposition. If the function t

<M>,(w) is absolutely continuous with

respect to Lebesgue measure for P-a.e. wen, then .290 is dense in .29 with respect to the metric of Definition 2.1. PROOF.

(a) If X e .29 is bounded, then Lemma 2.4 guarantees the existence of a bounded sequence {V")} of simple processes satisfying (2.6). From these


135

we extract a subsequence {X(mk)}, such that the set {(t, w) e [0, co) x SI; lim Xrnkga)) = Xt(co)}`

has product measure zero. The absolute continuity of t

<M>,(co) and

the bounded convergence theorem now imply [X(mk) - X] -+ 0 as k -+ co. (b) If X e .2 is not necessarily bounded, we define )(riga)) -4 Xi(a))1{1x,(,01.(n);

0 < t < co, a)E12,

and thereby obtain a sequence of bounded processes in 2. The dominated convergence theorem implies [X(n)

T

-

E

X,21{

nao 0

for every T > 0, whence lim, [X(n) - X] = 0. Each X(") can be approximated by bounded, simple processes, so X can be as well.

When t H <M>, is not an absolutely continuous function of the time variable t, we simply choose a more convenient clock. We show how to do this in slightly greater generality than needed for the present application. 2.7 Lemma. Let {A1; 0 < t < co} be a continuous, increasing (Definition 1.4.4) process adapted to the filtration of the martingale M = 1111.F,,; 0 < t < col. If X = {X..F,; 0 < t < oo} is a progressively measurable process satisfying T

E

X,2 dA, < oo

for each T > 0, then there exists a sequence {X(")}°11 of simple processes such that

sup lim E

1XP) - X,I2 dA, = 0.

T>0 nom

0

PROOF. We may assume without loss of generality that X is bounded (cf. part (b) in the proof of Proposition 2.6), i.e., (2.9)

IXt(o)))

C < co; V t > 0, a) ED.

As in the proof of Lemma 2.4, it suffices to show how to construct, for each fixed T > 0, a sequence {X(")},;°_, of simple processes for which T

lim E n-

!XI") - X,I2 dA, = O. 0

Henceforth T > 0 is fixed, and we assume without loss of generality that (2.10)

Xi(w) = 0; V t > 7', a) e


136

We now describe the time-change. Since A,(w) + t is strictly increasing in t > 0 for P-a.e. w, there is a continuous, strictly increasing inverse function T(w), defined for s > 0, such that V s > 0.

A ,;(,)(w) + Ts(w) = s;

In particular, Ts < s and {T, < t} = {A, + t > s} E. Thus, for each s > 0, Ts is a bounded stopping time for Taking s as our new time-variable, we define a new filtration IC by

S>0,

= and introduce the time-changed process

s > 0, wen,

YS(w) = XT,(0,)(0));

which is adapted to {Ws} because of the progressive measurability of X (Proposition 1.2.18). Lemma 2.4 implies that, given any e > 0 and R > 0, there is a simple process { V, Ws; 0 s < col for which E

(2.11)

- Y.,12 ds < 6/2.

But from (2.9), (2.10) it develops that CO

EJ

CO

1;2 ds = E

= E j'AT-1-7. X?..ds < C(EAT + T)< op, o

so by choosing R in (2.11) sufficiently large and setting V = 0 for s > R, we can obtain CO

E

J

IYS`- Y1Zds <e.

Now V is simple, and because it vanishes for s > R, there is a finite partition

< s < R with

0 = so < s,
d n-.co

s

d<M>d,

= E[lA

where the last equality follows from [V) - X], = 0. This proves that I(X) also satisfies (2.16) and, consequently, (2.14) and (2.15). Because X and

M are progressively measurable, Its

d<M>. is A- measurable for fixed

0 < s < t < co, and so (2.16) gives us (2.19). The validity of (2.17) for X, Y e also follows from its validity for processes in Yo, upon passage to the limit.

The process 1(X) for X e 2* is well defined; if we have two sequences

{X(")},711 and { Pnl,°,11 in 2'0 with the property lim, [X" - X] = 0, - X] = 0, we can construct a third sequence {Z(1:11 with this property, by setting Z(2"-1) = X(n) and Z(2") = Y(n), for n > 1. The limit 1(X)

of the sequence {/(Z("))},7°_, in dic2 has to agree with the limits of both sequences, namely {/(X("))}:=, and 11(P"))1°11.

2.9 Definition. For X e...9", the stochastic integral of X with respect to the martingale M e .112 is the unique, square-integrable martingale I(X) =

{Ii(X),g;; 0 < t < col which satisfies limn II/(X(")) - 1(X)II = 0, for every sequence {Vnl,;°_, g 210 with lim, [X(n) - X] = 0. We write .1,(X) = .1' XsdMs;

0 0, we have

ER T(X)j,,s] =

(2.21)

s(X),

a.s. P.

With X, Y e 21* we have, a.s. P: (2.22)

EUlt T(X) -

A s(X))(I t A AY)

- t A s(Y))1-Fs]

E[ft A T

=

t As

XYd<M> g.s],

and in particular, for any number s in [0, t],


140

(2.23)

E[(I,(X) - 15(X))(1,(Y) - Is(Y))1.F5] =

E[ ft X.Y.d<M>. gd. s

Finally,

1,,,T(x) = It(g) a.s.,

(2.24)

where 51,(w) g Xt(a))1{tr( .)} PROOF. We have already established (2.12)-(2.17) and (2.19). From (2.13) and the optional sampling theorem (Problem 1.3.24 (ii)), we obtain (2.21). The same

result applied to the martingale {/t2.(X) -Po XZ' d<M>., gr; t the identities

0} provides

E[(I(AT(X) -I , A s(X))2I.Fs] = EMT(X) - It2As(X)1,0

=

E[f

t AT

X! d<M>. .Fs], P-a.s.

I As

Replacing X in this equation, first by X + Y and then by X -Y, and subtracting the resulting equations, we obtain (2.22). It remains to prove (2.24). We write

/, A T(X) - Mg) = /tA T(X -JO -uf(S')_ it, Tan. Both {/tA T(X -fa A; t 0} and fli(.1) - 1,,,,,( ),gt; t 0} are in dic2; we show that they both have quadratic variation zero, and then appeal to Problem 1.5.12. Now relation (2.22) gives, for the first process,

E[(I(AT(X - i?) -Is T(X - 2))2I-Fs] tA r

= ELI

(X. - g.)2 d<M>. sAT

g;1= 0

a.s. P, which gives the desired conclusion. As for the second process, we have ,

E [(mg) -it A T(g))2] = E

[ftn.,' g! d<M>d = 0,

and since this is the expectation of the quadratic variation of this process, we again have the desired result.

2.11 Remark. If the sample paths t 1-- <M>,(co) of the quadratic variation process <M> are absolutely continuous functions of t for P-a.e. w, then Proposition 2.6 can be used instead of Proposition 2.8 to define 1(X) for every X E Y. We have I(X)e.112 and all the properties of Proposition 2.10 in this case. The only sticking point in the preceding arguments under these condi-

tions is the proof that the measurable process f; --4-- Ito Xs2 d<M>5 is {A}adapted. To see that it is, we can choose Y, a progressively measurable modification of X (Proposition 1.1.12), and define the progressively measur-


141

able process G, A Po Ys2 d<M>s. Following the proof of Lemma 2.4, step (c),

we can then show that P[F, =

= 1 holds for every t > 0. Because G, is A-measurable, and A contains all P-negligible events in F (the usual conditions), F is easily seen to be adapted to {A}. In the important case that M is standard Brownian motion with <M>, = t, the use of the unproven Proposition 1.1.12 can again be avoided. For bounded X, Problem 2.5 shows how to construct a sequence {X(k) }f_, of bounded, simple processes so that (2.8) holds; in particular, there is a subsequence, also called {Xm}f_, , such that for almost every to [0, T] we have Xs' ds = lim

F, 0

k-.0z,

f (X''))2 ds,

a.s. P.

0

Since the right-hand side is A-measurable and A contains all null events in F, the left-hand side is also A- measurable for a.e. t e [0, T]. The continuity of the samples paths of {Ft; t > 0} leads to the conclusion that this process is .y,- measurable for every t. For unbounded X, we use the localization technique employed in the proof of Proposition 2.6.

We shall not continue to deal explicitly with the case of absolutely continuous <M> and X e .29, but all results obtained for X e .29* can be modified

in the obvious way to account for this case. In later applications involving stochastic integrals with respect to martingales whose quadratic variations are absolutely continuous, we shall require only measurability and adaptivity rather than progressive measurability of integrands. 2.12 Problem. Let W = {W, A; 0 < t < co} be a standard, one-dimensional Brownian motion, and let T be a stopping time of {A } with ET < co. Prove the Wald identities E(WT) = 0, E(WT) = ET

(Warning: The optional sampling theorem cannot be applied directly because

W does not have a last element and T may not be bounded. The stopping time t n T is bounded for fixed 0 < t < co, so E(W,,,T) = 0, E(W t2 T) = E(t n T), but it is not a priori evident that (2.25)

lim E(W( A T)

=

EWT,

lim E(14

t-co

T) = E(WT2)-)

2.13 Exercise. Let W be as in Problem 2.12, let b be a real number, and let Tb be the passage time to b of (2.6.1). Use Problem 2.12 to show that for b 0 0,

we have ET = co.

C. A Characterization of the Integral Suppose M = {M

0, = J r X Yd <M,N) ;

(2.26)

0, P-a.s.

t

0

If X and Y are simple, then it is straightforward to show by a computation similar to (2.18) that for 0 < s < t < co,

E[e (X) - InX))(INY) -Is Y))1,9;]

(2.27)

=

E[1, X d<M, N> gd;

P-a.s.

This is equivalent to (2.26). It remains to extend this result from simple processes to the case of X e -99*(M), Y e.29*(N). We carry out this extension in several stages, culminating in Propositions 2.17 and 2.19 with a very useful characterization of the stochastic integral. 2.14 Proposition (An Inequality of Kunita & Watanabe (1967)). If M, N e dr2, X e Y*(M), and Y e Y*(N), then a.s.

E

Ysl d4,

(f

Xs

d<M>s)1/2

(f

1/2

ds)

where 4, denotes the total variation of the process

;

0 < t < co,

A <M, N> on [0,s].

PROOF. According to Problem 1.5.7 (iv), (w) is absolutely continuous with respect to cp(co) A A<M> ] (co) for every co e n with P(C1) = 1, and for every such w the Radon-Niko4m theorem implies the existence of functions fie , w): [0, co) -> 111; i = 1, 2, 3, such that <M> (w) =

(s,

cicos(0),

,(co) = <M, N>,(co) =

Consequently, for a, /3 elA and w e

(co) =

f f3 (s, co) dcps(w); t

f2(s,(0)403(c0),

0 < t < co.

C satisfying P(Cap) = 1, we have

0 < ,(co) - (co)

=

(cef,(s, w) + 2aflf3(s, w) + fl2f2(s, w)) chps(co); 0 < u < t < co.

This can happen only if, for every co en, there exists a set 7;3(co)e a([o, 00)) with IT.ocodcpi(co) = 0 and such that

3.2. Construction of the Stochastic Integral (2.28)

143

c(2./1(t, co) + 20f3(t, co) + ig2f21t, (0)

0

holds for every t T(w). Now let a _4 n,,EQQ,, poi, A U243 7 (04 so that P(C') = 1, $T(,,,) dpi(w) = 0; V wen. Fix wen; then (2.28) holds for every t T(w) and every pair (a, /1) of rational numbers, and thus also for every t T(w), (1,13)e R2; in particular, a 2 I XtMl 2f1 (t,

0;

2ocl X ,(a) Y(0)11 f 3(t, co)i + I Y*012 f 2(t, (0) Vt

T(w).

Integrating with respect to dp, we obtain s12 d<m>s

oz2

20c s

0

YI s C14 s +

0

1Y 12 d5

0;

0 < t < CO,

0

almost surely, and the desired result follows by a minimization over a. 2.15 Lemma. If M, N e diec2, X e .29*(M), and {X(")};'11 S .29*(M) is such that

for some T > 0, lim n

f

co

I

Xi(in) -

2 d<M> = 0;

a.s. P,

o

then

lira ,12

0

JT

IXL ")- XI2d <M>u =O.

co

Consequently, for each T > 0, a subsequence {;Y-(n)},;°_, can be extracted for which T

lim .1

n-:o

- X12 d<M>u = 0,

a.s.


144

But (2.26) holds for simple processes, and so we have ; 0 S t< T

0

almost surely; letting n -* co we obtain (2.29) from Lemma 2.15 and the Kunita-Watanabe inequality (Proposition 2.14). 2.17 Proposition. If M, N e dll, X e_r*(M), and Y e 2*(N), then the equivalent formulas (2.26) and (2.27) hold.

PROOF. Lemma 2.16 states that d<M, IN(Y)>,, = y, d<M, NX. Replacing N in (2.29) by /N(Y), we have , = f t X,,d<M, IN(Y)> o ft

=

X,,Y,,d<M, N>u;

t _. 0, P-a.s.

o

2.18 Problem. Let M = {M .9;; 0 S t < co} and N, = {Ng",; 0 < t < co} be in dr2 and suppose X e .9':,(M), Y e ..rt(N). Then the martingales /"'(X), /NY)

are uniformly integrable and have last elements km(X), /NY), the crossvariation , converges almost surely as t -) co, and E[P0M,(X)IZ (Y)] = E; 0 < t < co, a.s. P, o

for every N e dic2.

PROOF. We already know from (2.29) that (1) = /m(X) satisfies (2.30). For uniqueness, suppose (1) satisfies (2.30) for every N e ./11. Subtracting (2.29) from (2.30), we have

0, so X Y e 1 "(M). For any F/ e .1,1, (2.26) gives d, =

Xsd<M,R> and thus , =

f

i

XsYsd<M, R>,

o

=

ft Ysd, = ,. o

According to Proposition 2.19, IM(X Y) = IN(Y).

2.21 Corollary. Suppose M, R e ./e2, X e Y*(M), and 2 e 2 *(1C1), and there exists a stopping time T of the common filtration for these processes, such that for P-almost every co, X, A T(04(W) = it A T(c))(W),

Mt A T(u))(W) = lilt A T(u)/(W);

0 < t < CO.

Then

I7 T(,)(X)(co) = Ii 7-00(50(w);

0 5 t < 00, for P-a.e. w.

PROOF. For any N e ..11, we have <M - fa, N>t A T = 0; 0 < t < co, and so (2,29) implies , A T = 0; 0 < t < co. Setting N = IM(X) Im (1) and using Problem 1.5.12, we obtain the desired result. 111

D. Integration with Respect to Continuous, Local Martingales Corollary 2.21 shows that stochastic integrals are determined locally by the local values of the integrator and integrand. This fact allows us to broaden

the classes of both integrators and integrands, a project which we now undertake.

Let M = {M Ft; 0 5_ t < co} be a continuous, local martingale on a probability space (a, ,, P) with Mo = 0 a.s., i.e., M e dr I" (Definition 1.5.15). Recall the standing assumption that {F,} satisfies the usual conditions. We


146

define an equivalence relation on the set of measurable, {g,}-adapted processes just as we did in the paragraph preceding Definition 2.1. 2.22 Definition. We denote by 91 the collection of equivalence classes of all measurable, adapted processes X = {X.Fi; 0 < t < col satisfying (2.31)

P[f

d<M>, < co] = 1

for every T e [0, co).

We denote by 91* the collection of equivalence classes of all progressively measurable processes satisfying this condition. Again, we shall abuse terminology by speaking of 91 and g* as if they were classes of processes. As an example of such an abuse, we write g* g g, and if M belongs to lfZ (in which case both 2' and 2)* are defined) we write g g and ..c," g". We shall continue our development only for integrands in g*. If a.e. path t H <M>,(w) of the quadratic variation process <M> is an absolutely continuous function, we can choose integrands from the wider class g. The reader will see how to accomplish this with the aid of Remark 2.11, once we complete

the development for g". Because M is in .41"", there is a nondecreasing sequence {S},°=, of stopping

times of {A}, such that lim, S = oo a.s. P, and kmtAs,,,,Ft; 0 < t < col is in .41. For X e g", one constructs another sequence of bounded stopping times by setting R,,(w)

n A inf {0

t < co;

XR(o)d <M>s(co)

n}.

R = co,

This is also a nondecreasing sequence and, because of (2.31), a.s. P. For n > 1, co e Q, set (2.32) (2.33)

T(w) = R(w) A Sn(w), M")(w) A Mt A "jai)/ )011)(W) = Xt(01

TWO> t} ;

t
, = 0, in probability. Then

X(s") dM, - J X,dMs

sup 0.1EST

11-.00

*

PROOF. The proof follows immediately from Problem 1.5.25 and Proposition 2.24.

2.27 Problem. Let M e dic'" and choose X e g*. Show that there exists a sequence of simple processes {X(")},;°_, such that, for every T > 0, T

X: ")- X,I2 d<M>, =0

lim n- o0

0

and lim n-

sup I /,(X(")) - 4(X)I = 0

051ST

hold a.s. P. If M is a standard, one-dimensional Brownian motion, then the preceding also hold with X e .9.

2.28 Problem. Let M = W be standard Brownian motion and X e g. We

define for 0 <s s, s

f f'(Xs)dBs

0< t
0, the function Xs(w) is bounded for 0 < s 5 t, so f ' (X s(co)) is bounded on this interval. It follows that ft° f ' (X s) dM s is defined

150


as in the last section, and this stochastic integral is a continuous, local martingale. The other two integrals in (3.3) are to be understood in the Lebesgue-Stieltjes sense (Remark 1.4.6 (i)), and so, as functions of the upper limit of integration, are of bounded variation. Thus, {f (X,),,; 0 < t < col is a continuous semimartingale.

3.5 Remark. Equation (3.3) is often written in differential notation: (3.3)'

df(X,) = f '(X,)dM, + f ' (X,) dB, +

-1

2

f "(X,) d Of> ,

= f(Xt) dXt + -2 f "PO d 00,, 0 < t < oo. This is the "chain-rule" for stochastic calculus. PROOF OF THEOREM 3.3. The proof will be accomplished in several steps.

Step 1: Localization. In the notation of Definition 3.1 we introduce, for each n _>_ 1, the stopping time 0;

if I X0 I

T = { inf{ t co;

.. n,

0; I Mt 1

n or 4 _. n or <M>,

if I Xo I < n and { ... }

n};

if I Xo I < n,

= 0.

The resulting sequence is nondecreasing with lim, T. = co, P-a.s. Thus, if we can establish (3.3) for the stopped process X:0 .4--, --i X T ,: t > 0, then we obtain the desired result upon letting n co. We may assume, therefore, that X0(w) and the random functions M ,(co), 13,(w), and <M>,(w) on [0, co) x ("/ are

all bounded by a common constant K; in particular, M is then a bounded martingale. Under this assumption, we have I X*0)1 < 3K; 0 < t < co, wee, so the values of f outside [ - 3K, 3K] are irrelevant. We assume without loss of generality that f has compact support, and so f, f', and f" are bounded. Step 2: Taylor expansion. Let us fix t > 0 and a partition II = {to,ti, ...,t,,,} of [0, t], with 0 = to < t, < < t, = t. A Taylor expansion yields

.

./(Xt) -f(x0)=k=1 E {f(xt,)- fix,_,)}

. l = kE f'(x,_,)(x,- x,_,)+ 7,kE=1 f "(nk)(xt, =1

x(,_,)2,

where 1h(w) = X,k_t(w) + 0(co)(X,k(w) - X,k_i(w)) for some appropriate Ok (CO) satisfying 0 < Ok(w) < 1, co E n. We conclude that (3.4)

where

ffX() - fiXo) = -11 (II) + J2(1-1) + 4J3(H),

3.3. The Change-of-Variable Formula

J1(1-1) sE mE f

151

'

- Btk ,), -

.12(11) A E

k=1

E flnk)(xt, - x( 1)2.

J3(11)

k=1

It is easily seen that .11(11) converges to the Lebesgue-Stieltjes integral fof '(Xs)dBs, a.s. P, as the mesh 111111 = maxi , in 1.1 (SI, .97 , P) as 111111 - 0, it suffices to compare J:(11) to the approximating sum

.17(11) A i f"(x,_,)(<m),k- <m>,_,) k=i

Recalling the discussion just before Lemma 1.5.9, we obtain

WWI) -.17(1)12 =E

=

2

Lm f"(xi, )1(14,,- 14,_,)2 -(04),- <M>tk_,)}

k=1

E[X___1

[f "(Xtk)]2

- Alti,_,)2 - (<m)t - <m>tk_1)}2]

.

211f"ii2

.

'E[I, (M -M )4 +k=1E Km\ - <m>, k=1

211.i" II 2.D'E

[ Vt(4(1-1) + OW >t

i

)21

max (Oki/Xi, - <M>tk id'

1s

k-.0o

0

Thus, passing to the limit in (3.4), we see that (3.3) holds P-a.s. for each 0 < t < co. In other words, the processes on the two sides of equality (3.3) are

modifications of one another. Since both of them are continuous, they are indistinguishable (Problem 1.1.5).

I=1

We have the following, multidimensional version of Ito's rule.

3.6 Theorem. Let {M, A (Mr),

Mr)), ,F,; 0 < t < col be a vector of local ,Br)), g;; 0 5 t < col a vector of adapted processes of bounded variation with Bo = 0, and set X, = X0 + M, + B,; 0 < t < co, where X0 is an go- measurable random vector in Rd. Let f(t,x): [0, co) x Rd R be of class C"2. Then, a.s. P, ,

martingales in .11`''", {B, A (BP),

t

(3.5)

d

f -atf(s, Xs)ds + E

f(t, X,) = f(0, X0) +

o

a

d

+E 1=1

Oxi

1d

d

ox,

f(s, Xs)dBli)

xodmv 32

+5EE - i=1 j=1

f(s, X s) d <M"), Mul>s,

0,. o

The Ito calculus differs from ordinary calculus in that familiar formulas, such as the one for integration by parts, now have correction terms such as <M, N>, in (3.8). One way to avoid these corrections terms is to absorb them into the definition of the integral, thereby obtaining the Fisk-Stratonovich integral of Definition 3.13. Because it obeys the ordinary rules of calculus (Problem 3.14), the Fisk-Stratonovich integral is notationally more convenient than the HO integral in situations where ordinary and stochastic calculus interact; the primary example of such a situation is the theory of diffusions on


156

differentiable manifolds. The Fisk-Stratonovich integral is also more robust under perturbations of the integrating semimartingale (see subsection 5.2.D), and thus a useful tool in modeling. We note, however, that this integral is defined for a narrower class of integrands than the HO integral (see Definition 3.13) and requires more smoothness in its chain rule (Problem 3.14). Whenever the Fisk-Stratonovich integral is defined, the Ito integral is also, and the two are related by (3.9).

3.13 Definition. Let X and Y be continuous semimartingales with decompositions given by (3.7). The Fisk-Stratonovich integral of Y with respect to

Xis (3.9)

1

01

Ysdill, +

lc 0 dX, -A

YsdB, + 2 <M, N>,;

o

0

t < co,

o

where the first integral on the right-hand side of (3.9) is an Ito integral.

3.14 Problem. Let X =

,

V)) be a vector of continuous semi-

martingales with decompositions

= Xg) + Ati) + where each Mo) e class C3, then (3.10)

i=

BP;

d,

I" and each B(1) is of the form (3.2). If f: l -R is of

f(x,)= fixo) +

=i

ft ,a

0 ox;

fajodxr.

3.15 Problem. Let X and Y be continuous semimartingales and n = {t0,t1,...,tm} a partition of [0, t] with 0 = t0 < t1
, = t, we have Mti) Thus, the real and imaginary parts of {r, eio"-)divt-h, A; 0 < t < co} are not only in .1r1, but also in .1'. Consequently,

E[ f t eR.,xv) dmy)

= 0,

P-a.s.

For A E., we may multiply (3.13) by e-1".)1A and take expectations to obtain E[ein,,xt-xoi

'A] =

P(A) - -2

E[ei(". x '4) 1A] dv.

This integral equation for the deterministic function t

E[er( ",x,-X.)1A] is

readily solved:

E[eu,xi-xd.A, and (3.12) follows.

= P(A )e-(1124.112(i-s),

V

e

158


3.17 Exercise. Let W = (W('), W,(2), Wi(3)) be a three-dimensional Brownian motion starting at the origin, and define 3

X = n sgn(WP), Mu)

wi(i),

m:2)

pvt(2)

and

= M:3)

X W,(3).

Show that each of the pairs (M(1), M(2)), (M(1), M(3)) and (M(2), M(3)) is a two-dimensional Brownian motion, but (Mu), m(2), m(3)) is not a threedimensional Brownian motion. Explain why this does not provide a counterexample to Theorem 3.16, i.e., a three-dimensional process which is not a Brownian motion but which has components in d)" and satisfies (3.11).

3.18 Problem. Let W = 1W, = (Wu), ... , W(d)), g,; 0 < t < cdo I be a d-

dimensional Brownian motion starting at the origin, and let Q be a d x d orthogonal matrix (QT = 12-1). Show that 11/, A QW, is also a d-dimensional Brownian motion. We express this property by saying that "d-dimensional Brownian motion starting at the origin is rotationally invariant."

C. Bessel Processes, Questions of Recurrence Another use of the P. Levy Theorem 3.16 is to obtain an integral representation for the so-called Bessel process. For an integer d _. 2, let W = { W, = (IV,(1), ... , Wt(d)), ,i; 0 < t < col, {Px1,.. Rd be a d-dimensional Brownian family on some measurable space (SI, F). Consider the distance from the origin (3.14)

R, A II Hitli = Wt(1)/2 +

+ (Wi(d))2;

0 < t < oo,

so Px [Ro= Ilxli] = 1. If x, y e Rd and Ilxil = OA, then there is an orthogonal

matrix Q such that y = Qx. Under Px, (V = 111/, A QW ..F,; 0 < t < ool is a d-dimensional Brownian motion starting at y, but II afili = II Will, so for any F EM(C[0, co)), we have (3.15)

Px[R.e F] = Px[110 2, and let W = { W, .57;,; 0 < t < co}, { Px}x.R. be a d-dimensional Brownian family on (SI, ,). The process R = {R, = 11W,II, ..F,; 0 < t < co} together with the family of measures {P' }, >0 {Po .o.....o»>0 i (S/,.F) is called a Bessel family with dimension d. For fixed r > 0, we say that R on (SI, g", Pr) is a Bessel process with dimension d starting at r. .


159

3.20 Problem. Show that for each d > 2, the Bessel family with dimension d

is a strong Markov family (where we modify Definition 2.6.3 to account for the state space [0, co)). 3.21 Proposition. Let d > 2 be an integer and choose r > 0. The Bessel process

R with dimension d starting at r satisfies the integral equation 'd

R, ---- r + fo

(3.16)

-

1

2R,

ds + Bt;

0 < t < co,

where B = {B..; 0 5 t < co} is the standard, one-dimensional Brownian motion B Q > 13(1)

(3.17)

with

BY) A

I0' We R,

1 < i < d.

dWP);

PROOF. We use the notation of Definition 3.19, except we write P instead of P. Note first of all that R, can be at the origin only when Ws") is, and so the Lebesgue measure of the set {0 < s < t; Rs= 0} is zero, a.s. P (Theorem 2.9.6).

Consequently, the integrand (d - 1)/2R3 in (3.16) is defined for Lebesgue almost every s, a.s. P. Each of the processes B(i) in (3.17) belongs to .At e2, because E

f ' (-1Rs W( i))

2

ds < t;

0 < t < co.

0

Moreover, , =

1

J o R52

r

1

. . wscowsco d<wo, Ivo>5 = 45_____ wo Ivo) ds , u D2s s

0 ls

which implies

, =-- E 0, define 3

/-

+

gjy) =

3

,y 4,./e

1

y2;

8e,fi y

fi;

6,

SO g, is of class C2 and lime4.0 g,(y) = g(y) for all y > 0. Now apply Ito's rule

to obtain e(e) + .1,(e) + K,(e),

ge(Y,) = ge(r2) +

(3.18)

1=1

where 1

[1{r >0 -+ 1{r. 0 and m > 1, show that (3.25)

f: X tdW,

2m

(m(2m - 1))mrn-lE

(Hint: Consider the martingale {Mt = f o Xs dW,

1X,12'n dt.

0 < t < T }, and apply

Ito's rule to the submartingale IMt12m .)

Actually, with a bit of extra effort, we can obtain much stronger results. We 12m) be the increasing functions E(1 and E(<M>r), with the convention shall show, in effect, that for any M E (3.26)

itit* -A max IMsl,

o<st

have the same growth rate on the entire of [0, co), for every m > 0. This is the subject of the Burkholder-Davis-Gundy inequalities (Theorem 3.28). We present first some preliminary results. 3.26 Proposition (Martingale Moment Inequalities [Millar (1968), Novikov (1971)]). Consider a continuous martingale M which, along with its quadratic variation process <M>, is bounded. For every stopping time T, we have then (3.27)

E(1/147-12"1) < C;E(<M)7);

m>0

(3.28)

BmE(<M)7) < E(IMr12m);

m > 1/2

(3.29)

B,E(<M>7) _5 EUM1)21 5 CmE(<M>7); m > 1/2


164

for suitable positive constants B C., C,, which are universal (i.e., depend only on the number m, not on the martingale M nor the stopping time T). PROOF. We consider the process

Y, -A 6 + t<M>, + M? = 6 + (1 + t)<M>, + 2

f Ms c/M 0

t < oo,

o

where 6 > 0 and E z 0 are constants to be chosen later. Applying the changeof-variable formula to f(x) = xm, we obtain (3.30)

Y,'" = 6m + m(1 + E)

il Ysm-1 d<M>s + 2m(m - 1).1o Yr2M: d<M> o

+ 2m f i

YriMsdMs; 0

t < co.

o

Because M, Y, and <M> are bounded and Y is bounded away from zero, the last integral is a uniformly integrable martingale (Problem 1.5.24). The Optional Sampling Theorem 1.3.22 implies that E 11; Yri Ms dM, = 0, so taking expectations in (3.30), we obtain our basic identity T

(3.31)

EYE = 6m + m(1 + OE 1

Ysm-1 d<M>s

o T

+ 2m(m - 1)E f yr2ms2 d<m>s. o

Case 1: 0 < m < 1, upper bound: The last term on the right-hand side of (3.31) is nonpositive; so, letting 610, we obtain T

(3.32)

E[E<M>T + Mflm < m(1 + E)E J

(E<M>s + M:)"1-1 d<M>, 0

<M>;' d <M>s

MO + c)cm -1 E I o

= (1 + E)E'E(<M>7). The second inequality uses the fact 0 < m < 1. But for such m, the function f(x) = xm; x > 0 is concave, so (3.33)

x > 0, y > 0,

2"1-1(xm + ym) < (x + y)m;

E

and (3.32) yields: Em EVM >7) + E(IMTI2m) (3.34)

E(I MTI2m)

(1 + E)G

[(1 + E) (-21 m

E(< M >7), whence

- em1E(<M>7).

Case 2: m > 1, lower bound: Now the last term in (3.31) is nonnegative, and the direction of all inequalities (3.32)-(3.34) is reversed:


E(IMT121

165

[(1 + E) (c/m -1

- EmiE(01)7).

Here, c has to be chosen in (0,(2'1 Case 3:1 < m < 1, lower bound: Let us evaluate (3.31) with E = 0 and then let 810. We obtain (3.35)

f

E(IMTI2m) = 2m(m -

IM,12(m-1)d<M>s.

On the other hand, we have from (3.33), (3.31):

2m-1[emE(<M>7) + E(5 + Mi)m] E[E<M>T +

+ MD]m T

6" +

±

(6 + Ms2)m-1 d<M>so

Letting S 4 0, we see that

(3.36) 2' CemE(01>7) +

f

I MT 12m)] < m(1 +

I Ms12("1-1) d<M>s.

Relations (3.35) and (3.36) provide us with the lower bound -1

E(1/1/7.12m)

cm (12+:211 m

1)

E1<m>71

valid for all c > 0. Case 4: m > 1, upper bound: In this case, the inequality (3.36) is reversed, and we obtain E(IMTI2m) < Ern((1

+ c)21-m

2m - 1

1)-1 E(<M>7!),

where now c has to satisfy E > (2m - 1)2' - 1. This analysis establishes (3.27) and (3.28). From them, and from the Doob maximal inequality (Theorem 1.3.8) applied to the martingale IMTAt, -Ft; 0 < t < oo}, we obtain for m > 1/2: B,E(<M>TA

E(IMT A il2m) < EUM1 A t/21

( 2m ) 2171

- 1)2

1"

IMTAtI2m)

< C; (2m 2m_ 1)21"E(<M>7,); which is (3.29) with T replaced by T n t. Now let t and use the monotone convergence theorem.

co in this version of (3.29) 11:1


166

3.27 Remark. A straightforward localization argument shows that (3.27), (3.29) are valid for any M edrI". The same is true for (3.28), provided that the additional condition E( <M >) < co holds. We can state now the principal result of this subsection. 3.28 Theorem (The Burkholder-Davis-Gundy Inequalities). Let M e dric'c and recall the convention (3.26). For every m > 0 there exist universal positive constants km, Km (depending only on m), such that (3.37)

kmE(<M >) < E[(M1)21 < KmE(<M>7)

holds for every stopping time T PROOF. From Proposition 3.26 and Remark 3.27, we have the validity of (3.37) for m > 1/2. It remains to deal with the case 0 < m < 1/2; we assume without loss of generality that M, <M> are bounded. Let us recall now Problem 1.4.15 and its consequence (1.4.17). The righthand side of (3.29) permits the choice X = (M*)2, A = C, <M> in the former, and we obtain from the latter

2 -m

EL(MP)21
7)

for every 0 < m < 1. Similarly, the left-hand side of (3.29) allows us to take X = B, <M >, A = (M*)2 in Problem 1.4.15, and then (1.4.17) gives for 0 < m < 1:

1 -m

2 -m

0

BTE(<M>7)
,; 0 ,=,

t < cc.

Show that for any m > 0, there exist (universal) positive constants ,I.m, Am such

that (3.38)

AmE(AT) < E(11M111)2m < AmE(AT)

holds for every stopping time T 3.30 Remark. In particular, if the AO in Problem 3.29 are given by MP )

=E tr=1 jot

n,Pwsch d,

where { W = (W,(1), ... , W,(')), .Ft; 0 < t < co} is standard, r-dimensional

Brownian motion, {X, = (VP); 1

__

i < d, 1 < j 5 r,0 < t < co} is a matrix


167

of measurable processes adapted to (3.38) holds with (3.39)

and 11Xt112 A Ef.-.1

(XP.J.))2, then

AT = o

E. Supplementary Exercises 331 Exercise. Define polynomials H(x, y); n = 0, 1, 2, ... by 0"

H(x, y) =

ace

exp ax - -1 (x2 y) 2

x, y e 11;t

a=0

Ho(x, y) = 1, H, (x, y) = x, 112(x, y) = x2 - y, H3(x, y) = x3 - 3xy, H4(x, y) = x4 - 6x2y + 3y2, etc.). These polynomials satisfy the recursive relations

(e.g.,

-

(3.40)

=

ax

n = 1, 2, ...

y);

as well as the backward heat equation (3.41)

a

az H,,(x, v)

+

axe

n = 0, 1, ...

= 0;

H.(x,

For any M e.A,,,k,c, verify

(i) the multiple Ito integral computation t1

o

dMin

41

o

.

. dMt2 dMt, =

1

n!

H,,(M <M>,),

(ii) and the expansion exp (xM,

-

2

<M>, )

ot.

= E -H(M<M>,). ,, =0

n!

(The polynomials H(x, y) are related to the Hermite polynomials h ^(x)

(-1r ex2/2 p-ir

dn

dx"

e-x2a

by the formula H(x, y) = .111! y"/2 h(x/fy).) 3.32 Exercise. Consider a function a:

(0, co) which is of class C1 and such

that 1/a is not integrable at either +co. Let c, p be two real constants, and introduce the (strictly increasing, in x) function f(t, x) = e" f o dy /a(y); 0 < t < co, x e II and the continuous, adapted process = 4 + p roe" ds + c'oe" dW, A; 0 < t < co. Let g(t, -) denote the inverse of f(t, -). Show that the


168

process X, = g(t,i) satisfies the stochastic integral equation X, = X0 + 1 t b(Xs) ds +

(3.42)

ft 01)0 dWs;

0 < t < oo

o

o

for an appropriate continuous function b: gl -> DI, which you should determine.

333 Exercise. Consider two real numbers .5, it; a standard, one-dimensional

Brownian motion W; and let W,(") = W, + pt; 0 < t < co. Show that the process

X, = f t exp[ol W,(P) - Ws(P)} - +62(t - s)]ds;

0 < t < co

o

satisfies the Shiryaev-Roberts stochastic integral equation

X, =

it (1 + .3,1Xs)ds + .3 o

f

XsdWs.

o

334 Exercise. Let W be a standard, one-dimensional Brownian motion and 0 < T < co. Show that lim sup p-00 05t5T e-st

fefts

diV,

= 0, a.s.

o

335 Exercise. In the context of Problem 2.12 but now under the condition E.,/T < co, establish the Wald identities E(WT) = 0,

E(W7) = ET.

336 Exercise (M. Yor). Let R be a Bessel process with dimension d z 3, starting at r = 0. Show that {M, A (1/Rd,-2); 1 < t < co} (i) (ii)

(iii)

is a local martingale,

satisfies sups ,,,

f(g.))dgio.

lim

0

0

2

0

holds a.s. P, for every fixed t > 0.

3.4. Representations of Continuous Martingales in Terms of Brownian Motion In this section we expound on the theme that Brownian motion is the fundamental continuous martingale, by showing how to represent other continuous martingales in terms of it. We give conditions under which a vector of d continuous local martingales can be represented as stochastic integrals with respect to an r-dimensional Brownian motion on a possibly extended probability space. Here we have r < d. We also discuss how a continuous local

martingale can be transformed into a Brownian motion by a random timechange. In contrast to these representation results, in which one begins with a continuous local martingale, we will also prove a result in which one begins

with a Brownian motion W = { W Ft; 0 < t < co} and shows that every continuous local martingale with respect to the Brownian filtration {..F,} is a stochastic integral with respect to W. A related result is that for fixed 0 < T < co, every g-r-measurable random variable can be represented as a stochastic integral with respect to W. We recall our standing assumption that every filtration satisfies the usual conditions, i.e., is right-continuous, and ..F0 contains all P-negligible events.

4.1 Remark. Our first representation theorem involves the notion of the extension of a probability space. Let X = {X.Ft; 0 < t < co} be an adapted process on some (0,

P). We may need a d-dimensional Brownian motion

independent of X, but because (0,,,P) may not be rich enough to support this Brownian motion, we must extend the probability space to construct this.

Let (0, 327, P) be another probability space, on which we consider a ddimensional Brownian motion B----{B A; 0 < t < co }, set nAo.n,

g

"ost-,15.4p x P, and define a new filtration by gt

g; ®A. The

latter may not satisfy the usual conditions, so we augment it and make it right-continuous by defining


170

A A n aosu s>,

where Ai is the collection of P-null sets in g. We also complete g by defining

= cr(g u .4'). We may extend X and B to {A}-adapted processes on P) by defining for (w,

E

= ). Then 11 = {R A; 0 < t < col is a d-dimensional Brownian motion, independent of 5C-, A; 0 < t < col. Indeed, B is independent of the exten= x ,(0)),

gi(0),

A(co

sion to a of any ,-measurable random variable on U. To simplify notation, we henceforth write X and B instead of fe and 11 in the context of extensions.

A. Continuous Local Martingales as Stochastic Integrals with Respect to Brownian Motion Let us recall (Definition 2.23 and the discussion following it) that if W = ig .Ft; 0 < t < col is a standard Brownian motion and X is a measurable, adapted process with P[yo Xs ds < oo] = 1 for every 0 < t < co, then the stochastic integral /,(X) = Po X sc11415 is a continuous local martingale with quadratic variation process 1 ro X s2 ds, which is an absolutely con-

tinuous function of t, P a.s. Our first representation result provides the converse to this statement; its one-dimensional version is due to Doob (1953).

4.2 Theorem. Suppose M = {M, = (MP), , AV)), A; 0 < t < col is defined on (a, , P) with Mt') E e' I", 1 ,(w) is an absolutely continuous function of t for P-almost every w. Then there is an extension P) of (S2, P) on which is defined a d-dimensional Brownian motion W = {W, = (W,(1), , W(°)), A; 0 < t < col, and a matrix X = {(X;'" )°,k =1, A; 0 < t < co} of measurable, adapted processes with (4.1)

p

[f

col=

(xli,k))2 ds

1
,=

E k =1 d

x , k) d ws(k);

Ji'0

1 < I < d, 0 < t < co,

x!")XP'k) ds ;

1 < j < d,

0

t < oo.

PROOF. We prove this theorem by a random, time-dependent rotation of coordinates which reduces it to d separate, one-dimensional cases. We begin by defining

3.4. Representations of Continuous Martingales in Terms of Brownian Motion 171

(4.4)

. .

=

=d

dt

= lim nUM(`), Mtn >, - <M(i), Mcn>0_0 f)).],

so that the matrix-valued process Z = {4 = (4.1;=,,A; 0 < t < col is symmetric and progressively measurable. For a = (a,,..., ota)e Rd , we have d

d

Ed

E ccizIja = - E aior) i=i dt

0,

so 4 is positive-semidefinite for Lebesgue-almost every t, P-a.s. Any symmetric, positive-semidefinite matrix Z can be diagonalized by an = QT, so that ZQ = A and A is diagonal orthogonal matrix Q, i.e., with the (nonnegative) eigenvalues of Z as its diagonal elements. There are several algorithms which compute Q and A from Z, and one can easily verify that these algorithms typically obtain Q and A as Borel-measurable functions

of Z. In our case, we start with a progressively measurable, symmetric, positive-semidefinite matrix process Z, and so there exist progressively measurable, matrix-valued processes {Qt(co) = (q1-1(co))1.;_,; gf,; 0 < t < co} and {A,(w) = (.5u4(a)))1,;=, , 0 < t < co} such that for Lebesgue-almost every t, we have

E

gc.i1k.J

k=1 d

(4.6)

E d

d

(4.5)

1:5_

Si;

k=1

d

EE

(5v1.1 > 0 ;

1

j < d,

k=1 1=1

a.s. P. From (4.5) with = j we see that (e1)2 'i ft

s < <Mm>, < co,

and we can define continuous local martingales by the prescription (4.7)

Nio) A

1 < i < d, 0 < t < co.

dIVIlk); k=1

0

From (4.4) and (4.6) we have, a.s. P, d

(4.8)

d

, = E E k=1 1=1 Jo d

k .i qs qs j d<M(k), Al")).s

.t

d

=EE

q! i 4 I 11.! i ds

k=1 1=1

= Si;

We see, in particular, that

Aids.


172

(4.9)

Asi ds = ,

d

E E AV(0),

Ai(w)

1=1 ;=,

where AP) denotes total variation of A(Li) on [0, t]. Let 7;(0)) be the inverse of the function At(w) + t, i.e., il,s(0(co) + Ts(co) = s; 0 5 s < co. (i) Show that for each s, T., is a stopping time of (ii) Define Ws A ..f.T.; 0 5 s < co. Show that if {.Fi} satisfies the usual conditions, then {Ws} does also. (iii) Define

Nr

MP,

15i

d;

0 < s < oo.

Show that for each 1 5 i 5 d: N(i) drloc, and the cross-variation <MI), /NM>, is an absolutely continuous function of s, a.s. P.

B. Continuous Local Martingales as Time-Changed Brownian Motions The time-change in Exercise 4.4 is straightforward because the function A, + t

is strictly increasing and continuous in t, and so has a strictly increasing, continuous inverse T. Our next representation result requires us to consider the inverse of the quadratic variation of a continuous local martingale; be-


174

cause such a quadratic variation may not be strictly increasing, we begin with a problem describing this situation in some detail.

4.5 Problem. Let A = IA(t); 0 < t < co } be a continuous, nondecreasing function with A(0) = 0, S -4a', A(co) 5 co, and define for 0 < s < co: T(s) =

05s<S

{in* > 0; A(t) > s };

s > S.

oo;

The function T = { T(s); 0 5 s < co } has the following properties:

(i) T is nondecreasing and right-continuous on [0, S), with values in [0, co). If A(t) < S; V t .- 0, then limsts T(s) = co. (ii) A(T(s)) = s A S; 0 _< s < op. (iii) T(A(t)) = supIr t: A(r) = A(t)}; 0 5 t < co. (iv) Suppose cp: [0, co) -> R is continuous and has the property A(t1) = A(t)

cp(ti)= p(t).

for some 0 < t1 < t

Then p(T(s)) is continuous for 0 < s < S, and (4.13)

p(T(A(t))) = cp(t);

0 < t < c0.

(v) For 0 5 t, s < co: s < A(t).. T(s) < t and T(s) S t s < A(t). (vi) If G is a bounded, measurable, real-valued function or a nonnegative, measurable, extended real-valued function defined on [a, b] c [0, co), then A(b)

(4.14)

G(t) dA(t) =

1

G(T(s))ds.

A(a)

4.6 Theorem (Time-Change for Martingales [Dambis (1965), Dubins & Sch-

warz (1965)]). Let M = {M.9;;0 5 t < co } e../r1" satisfy lim,o, <M>, = co, a.s. P. Define, for each 0 < s < co, the stopping time (4.15)

T(s) = inf {t > 0; <M>, > s }.

Then the time-changed process (4.16)

B, '' MT(,),

-41.. LFT(s);

0 ''.

S < 00

is a standard one-dimensional Brownian motion. In particular, the filtration {W,} satisfies the usual conditions and we have, a.s. P: (4.17)

M, = B<m>,;

0 5 t < co.

PROOF. Each T(s) is optional because, by Problem 4.5 (v), { T(s) < t} = I<M>, > sl e .5c and {..F,} satisfies the usual conditions; these are also satisfied by Al just as in Problem 4.4 (ii). Furthermore, for each t, <M>, is a stopping time for the filtration Is} because, again by Problem 4.5 (v), { <M>, 5 s} = { T(s)

t} egT(s) .--- Is;

0 ..' S < 0 0 .


Let us choose 0 5 s, < s2 and consider the martingale {R, = M T(s2), .57t; 0 < t < co}, for which we have , = <M>, A T(S2)

<M>T(s2) = s2;

0 5- t < GO

by Problem 4.5 (ii). It follows from Problem 1.5.24 that both Cf and SI2 - are uniformly integrable. The Optional Sampling Theorem 1.3.22 implies, a.s. P:

E[Bs2 -

=

- KIT(si).LFT(s,)] = 0,

E[(13,2 - Bs,)20si] = EURT(s2)- MT00 1

I .-Fro,)]

= ECps2)- T401,-FT(so] = S2 -S I s < co} is a square-integrable martingale with quadratic variation , = s. We shall know that B is a standard Brownian Consequently, B = {Bs, Ws; 0

motion as soon as we establish its continuity (Theorem 3.16). For this we shall use Problem 4.5 (iv). We must show that for all co in some Sr f2 with P(S2*) = 1, we have: (4.18)

<M>ti(w) = <M>,(w) for some 0 < t, < t

mii(w) = mt(c0).

If the implication (4.18) is valid under the additional assumption that t, is rational, then, because of the continuity of <M> and M, it is valid even without

this assumption. For rational t > 0, define o- = inf t > t,: <M>, > <M>,, I,

N, = M(,,), - M,,, 0 < s < co, so {N,, ..F,1; 0 < s < co} is in dr'1" and , =

- <M>,, = 0, a.s. P.

It follows from Problem 1.5.12 that there is an event f2(t, ) s 12 with P(C1(t,))= 1 such that for all co e S1(t,), <M>,,(co) = <M>,(co), for some t > t,

M,,(w) = Mi(co).

The union of all such events fl(t, ) as t, ranges over the nonnegative rationals will serve as fr, so that implication (4.18) is valid for each co E sr. Continuity of B and equality (4.17) now follow from Problem 4.5 (iv). 11:1

4.7 Problem. Show that if P[S Q <M>

co] > 0, it is still possible to define a Brownian motion B for which (4.17) holds. (Hint: The time-change T(s) is now given as in Problem 4.5; assume, as you may, that the probability space has been suitably extended to support an independent Brownian motion (Remark 4.1).)

The proof of the following ramification of Theorem 4.6 is surprisingly technical; the result itself is easily believed. The reader may wish to omit this proof on first reading.


176

4.8 Proposition. With the assumptions and the notation of Theorem 4.6, we have

the following time-change formula for stochastic integrals. If X = {X.; 0 < t < co} is progressively measurable and satisfies

.

(4.19)

X; d<M>t < co

a.s.,

then the process (4.20)

X2(,), W's;

Ys

0

<s. A

s}


= {n A t Lc. T(s)} = {n 5 T(s), <J>m) <M>n

E 'T(s).

By

assumption,

= oo a.s.; it follows that .7 is in Jr' (relative to {C).

Using this fact, we may repeat the preceding argument to show that Joy>, = J, is a continuous local martingale relative to the filtration Mm>,}, which contains and may actually be strictly larger. In fact, we can choose an arbitrary continuous local martingale N relative to {Ws} and construct Art = Rof>,, a continuous local martingale relative to

{W00,}. If we take N = B, then N = M from (4.17) and so M is in dr'" relative to { 0 such that IC,C + <M>, < pt, V t 0 is valid almost surely. Show that for fixed T > 0 and sufficiently large n > 1, we have F'[max

IX,I > n] < exp {

0

-n2 1.

18p T

C. A Theorem of F. B. Knight Let us state and discuss the multivariate extension of Theorem 4.6. The proof will be given in subsection E.

4.13 Theorem (F. B. Knight (1971)). Let M = {M, = (MP), t < co} be a continuous, adapted process with M") diecmc, a.s. P, and <M"),110>, = 0;

(4.31)

1

,

Md)),F,; 0 < = 00;

j < d, 0 < t < CO.

Define

Ti(s) = ingt

0; <M")>, > s};

0 < s < co, I < i < d,

so that for each i and s, the random time Ti(s) is a stopping time for the (right-continuous) filtration {.9;;}. Then the processes

Br

M)(s);

0 < s < co,

1

d,

are independent, standard, one-dimensional Brownian motions.

Discussion of Theorem 4.13. The only assertion in Theorem 4.13 which is not

already contained in Theorem 4.6 is the independence of the Brownian motions B(i); 1 < i < d. Theorem 4.6 states, in fact, that B(1) is a Brownian motion relative to the filtration {(e) -4- ,T4 ),,o, but, of course, these filtrations are not independent for different values of i because g!;) = Fes; 1 < i < d. , For are independent, The additional claim is that the a-fields Fr, where {.F.,B1 is the filtration generated by B(i). This would follow easily if the assumption (4.31) were sufficient to guarantee the independence of M(I), Mu)

for i j; in general, however, this is not the case. Indeed, if W = { 0 < t < co} is a standard Brownian motion, then with AV)

1{w,> o} dWs,

we have M(I), M(2) e ..fiq and

0

t < co,


180

<M(1), M(2)>, =

f

1 {w,} 1 {w. and <M(2)> would also be independent. On the contrary, we have <M")>, + <M(2)>, =

1{,.>

ds +

0 < t < aD.

1{w. = 0. It follows from Problem 1.5.12 that P[Z, = 0, V 0 < t < oo] 1. It suffices, therefore, to establish the decomposition for every finite timeinterval [0, T]; by uniqueness, we can then extend it to the entire half-line [0, co). Let us fix T > 0, let gt, be the closed subspace of P) defined is in

by (4.32), and let giq- denote its orthogonal complement. The random variable MT is in /2(52, 9T, P), so it admits the decomposition (4.35)

MT = IT(Y)

ZT,

where Y e 2' and ZT e L2 (SI, .97T, P) satisfies

E[ZTIT(X)] = 0; V X e

(4.36)

Let us denote by Z = {Z Ft; 0 < t < col a right-continuous version of the martingale E(ZTigc)(Theorem 1.3.13). Note that Z, = ZT for t > T Obviously Z e.42 and, conditioning (4.35) on .F we obtain Ait = 4(Y) + Zt;

(4.37)

0 < t < T, a.s. P.

It remains to show that Z is orthogonal to every square-integrable martingale of the form I(X); X e-C1, or equivalently, that {Z,/,(X), A; 0 < t < T} is a martingale. But we know from Problem 1.3.26 that this amounts to having E[ZsIs(X)] = 0 for every stopping time S of the filtration {A}, with S Lc. T. From (2.24) we have Is(X) = IT(X), where Xi(w) = Xi(o.01{, <s(,)} is a process in Therefore,

E[ZsIs(X)] = E[E(ZTI,$)ls(X)] = E[ZTIT(X)] = 0 by virtue of (4.36).

It is useful to have sufficient conditions under which the classes llZ and ./#1 actually coincide; in other words, the component Z in the decomposition of Proposition 4.14 is actually the trivial martingale Z = 0. One such condition is that the filtration {A} be the augmentation under P of the filtration {.F,W} generated by the Brownian motion W. (Recall from Problem 2.7.6 and Proposition 2.7.7 that this augmented filtration is continuous.) We state and


182

prove this result in several dimensions. A martingale relative to this augmented filtration will be called Brownian.

4.15 Theorem (Representation of Brownian, Square-Integrable Martingales W,(d)),..F,; 0 < t < ool be a as Stochastic Integrals). Let W = tW, = (W,"), d-dimensional Brownian motion on (S/,..F, P), and let {..F,} be the augmentation under P of the filtration {..F,w } generated by W. Then, for any square-integrable

martingale M = {M, 0 < t < co) with Mo = 0 and RCLL paths, a.s., there exist progressively measurable processesYoe Y = {Y,(1), g. 0 < t < col such that (4.38)

E

j

(Y,i)2 dt < oo;

*

j< d

1

for every 0 < T < co, and t

(4.39)

M, = E

ysodwp; 0 < t < CO .

J=1

0

In particular, M is a.s. continuous. Furthermore, if fth; 1 < j < d, are any other progressively measurable processes satisfying (4.38), (4.39), then

f

- fru) 12 dt = 0,

I

a.s.

.i=1

PROOF. We first prove by induction on m, where m = processes Y(1),..., Y(m) in .29* such that

d, that there are

m

(4.40)

Z, A M, - E

o < t < co

ysti)dwp;

.i=1

is orthogonal to every martingale of the form ET_1 f o XS dW,,u), where X(J e

2*; 1 _< j m. If m = 1, this is a direct consequence of Proposition 4.14. Suppose such processes exist for m - 1, i.e.,

Z A M,

-

m-i

ypdwp; o < t < cc,

E

J=1

0

is orthogonal to ET_-11 Ito xpdwso for all Xul e 2*; 1 sjsm - 1. Apply Proposition 4.14 to write

Z=

fYs(m) d Ws(m) + 4;

0 < t < oo,

0

for some 11(" e 2*, where Z is orthogonal to 1.0 gn) d Ws(m) for all X(m) e

2*. For 1 s j< m- 1 and X01 e 2*, we have = 0. Thus, we have the decomposition (4.40) for M. In particular, with m = d,


0 5 t < co,

<M, Hi(h>, = I Yu) ds;

(4.41)

1 < j < d.

0

Following Liptser & Shiryaev (1977), pp. 162-163, we now show that, in the notation of (4.40) with m = d, we have P-a.s. that Z,

0 < t < co.

0;

First, we show by induction on n that if 0 = so < s, < < s < t, and if the functions fk: C, 0 < k < n are bounded and measurable, then

E[Z, fl fk(Wsk)] = 0.

(4.42)

k=0

When n = 0, (4.42) can be verified by conditioning on .90 and using the fact Zo = 0 a.s. Suppose now that (4.42) holds for some n, and choose s < t. For 1: 0 = (0,, . . ,0)E Rd fixed and s. < s < t, define with i = co(s) -4

E[Z,

n fk(Wsdeimwd= E[Z k=0 n fk(wsdeimwd.

k=0

Using Ito's rule to justify the identity el("'w.)

=

d

s

+ E ie.

110112

ei(e'w-i dly(i)

2

.i=--1

s

e"'w-) du,

s"

we may write (4.43)

E[Zse"-woi,s.] = zs.ei(t 3 'W..) d

+ E le i E[Zs i=i

f

s

ei(e'w-) dWu)

3,,

110112 Er Zs 2 L

f eimw.)d

j

,

u

But (4.40) and (4.41) imply

E[Z, f

s

et(9'w-) dW,P)

grd = E [(Z., - Zs)

Js

eimw-) dWli) .FS.1 = 0.

Multiplying (4.43) by FP,:_o f(Wsk) and taking expectations, we obtain (4.44)

co(s) = Os.)

110112 f s

2

s

E[Zs- fl f5(14/5,)e"w-)]du k =1

110112

= co(s.) -

2

4910 du; s

s

t.

By our induction hypothesis, cp(s) = 0, and the only solution to the integral equation (4.44) satisfying this initial condition is 9(s) = 0; s < s < t. Thus


184

(4.45)

= 0;

E

v e Rd.

k=o

With D A Zt irk'=o fk(Wsk), we define two measures on (Old, M(Rd)) by

11±(F) = ELD± ir(Ws)i;

Fe M(fre).

Equation (4.45) implies c."'x)

e'(°xl

(dx)

(dx);

V0

fo;sa

and by the uniqueness theorem for Fourier transforms, we see that it+ --= Thus

E[D+ f (Ws)] = E[D f(Ws)]; s < 4+1 g s < t, for any bounded, measurable f:01" -> C. This proves (4.42) for n + 1 and completes the induction step. A standard argument using the Dynkin System Theorem 2.1.3 now shows that we have

E[Z,fl = 0

(4.46)

for every A w-measurable indicator and thus, for every 547,w-measurable, bounded Since A differs from Aw only by P-null sets, (4.46) also holds for every A-measurable, bounded Setting = sgn(Z,), we conclude that Z, = 0 a.s. P for every fixed t, and by right-continuity of Z, for all t e [0, oo) simultaneously. The uniqueness assertion in the last sentence of the theorem is proved by observing that the martingale El=, f o (Yso )dHis(i) is identically zero, and so is its quadratic variation. 4.16 Problem. Let W = {W, =

,

Wi(d)), A; 0 < t < a)} be a d-

dimensional Brownian motion as in Theorem 4.15. Let M = {M A; 0 < t < oo } be a local martingale with M, = 0 and RCLL paths, a.s. Then there exist progressively measurable processes Y(i) {Y,(j), A; 0 < t < co} such that ('T (Y,

dt < co;

<j

d,

0

T < op ,

and

lye = E

yso) dwsu);

J'io In particular, M is a.s. continuous.

0 < t < co.

J=1 a

4.17 Problem. Under the hypotheses of Theorem 4.15 and with 0 < T < co, let c be an .97T-measurable random variable with EV < co. Prove that there


are progressively measurable processes Y"),... , I'm satisfying (4.38), and such

that d

(0 + E = E(0

(4.47)

Ystb d wsch ;

J=1

a.s. P.

o

E. Brownian Functionals as Stochastic Integrals We extend Problem 4.17 to include the case T = co. Recall that for M e we denote by ..C1(M) the class of processes X which are progressively measurable with respect to the filtration of M and which satisfy E f o X,2 d<M>, < oo. According to Problem 1.5.24, when X e Yt(M), we have .1`6° X dM, defined a.s.

P. If W is a d-dimensional Brownian motion, we denote by ..r:(W) the set of processes X which are progressively measurable with respect to the (augmented) filtration of W and which satisfy E roc' X,2 dt < co.

4.18 Proposition. Under the hypotheses of Theorem 4.15, let be an measurable random variable with EV < oo. Then there are processes Y(1), ..., Y(') in ..TVW) such that d

CO

= E() + E I .i=i

yso d Ws° ;

a.s. P.

o

PROOF. Assume without loss of generality that E() = 0, and let M, be a right-continuous modification of E(ig;). According to Theorem 4.15, there exist progressively measurable Y"),..., Y(d) satisfying (4.38) and (4.39). Jensen's

inequality implies MI S E(215";), so E i=1

i't (Ysti))2 ds = E<M>, = E(M,2)

E(V) < co; 0 < t < co.

o

Hence, Yti)E .29,(W) and M.

1.3.20 shows that M. =

f;°, Ysth MP.) is defined for 1 < j < d. Problem

= c.

We leave the proof of uniqueness of the representation in Proposition 4.18 as Exercise 4.22 for the reader. In one dimension, there is a representation result similar to that of Proposi-

tion 4.18 in which the Brownian motion is replaced by a continuous local martingale M. This result is instrumental in our eventual proof of Theorem 4.13. Uniqueness is again addressed in Exercise 4.22.

4.19 Proposition. Let M = {M; 0 < t < co} be in Jr' and assume that = co, a.s. P. Define T(s) by (4.15) and let B be the one-dimensional Brownian motion Mr(s), es;

0

s < co


186

as in Theorem 4.6, except now we take the filtration AI to be the augmentation

with respect to P of the filtration {AB} generated by B. Then, for every go,-measurable random variable satisfying E2 < oo, there is a process X e 43.(4) for which CO

(4.48)

= E()

J. X, dM,;

a.s. P.

PROOF. Let Y = { Ys, gs; 0 < s < co} be the progressively measurable process of Proposition 4.18, for which we have

f

(4.49)

E

(4.50)

= E() +

ys2 ds < co,

J

Ys dBs.

Define X,= Yol>,; 0 t < We show how to obtain an {.97,}-progressively measurable process X which

is equivalent to X. Note that because {A} {,T(01 contains {,23} and satisfies the usual conditions (Theorem 4.6), we have es A; 0 5 s < oo. Consequently, Y is progressively measurable relative to If Y is a simple process, it is left-continuous (c.f. Definition 2.3), and it is straightforward to show, using Problem 4.5, that { Yoki>,; 0 < t < cc} is a left-continuous process adapted to {.F(}, and hence progressively measurable (Proposition 1.1.13). In the general case, let { Y('')}c°_, be a sequence of progressively measurable (relative to {6'0), simple processes for which

lim E n- co

I Ys(") -Ys 12 ds = O. o

(Use Proposition 2.8 and (4.49)). A change of variables (Problem 4.5 (vi)) yields (4.51)

lim E

IX:") -

J

,1 2 d<M>, = 0,

o

where X:n) A Vn>,. In particular, the sequence {X(")},13_, is Cauchy in 2 t(M), and so, by Lemma 2.2, converges to a limit X e _KAM). From (4.51) we must have E

12, - X,12 d<M>, = 0,

which establishes the desired equivalence of X and X. It remains to prove (4.48), which, in light of (4.50), will follow from foc°

Ys dB, =

f

X, dM,;

a.s. P.

This equality is a consequence of Proposition 4.8.

3.4. Representations of Continuous Martingales in Terms of Brownian Motion 187 PROOF OF F. B. KNIGHT'S THEOREM 4.13. Our proof is based on that of Meyer

(1971). Under the hypotheses of Theorem 4.13, let {dl')} be the augmentation

of the filtration {..fn generated by B"); 1 5 i 5 d. All we need to show is that el'2), ... , dr are independent. For each i, let V') be a bounded, (IV-measurable random variable. According to Proposition 4.19, there is, for each i, a progressively measurable process

X") = {X1'),.; 0 < t < co} which satisfies 00

f(r)2 d<M(')>, < co; 1 < i < d, E

o

and for which

(i) = E(V)) + foo XP) dMP;

1 5i < d.

o

Let us assume for the moment that

1 5 i < d,

E(V)) = 0;

(4.52)

and define the (5,}-martingale ) -4-

ft Xli) dM11);

0 < t < co ,

1 s< co; 1 < i, d.

o

Repeated application of Holder's inequality yields: f1

E

0

2

( n ix)) d<m(0>s Joi

(VI)2] ,E{n[ Joi sup :)..s..t

-

<e>,}

1/2

[E sup

0 <s 712; 0 < t t < oo.

For m > 1, Doob's maximal inequality (Theorem 1.3.8 (iv)), and Jensen's inequality (Proposition 1.3.6) give


188

( 2m 2m- 1

E[ 11:),s..t sup Ill)12"11

El ai)12m

2m

2m

(2m

2m

- 1)

El Vi)I2m < a).

We have from Proposition 3.26 (see also Remark 3.27): I

B E f J---1

cc(i)dw,u),

a.s.

0

must also satisfy CO

Jo

d

E yi(j) - i7;012 dt = 0,

a.s.

;=-1

(ii) In the setting of Proposition 4.19, show that X is unique in the sense that any other process X E .99*(M) which permits the representation

= E() + J

17, dM a.s., 0

must also satisfy ro°

IX: - g:12 d<M>: = 0,

a.s.

(iii) Find a progressively measurable process Y such that 1

0 < j. Y,2 dt < oo, a.s., o

1

but

Y, dWi = 0,

a.s.

o

In particular, there can be no assertion of uniqueness of Y in Theorem 4.20.

4.23 Exercise. Is the following assertion true or false? "If M = {M .Ft; 0 5 t < co} and N = {N Ft; 0 < t < co} are in ../1"" and <M> , then M and N have the same law."


190

4.24 Exercise (Hajek (1985)). Consider the semimartingales

X, = X0 + j't ji ds + j't as dWs,

m(s)ds +

Y = Yo +

p(s)dVS

05t 0. (Hint: By extending the probability space if necessary, take W to be a Brownian motion independent of W, and consider the continuous semimartingales Y(i) A Y0 +

J

m(s)ds +

J

2 (s)

as dl/K + (-1)i

0.s2 dws;

i = 1, 2.)

3.5. The Girsanov Theorem In order to motivate the results of this section, let us consider independent Z,, on (52,,F, P) with EZi = 0, EZ? = 1. normal random variables Z1, Given a vector (pi,. .., un)egr, we consider the new probability measure P on (s2,9%) given by 1

n

piZi(w) - - E

P(dw) = exp

2 t =1

id

P(dw).

Then 15[Z le dz1,.. .,Zne dzn] is given by exp [tE

- - E id 1

= (2n)-"12 exp [ -

1

n

P [Z1 e dz 1,

- E (zi - pi)2]dz 2 i=t

, Zn e dzn]

dzn.

Therefore, under P the random variables Z1, Zn are independent and normal with EZi = pi and £[(Z1 - yi)2] = 1. In other words, {Z = Zi 1 < i < n} are independent, standard normal random variables on (s2, y, P). The Girsanov Theorem 5.1 extends this idea of invariance of Gaussian finitedimensional distributions under appropriate translations and changes of the

3.5. The Girsanov Theorem

191

underlying probability measure, from the discrete to the continuous setting. Rather than beginning with an n-dimensional vector (Z1,...,4) of independent, standard normal random variables, we begin with a d-dimensional Brownian motion under P, and then construct a new measure P under which a "translated" process is a d-dimensional Brownian motion.

A. The Basic Result Throughout this section, we shall have a probability space (n, P) and a d-dimensional Brownian motion W = {W = Wm), A; 0 < t < co} defined on it, with P[Wo = 0] = 1. We assume that the filtration {F,} satisfies the usual conditions. Let X = {X, = (V), , V)), A; 0 < t < ool be a vector of measurable, adapted processes satisfying (5.1)

dt < co ]=

P

1 < i < d, 0 5 T < co.

1;

Then, for each *, the stochastic integral /"(X(')) is defined and is a member of eirioc. We set (5.2)

Zi(X)

1

it t=1 a

exp [

dWs(i)

2

f

11Xsil 2

o

dd.

Just as in Example 3.9, we have =1

(5.3)

d

(1),

WZ,(X)

which shows that Z(X) is a continuous, local martingale with Zo(X) = 1. Under certain conditions on X, to be discussed later, Z(X) will in fact be a martingale, and so EZ,(X) = 1; 0 < t < co. In this case we can define, for each 0 < T < co, a probability measure PT on FT by (5.4)

E[lAZT(X)]; AE FT T.

PT(A)

The martingale property shows that the family of probability measures {PT; 0 < T < col satisfies the consistency condition PT(A)

(5.5)

A(A); AE

0 < t < T.

5.1 Theorem (Girsanov (1960), Cameron and Martin (1944)). Assume that Z(X) defined by (5.2) is a martingale. Define a process W = {W, =

Ito)), A; 0 < t < col by (5.6)

It(l)

Wt(i) -

ds;

1 < i < d, 0 < t < co.

For each fixed T e [0, co), the process {It, A; 0 < t < T} is a d-dimensional Brownian motion on (CI, FT, PT).


192

The preparation for the proof of this result starts with Lemma 5.3; the reader may proceed there directly, skipping the remainder of this subsection on first reading. Discussion. Occasionally, one wants 1V, as a process defined for all t e [0, op), to be a Brownian motion, and for this purpose the measures {PT; 0 < T < co} are inadequate. We would like to have a single measure P defined on so that P restricted to any ..FT agrees with PT; however, such a measure does not exist in general. We thus content ourselves with a measure P defined only on Fcw, the o--field generated by W, such that P restricted to any ..*-TW agrees with PT, i.e., (5.7)

P(A) = E[1AZT(X)]; AE

0 5_ T < co.

,

If such a P exists, it is clearly unique. The existence of P follows from the Daniell-Kolmogorov consistency Theorem 2.2.2. We show this when d = 1; only notational changes are required for the multidimensional case. Let t = (t . . . , t) be a finite sequence of distinct, nonnegative numbers, as in Definition 2.2.1, and let t = max {t,,... , to. Define

Qt(r) =

[WE CI; Mit i((0),

Wtn(a)))

r];

re.6$(R ").

Then {$21} is a consistent family of finite-dimensional distributions, so there is a probability measure Q on (RE°''), gi(Rtc"'))) such that Q,(1") = Q[w e

(w(t ,),

, w (t )) e 11;

But the typical set in ..ny has the form {co e

e .4(01").

W(w)e B}, where B e Mi(Rt"')).

Consequently, Q induces a probability measure P on F' defined by Flu) e n; WOO e B] A Q(B);

B e .4(011°'')),

and this measure satisfies (5.7).

The process 1;17 in Theorem 5.1 is adapted to the filtration {",}, and so is the process {ft, Xr ds; 0 < t < co}; this can be seen as in part (c) of the proof of Lemma 2.4, which uses the completeness of .97,. However, when working with the measure P which is defined only on Fes, we wish 1,r to be adapted to Ign. This filtration does not satisfy the usual conditions, and so we must impose the stronger condition of progressive measurability on X. We have the following corollary to Theorem 5.1.

5.2 Corollary. Let W = {W, gft; 0 < t < co} be a d-dimensional Brownian motion on (12, P) with P[Wo = 0] = 1, and assume that the filtration {.F,} satisfies the usual conditions. Let X = {X

; 0 < t < co} be a d-dimensional, progressively measurable process satisfying (5.1). If Z(X) of (5.2) is a martingale, then 1,V = {14-7 ; 0 < t < co) defined by (5.6) is a d-dimensional Brownian motion on (11," cow , P).


PROOF. For 0 5 ti
s, .97,; 0 5 t< T 1=1

o

is in ,thl". If N e .11,:l" and

IC I, A N, -i

ft XV d s;

i=1

0 5 t 5 T,

o

then , = <M, N>,;

0 < t < T, a.s. P and PT,

where the cross-variations are computed under the appropriate measures.

PROOF. We consider only the case where M and N are bounded martingales with bounded quadratic variations, and assume also that Z,(X) and 11=i ro (XS )2 ds are bounded in t and w; the general case can be reduced to this one by localization. From Proposition 2.14 2 ft

< <M>, 1 t

XV d<M,14/IiI>,

(X.1`))2 ds,

o

and thus SI is also bounded. The integration-by-parts formula of Problem 3.12 gives

Z,(X)S1, =

1

Zu(X)d111,, + > i=1

o

ft 1171.XLIZAX)d14;V, o

which is a martingale under P. Therefore, for 0 < s < t < T, we have from Lemma 5.3:

EAR tl.F.,1 =

Z,(X)

[E Z,(X)1171,1,j = las,

a.s. P and 15T.

It follows that AI e 2'1". The change-of-variable formula also implies:

AR, - <M, N>, =

ft M dN, + ft IV. dM, -i [ ft 111XLII d. o

+

i=1

o

o

ft I1 XV d<M, Ww>.1 o

as well as Z,(X)[1171,1C,

- <M, N> J =

ft Zu(X)111dN. + ft Zu(X)R. dM, o

o

f t [M.N. - <M, N >] XV 4,(X) dW,V .

+ i=1

0


195

This last process is consequently a martingale under P, and so Lemma 5.3

implies that for 0 .s5t5T

-

EAR iNt - t1.97s] =

N> s;

a.s. P and PT.

This proves that <M, N>, = <M, N),; 0 5 t < T, a.s. PT and P. PROOF OF THEOREM 5.1. We show that the continuous process Won (II, ,FT, PT)

satisfies the hypotheses of P. Levy's Theorem 3.16. Setting M = Wth in Proposition 5.4 we obtain SI = IP) from (5.8), so Wu) e i/V". Setting N = HM), we obtain

Let {M

<W-1),W(``)>, = Sj.kt;

0 < t < T, a.s. PT and P.

0 < t < T} be a continuous local martingale under P. With

the hypotheses of Theorem 5.1, Proposition 5.4 shows that M is a continuous semimartingale under PT. The converse is also true; if {114 .Ft; 0 < t
0.

5.10 Exercise. Show that

E(u)e-ar, = exp(ab - Ibl,//22 + 2a), a > 0. 5.11 Exercise (Robbins & Siegmund (1973)). Consider, for v > 0 and c > 1, the stopping time of {77,"1:


198

=

R, = inf {t > 0; exp(v W,

Show that

P[R, < co]

1

E(Y)R, -

2log c

c

.

v2

D. The Novikov Condition In order to use the Girsanov theorem effectively, we need some fairly general conditions under which the process Z(X) defined by (5.2) is a martingale. This process is a local martingale because of (5.3). Indeed, with T. A inf It > 0; max i, is a stopping time of {/s}. It follows that for b < 0: (5.19)

E[1(sb ,)] = 1.

The first expectation in (5.19) is bounded above by ebE[exp(Z <M>,)], which

converges to zero as b1 -co, thanks to assumption (5.18). As b. -co, the second expectation in (5.19) converges to EZ, because of the monotone convergence theorem. Therefore, EZ, = 1; 0 < t < co.

5.13 Corollary (Novikov (1972)). Let W = {W, = (W,(1),..., W,(d)), A; 0 r)), t < co) be a d-dimensional Brownian motion, and let X = pc, = (Xi A; 0 < t < co) be a vector of measurable, adapted processes satisfying (5.1). If (5.20)

E[exp (-21 f To

11Xs 112

ds)]< co;

0 < T < co,

then Z(X) defined by (5.2) is a martingale.

5.14 Corollary. Corollary 5.13 still holds if (5.20) is replaced by the following assumption: there exists a sequence {t},T=0 of real numbers with 0 = to < < tni co, such that t1 < (5.21)

E [exp (-1 f in 2

IlXs112ds)1 < oo;

do

1.

tn -1

PROOF. Let X,(n) = (X)1)1i,n_1.,)(t),..., Xrlitn_i.t.)(0), so that Z(X(n)) is a martingale by Corollary 5.13. In particular, E[Z,.(X (n))1,,._i] = Z,,,_i(X(n)) = 1;

n > 1.

But then,

E[Z,(X)] = E[Z,_1(X)EIZi(X(n))1A_,}i = E[4_1(X)], and by induction on n we can show that E[Z,.(X)] = 1 holds for all n > 1. Since E[Z,(X)] is nonincreasing in t and limn_o, t = co, we obtain (5.17). CI

5.15 Definition. Let C[0, cod be the space of continuous functions x: [0, oo) ->

Rd. For 0 < t < co, define % A r(x(s); 0 < s < t), and set / = /co (cf. Problems 2.4.1 and 2.4.2). A progressively measurable functional on C[0, ce)d is a


200

mapping /2: [0, oo) x C[0, oar -> IR which has the property that for each fixed 0 5 t < co, ti restricted to [0, t] x C[0, oor is .4([0, t]) %/.4([11)-measurable.

Ifµ =

tion) is a vector of progressively measurable functionals on

C[0, °or and W =

Wt(d)), 37".,; 0 < t < co} is a d-dimensional

= (W,(1),

Brownian motion on some

P), then the processes

,qi)(u)) A p(l)(t, W(oo));

(5.22)

0 < t < co,

1 < i < d,

are progressively measurable relative to 5.16 Corollary (Beneg (1971)). Let the vector pi =

,

!PI)) of progressively

measurable functionals on C[0, cor satisfy, for each 0 < T < oo and some KT > 0 depending on T, the condition ti(t, x)II < K T(1 + x*(t));

(5.23)

0 0, we can find {to, , tnal} such that 0 = to < = T and (5.21) holds for 1 n < n(T), then we can construct t1 < < a sequence ItnI,T.0 satisfying the hypotheses of Corollary 5.14. Thus, fix T > 0. We have from (5.22), (5.23) that whenever 0 < ti_, < t,, < T, then

t

11)(3112 ds < (t,, - 4_1)1(1(1 + WT*)2, Itt,_,

where W; A maxor II NI. According to Problem 1.3.7, the process Y, A exp[(t. )K + II Wt )2/4] is a submartingale, and Doob's maximal

-

inequality (Theorem 1.3.8(iv)) yields

E exp[f(t -

)1q(1 + WT*)2] = EI max Yi2 < 4EY,4, 0 0. Problem 6.7 now implies that for each w e 03, (6.7) holds for every Borel function f: fll -) [0, co). Recall finally that LI = C[0, co) and that P assigns probability one to the event S2z -A {w e SI; w(0) = z }. We may assume that C23 g 00, and redefine Lt(x, w) for w 0 Slo by setting

Li(x, w) A Lt(x - w(0), w - w(0)).

We set SP = {w ell; w - w(0) e M}, so that MCP) = 1 for every z e DI (c.f. (6.3)). It is easily verified that L and Q* have all the properties set forth in Definition 6.3. 1:1 6.12 Problem. For a continuous function h: R -> [0, co) with compact support, the following interchange of Lebesgue and Ito integrals is permissible: (6.24)

f 1 h(a)( f :

=

1 (,,,,,,)(Ws) d Ws)

da

ft (..ix' h(a)1()(Ws)da)dWs, a.s. P°. -co

o

6.13 Problem. We may cast (6.13) in the form (6.25)

I Hit - al = I z - al - 13,(a) + 24(a); 0 < t < co,

where B,(a) A -Po sgn( Ws - a) digs, for fixed a e R.

(i) Show that for any z e R, the process B(a) = {B,(a), .97i; 0 .< t < col is a Brownian motion under P, with Pz [B0(a) = 0] = 1. (ii) Using (6.25) and the representation (6.2), show that L(a) = {L,(a), ,-,; 0 < t < co} is a continuous, increasing process (Definition 1.4.4) which satisfies (6.26)

J

101\{.}(NdLi(a) = 0;

a.s. P=.

1o

In other words, the path t'--> L,(a, w) is "flat" off the level set Y,,,(a) = t < co; W(w) =a }. {0 (iii) Show that for P°-a.e. w, we have L,(0, w) > 0 for all t > 0. (iv) Show that for every z e R and Pz-a.e. w, every point of Yo,(a) is a point of strict increase of ti--+ L ,(a , w).


210

C. Reflected Brownian Motion and the Skorohod Equation Our goal in this subsection is to provide a new proof of the celebrated result of P. Levy (1948) already discussed in Problem 2.8.8, according to which the processes (6.27)

{4w - w, A max W- W; 0 -t 1

(Hint: Use (6.38) extensively.)

6.20 Problem. Let the function (p: R

O

be nondecreasing, and define

(p+(x) = lim ('(y), 1'(x) =

x

(u) du.

d0

y--uc±

(i) The functions cp.,. and (p_ are right- and left-continuous, respectively, with (6.43)

x e R.

(p_(x) < cp(x) < (p_,_(x);

(ii) The functions cp., have the same set of continuity points, and equality holds in (6.43) on this set; in particular, except for x in a countable set N, we have (14 (x) = p(x). (iii) The function cl) is convex, with D-(1)(x) = (p_(x) < cp(x) < (p_,.(x) = D+ 10(x);

(iv) If f: R (6.44)

x e R.

R is any other convex function for which D- f(x) < cp(x) < D+ f(x);

x e R,

then we have f(x) = f(0) + (1)(x); x e R.

R, there is a countable set 6.21 Problem. For any convex function f : O N l such that f is differentiable on R \ N, and (6.45)

f '(x) = D+ f(x) = D-f(x);

x e R\ N

Moreover (6.46)

f(x) - f(0) =

f f '(u) du = J x D-1 f(u) du;

x e R.

The preceding problems show that convex functions are "essentially" differ-

entiable, but Ito's rule requires the existence of a second derivative. For a convex function f, we use instead of its second derivative the second derivative measure p on (R, AR)) defined by (6.47)

p([a, b)) g D-f(b) - D- f(a);

-co < a < b < co.

214


Of course, if f " exists, then it(dx) = f "(x) dx. Even without the existence of f ", we may compute Riemann-Stieltjes integrals by parts, to obtain the formula g(x)/1(dx) =

(6.48)

for every function g R

-f

g'(x)D-f(x)dx and has compact support.

R which is piecewise

R 6.22 Theorem (A Generalized Ito Rule for Convex Functions). Let f: be a convex function and it its second derivative measure introduced in (6.47).

Then, for every z e R, we have a.s. Pz: (6.49)

f(W) = f(z) +

D f (Ws) digs +

J

0 < t < co.

Li(x)it(dx)

J

o

PROOF. It suffices to establish (6.49) with t replaced by t n T_ n 1,, and by such a localization we may assume without loss of generality that D+f is of (6.18) to obtain uniformly bounded on R. We employ the mollifiers convex, infinitely differentiable approximations to f by convolution: (6.50)

f

fn(x)

CO

p(x - y)f(y)dy;

n

1.

CO

It is not hard to verify that fn(x) = f°200 p(z)f(x - (z /n)) dz and (6.51)

lim fn(x) = f(x),

lim f,;(x) = Erf(X)

n-,co

n-co

hold for every x e R. In particular, the nondecreasing functions D f and {f'} 11_, are uniformly bounded on compact subsets of R. If g: R C1 and has compact support, then because of (6.48), lim n- co

J-.

g(x)f"(x)dx = -lim n-.co

=-

R is of class

g'(x)f,,'(x)dx -co co

g'(x)D-f(x)dx =

J

g(x)it(dx).

A continuous function g with compact support can be uniformly approximated

by functions of class 0, so that (6.52)

lim n- co

1

g(x)f"(x)dx = 1

-co

g(x)p(dx).

-co

We can now apply the change-of-variable formula (Theorem 3.3) to f(Ws), and obtain, for fixed t e (0, co): `f

1f

in(Wt) - fn(z) = .1on1WOdWs + -2

0

fn" (Ws) ds,

a.s. Pz.

When n -, co, the left-hand side converges almost surely to f(W,) - f(z), and the stochastic integral converges in L2 to fic, a 1(Ws)d Ws because of (6.51) and

3.6. Local Time and a Generalized Ito Rule for Brownian Motion

215

the uniform boundedness of the functions involved. We also have from (6.7) and (6.52):

f,"(x)Li(x)dx = 2

fn"(Ws)ds = 2 lim

lira "OD

L,(x)p(dx),

a.s. P2

n-.co

0

because, for P2-a.e. co GS-2, the continuous function x w) has support on the compact set [min, Ws(co), Ws(co)]. This proves (6.49) for each fixed t, and because of continuity it is also seen to hold simultaneously for all t e [0, co), a.s. PZ. El

6.23 Corollary. If f: R

R is a linear combination of convex functions, then (6.49) holds again for every z e R; now, p defined by (6.47) is in general a signed measure with finite total variation on each bounded subinterval of R.

6.24 Problem. Let a, < a2 < {a, , .

< an be real numbers, and denote D =

P is continuous and f' and f" exist and are continuous on R \D, and the limits , an}

.

Suppose that f:111

f ' (a, ±) s lim f '(x), f "(a, ±) = lim f "(x) x-wk +

x--*ak ±

exist and are finite. Show that f is the difference of two convex functions and, for every z e R, (6.53)

f(W,) = f(z) +

f'(W)dW, + -1 ft f"(W)ds 2

0

+ E Li(a,)[f ' (a, +) -f ' (a, - )]]; 0 < t < co, a.s. P . k =1

6.25 Exercise. Obtain the Tanaka formulas (6.11)-(6.13) as corollaries of the generalized Ito rule (6.49).

E. The Engelbert-Schmidt Zero-One Law Our next application of local time concerns the study of the continuous, nondecreasing additive functional A1(w) = fg f(Wn(co))ds;

0

t < co,

0

where f: P

[0, co) is a given Borel-measurable function. We shall be interested in questions of finiteness and asymptotics, but first we need an auxiliary result.

6.26 Lemma. Let f: P

[0, cc) be Borel-measurable; fix x e P, and suppose there exists a random time T with


216 T

Pq0 < T < co] = 1, P° [ I. f(x + Ws) ds < col>

O.

o

Then, for some e > 0, we have

Lf(x + y)dy < co.

(6.54)

PROOF. From (6.7) and Problem 6.13 (iii), we know there exists an event SP

with infe) = 1, such that for every co en*:

I T" f(x + Ws(w)) ds = 2 o

a°

--.

f(x + y)LT(a,)(y, co) dy

and LT())(0, co) > 0. By assumption, we may choose co e SI* such that fl-"f(x + Ws(w)) ds < cc as well. With this choice of w, we may appeal to the continuity of LT(0)( , w) to choose positive numbers c and c such that LT(.)(y, co)

c whenever IA 5 E. Therefore, T(w)

I.e

f(x + y) dy 5 j. 2c

-E

f(x + Ws(co))ds < co,

0

D

which yields (6.54).

6.27 Proposition (Engelbert-Schmidt (1981) Zero-One Law). Let f: l [0, co) be Borel-measurable. The following three assertions are equivalent:

(i) P°[IL f(His)ds < co; V 0 -_ t < co] > 0, (ii) P° [I:, f(Ws)ds < oo; V0 5 t < oo] = 1, (iii) f is locally integrable; i.e., for every compact set K c R, we have JKf(y)dy < cc.

PROOF. For the implication (i) (iii) we fix be R and consider the first passage time Tb. Because P° [Tb < co] = 1, (i) gives P°[ f o Tb f(Ws)ds < co; V 0 < t < co] > 0. But then ,4- ori,coo

,-4- Tb(eol

f(47(0))ds > JO

f(Ws(w))ds = j. f(b + Bs(w))ds, fTb(w)

0

where Bs(w) A 14/,+71(.)(co) - b; 0 < s < oo is a new Brownian motion under P°. It follows that for each t > 0, P°[ f 10 f(b + Bs) ds < co] > 0, and Lemma 6.26 guarantees the existence of an open neighborhood U(b) of b such that Su(of (Y1dY < oo. If K R is compact, the family {U (0}4. lc, being an open covering of K, has a finite subcovering. It follows that Ix f 61 dy < co. For the implication (iii) (ii) we have again from (6.7), for P°-a.e. w e ft Jo

f(Ws(w))ds = 2 I

-.

5[

f(y)Lt(y, co) dy = 2

I

mt(co)

f(y)Lt(y, w) dy

m,(.) Mt(w)

max

?NO)).

A 1 t(0))

24(y, co)]

-

.1 m,(0))

f(y)dy; 0 5 t < co,

3.7. Local Time for Continuous Semimartingales

217

where ?Ow) = min° ss Ws(co), Mt(co) = max05,,, Ws(w). The last integral is finite by assumption, because the set K = [mi(co), A li(co)] is compact. 6.28 Corollary. For 0 < a < co, we have the following dichotomy: p0 [f E

wdSla

V 0 < t < 00 =

< 00;

1;

if 0 < a < 11

0;

if a _._

f

1

6.29 Problem. The conditions of Proposition 6.27 are also equivalent to the following assertions: (iv) P° [po f (Ws) ds < co] = 1, for some 0 < t < co; (v) Px[fo f(Ws) ds < co; V 0 < t < co] = 1, for every x e IR;

(vi) for every x e R, there exists a Brownian motion IB g,; 0 < t < col and a random time S on a suitable probability space (0, , Q), such that Q [Bo = 0, 0 < S < co] = 1 and

f Q[

s

f(x + 13s)ds
O.

o

(Hint: It suffices to justify the implications (ii) = (iv) the first and last of which are obvious.)

(vi)

(iii)

(v)

(vi),

630 Problem. Suppose that the Borel-measurable function f: R --9 [0, oo) satisfies: measty e 01; f(y) > 01 > 0. Show that (6.55)

Px[u) e SI;

f

.3

f (Ws(w))ds =

xi=

1

o

holds for every x e Fl. Assume further that f has compact support, and consider the sequence of continuous processes 1

X:n)

A.\/n

f(Ws)ds;

0

t < co, n > 1.

o

Establish then, under P°, the convergence (6.56)

X (^) --. X

in the sense of Definition 2.4.4, where X, -4 2 V f ll 1 L,(0) and Vfil, A ff. f(y) dy > 0.

3.7. Local Time for Continuous Semimartingalest The concept of local time and its application to obtain a generalized Ito rule can be extended from the case of Brownian motion in the previous section to that of continuous semimartingales. The significant differences are that f This section may be omitted on first reading; its results will be used only in Section 5.5.


218

time-integrals such as in formula (6.7) now become integrals with respect

to quadratic variation, and that the local time is not necessarily jointly continuous in the time and space variables. We shall use the generalized Ito rule developed in this section as a very important tool in the treatment of existence and uniqueness questions for one-dimensional stochastic differential equations, presented in Section 5.5. Let

X, = X0 + M, + Vt; 0 < t < oc

(7.1)

be a continuous semimartingale, where M = IM Ft; 0 < t < col is in dr', V = { V g;; 0 < t < oo} is the difference of continuous, nondecreasing, adapted processes with Vo = 0 a.s., and {,,} satisfies the usual conditions. The results of this section are contained in the following theorem and are inspired by a more general treatment in Meyer (1976); they say in particular that convex functions of continuous semimartingales are themselves continuous semimartingales, and they provide the requisite decomposition. 7.1 Theorem. Let X be a continuous semimartingale of the form (7.1) on some probability space (Q, .0-i-, , P). There exists then a semimartingale local time for

X, i.e., a nonnegative random field A = lAt(a,w);(t,a)e [0, co) x R, wen} such that the following hold: (i) The mapping (t, a, w) F- At(a,w) is measurable and, for each fixed (t, a), the random variable Ada) is .97,-measurable.

(ii) For every fixed a e R, the mapping ti--11,(a,w) is continuous and nondecreasing with Ao(a, w) = 0, and (7.2)

fom 1 R\ {}(X,(co)) d A,(a, co) = 0,

for P-a.e. wen.

[0, op), the identity

(iii) For every Borel-measurable k: IR t

(7.3)

1 k(Xs(co))d<M> (co) = 2 1

k(a)A,(a, co) da;

0 < t < co

o

holds for P-a.e. we a (iv) For P-a.e. we Q, the limits lim At(b, co) = At(a, co)

and

At(a -, w) '' lim At(b, co)

r-,1 b.i.a

S -./

bt a

exist for all (t, a) e [0, cc) x R. We express this property by saying that A is a.s. jointly continuous in t and RCLL in a. (v) For every convex function f:R R, we have the generalized change of variable formula t

(7.4)

f(X,) =

f(X0) +

1 Df(Xs)dM, + 1 Df(Xs)dV, o

o

CO

+I

At(a)tt(da);

0 < t < oo, a.s. P,

3.7. Local Time for Contiauous Semimartingales

219

where D- f is the left -hand derivative in (6.40) and II is the second derivative measure (6.47).

7.2 Corollary. If f: R -- R is a linear combination of convex functions, (7.4) still holds. Now p defined by (6.47) is a signed measure, finite on each bounded subinterval of R.

7.3 Problem. Let X be a continuous semimartingale with decomposition (7.1) and let f: R --- FR be a function whose derivative is absolutely continuous. Then f" exists Lebesgue-almost everywhere, and we have the ItO formula: t

fiXt) = MO

:

+ f f(Xs)dM, + f r(X0dV, o

1

+2

o

0 s t
b in some neighborhood of a (cf. (4.20)', Remark 4.9). We shall prove the existence of a continuous modification

of H by using the extension of the Kolmogorov-entsov theorem (Problem 2.2.9). According to the latter, it suffices to show (7.14)

Ellk(a) - Hs1(b)1' 5 C[(s2 - s1)2 + (b - a)271+s;

0<s, <s2< co, a,beR for suitable positive constants a, /3, and C. Note that (7.15)

E I 11,2(a) -Hs 1(b)18 < 2' E I 11,2(a) - 11,2(b)11 + 2' E 111, 2(b) -It i(b)18,

and, according to the martingale moment inequality (3.37), we may bound the latter expectation by E111,2(b) - 11,,(b)12

C.E[M(b)>s, - sj212 5 Cs(s2 - s1)a'2.

When a > 4, this bound is of the type required by (7.14). Thus, it remains only to deal with the first expectation on the right-hand side of (7.15), i.e., to show (7.16)

Ellis(a) - 115(b)12 5 C(b - a)2("); 0 < s < co, -co < a 4, /3 > 0, and C is a positive constant. We fix a < b and introduce the convex function f (x) =

f

X

o

4,,,i(z)dz dy, o

for which I f 'I is bounded by b -a and 1)--f' = 10,,bi. In particular, passage to the limit in (7.5) yields

i f'(X0)d1148 + f f'(X0)dV0 +,

f(X,) = f(X0) +

0

,

o

2

' J 0

10, bi(X8)d<M>o. '


222

Assumption (7.11) and Problem 1.5.24 show that X, and all the preceding integrals have limits as t oo, so we may replace t by T(s) (which may be infinite) to obtain, for every k > 1, the bound (7.17) rT(s)

T(s)

k

G 6k I f(XT(s))- f (x 0)1k + 6k j.

k

f'(X9)dMo

Jo rT(s)

k

f'(X,i)dii

+ 6k

Jo

We bound the terms on the right-hand side of (7.17). The mean-value theorem implies I f (x T(o)) -f (x 0)1k < (b - a)k I XT(S) -X Olk

2k Kk (b

- a)k,

and it is also clear that T(s)

k

f'(X9)d1/9 < Kk(b - a)k.

io

Applying the martingale moment inequality (3.37) to the stochastic integral in (7.17), we obtain the bound To)

E

J.0

k

k

f ' (X v)ditiv < lim E fo(s)At f'(XJdMt, G Ck E

[1

oo

I f ' (x v)I2 d<M>v o

< CkKki2(b - a)k. We conclude from these considerations that there exists a constant C, depending

only on k and on the bound K in (7.11), such that T(s)

k

1(,,,)(Xt,)d<M>t,

E

fo

C(b - a)k;

0 < s < co,

-co < a 2. From (7.12), (7.13) we see that It(a)= Hoo,(a), and since <M> is continuous, the existence of a continuous modification for H implies the same for I. The Lebesgue-Stieltjes integral (7.18)

J,(a) --4 1[ 1(,00)(Xs)dVs o

appearing in (7.7) can fail to be jointly continuous in (t, a); see Exercise 7.9. However, it is jointly continuous in t and RCLL in a; the proof is left as a problem for the reader.

3.7. Local Time for Continuous Semimartingales

223

7.6 Problem. Let X be a continuous semimartingale with decomposition (7.1). For P-almost every co e 1, we have for all (t, a) e [0, co) x (7.19)

lim .1,(b, co) = b

=

lim

(7.20)

tt

f

lia,c0(X,(0)dVs(co).

0

bfa

PROOF OF THEOREM 7.1. Using Lemma 7.5 and Problem 7.6, we choose a modification of Cf+ (a) in (7.7) which is jointly continuous in t and RCLL in a. We take Ai(a) and C, ±(a) to satisfy (7.10) for every t, a, and co. In particular, Ai(a, co) is jointly a( [0, co)) 0 MR) F-measurable and satisfies (iv), and the other measurability claims of (i) and (ii) hold. In particular, it follows from (7.19), (7.20) that

A,(a) - A,(a -) = fo 1{}(Xs)dVs.

For the proof of (7.2), consider any two rational numbers 0 < u < v < co and the event Ha A {co ED; Xs(co) < a,

V se [u, v] }.

From relation (7.7) we have on H: Av(a, co) = A (a, co), except for co in a null set N,,,,. Let N M Na and fix co e 12\N. The set S(a, co) A ,_, 0 u,v e Q {0 < t < co; X,(co) < a} is open and, as such, is the countable union of disjoint open intervals. Let (a, /3) be such an interval. If a < u < v < /1, where u and v are rational, then A(a, co) = A (a, co). It follows that 1,.13) d A,(a, co) = 0, and thus IW 1,_,,o(X,(co))d A,(a, co) = 0.

(7.21)

0

A similar argument based on (7.8) shows that (7.22)

f.

1,,)(X,(co))dA,(a, co) = 0,

a.s.

which establishes (7.2).

For the proof of (7.4), we may assume, by the usual localization argument, that there exists a constant K > 0 such that (7.11) holds. Consequently, we

may also assume without loss of generality that D-f is constant outside (- K, K), so the second derivative measure p has support on [ - K, K). Let x e [ - K, K] be fixed and introduce the function

a < -K - 1, (x + K)(a + K + 1); -K - 1 < a < -K, 0;

g(a) =

x - a;

-K < a < x,

0;

x < a.


224

According to (6.48), (6.46), we have for -K < x < K, (x - a) +µ(da) =

(7.23)

J

-f

g'(a)D- f (a) da -K

= -(x + K)

= -(x

D- f (a) da + -K-1

J

X

D- (a) da

-K

K) D-f(- K) + f(x) - f(- K),

and from the definition of it: (7.24)

K, x)) = D- f(x) - D-f(- K).

1,,,,,,o(x)y(da) =

cc'

We may now integrate with respect to it in the Tanaka-Meyer formula (7.7) and use (7.23) to obtain (7.25)

f(X,) = (X, - X0) D- f( - K) + f(X0) +

J

J t

-co co

+

i(,00(Xs)dVsit(da) + J _co

f

10,,,,)(X0d114,p(da)

0

co

Ai(a)it(da).

-co

0

Fubini's theorem and (7.24) allow us to write (7.26)

fc° f t -a)

10,,00)(Xs)dV, Oda) = f i 1)- f (X ,) dils -V, 1)- f ( - 10.

0

0

A similar interchange of the order of integration in the integral ('t

1(..)(X,) dM, y(da) =

(7.27) 00

D- f(Xs) d114, -M, D- f(- K)

0

is justified by Problem 7.7 following this proof. Substitution of (7.26), (7.27) into (7.25) results in (7.4). Finally, let us consider (7.4) in the special case f(x) = x2. Then µ(da) = 2 da, and comparison of (7.4) with the result from the usual Ito rule reveals that 00

<M>, = 2

J

Ai(a)da; 0 < t < co, a.s.

Thus, for any measurable function h(s, co): [0, co) x CO

OD

OD

h(s, co) d <M>,(co) = 2 o

holds for P-a.e. wee. Now if k:

[0, co,

h(s, co) d As(a, co) da -co

o

-* [0, co) is measurable, we may take

h(s,(o) = 110,11(s)- k(X Jo))) and obtain for P-a.e. to:

3.7. Local Time for Cc ntinuous Semimartingales

k(X (w))d<M>s(co)= 2

J o

s

_.

=2J

225

k(Xs(w))dAs(a,w)da J.` k(a)Ai(a, co) da;

0 5 t< co,

thanks to (7.2). This completes the proof. 7.7 Problem. Let X be a continuous semimartingale with decomposition (7.1), it be a 6- finite measure on (DI, .4(01)), and h: R [0, oo) be a continuous function with compact support. Then

.1°3 h(a)(1 10,,,)(Xs)dMs)p(da)=

(.1

h(a)10,.)(Xs)p(da))01,.

7.8 Remark. The proof of Theorem 7.1 shows that the semimartingale local time Azi" (a) for a continuous local martingale M is jointly continuous in (t, a), because the possibly discontinuous term .1,(a) of (7.18) is not present. In particular, (7.9) becomes then a.s. P: (7.28)

IM, - al =1M0 -

+

sgn(Ms - a)dMs + 2Ana); 0 < t < co.

Comparison of (7.28) with the Tanaka formula (6.13) shows that the semimartingale local time A"(a) for Brownian motion W coincides with the local time L(a) of the previous section. If M e,C, then for any stopping time t, (7.29)

BIM, Arl = 2E[e, r(0)]; 0 < t < co.

7.9 Exercise. Show by example that Jt(a) defined by (7.18) can fail to be continuous in a.

7.10 Exercise. Let X be a continuous semimartingale with decomposition (7.1). Show that for every a e R,

1{.}(Xs)d<M>s= 0, a.s. P. 0

7.11 Exercise. Show that the semimartingale local time of a continuous process of bounded variation is identically zero.

7.12 Exercise (LeGall (1983)). Let X be a continuous semimartingale with decomposition (7.1), and suppose that there exists a Borel-measurable function k: (0, co) (0, co) such that .1(0,0 (du /k(u)) = po,V s > 0, but for every t e (0, co) we have

226


it d<M> jo k(X5) i{"} < c°'

a.s. P.

Then the local time A(0) of X at the origin is identically zero, almost surely.

7.13 Exercise. Consider a continuous local martingale M and denote St 4 maxo <s t,

S, - Mt, St) are local martingales. Deduce also the following:

(i) The process (S, - M,)2 - <M>, is a local martingale. (ii) For every real-valued function g of class C1(R), the processes

g(L) - 141 04)

and

g(S,) - (S, - M,) g'(S,)

are local martingales.

7.14 Exercise. For a nonnegative, continuous semimartingale X of the form (7.1) with X0 -..= 0, the following conditions are equivalent: (i) 1/ is flat off {t > 0; X, = 0}. (ii) The process P01{,s00} dXs; 0 < t < co belongs to .ilic I". (iii) There exists N edr.1" such that X, = maxo 0, we can find m > 1 so that with X' A g(m) we have E Jo IX: - x(12 dt < O. The Minkowski inequality leads then to


(E fr

227

- Xt_hl2 dE)"2

0 1/2

(E

IX, - XN2 dE)

+ E

IX _h -

dt)

(E

f

- X:_hI2 dt

13

< 2E +

(E

- X:_h12 CIEY I)

We can now let h j 0 and conclude, from the continuity of X' and the bounded convergence theorem, that T

liM E /40

j. IX, -

dt < 4E2.

0

(c) Let i be any nonnegative integer. As s ranges over [1/2", (1 + 1)/2"), tp(t - s) + s ranges over [t - (1/2"), t). Therefore, T

T

IX, -

- X,I2 ds dt = 2"E

E 0

JO

0

= 2"

dhdt

0

s2- [E f T IX, - Xr_hI2 dtidh 0

13

T

max E

0112-.

IX, - X,_12 dt, 0

which converges to zero as n -0 co because of (b). (d) From (c) there exists a sequence {n, }ck°=, of integers, increasing to infinity

as k

co, such that for meas x meas x P-a.e. triple (s,t,co) in [0,1] x

[0, T] x SI, where meas means "Lebesgue measure," we have

lim IXPk')(co) - X,0)I2 = 0.

(8.1)

k-.co

Therefore, we can select se [0,1] such that for meas x P-a.e. pair (t, co) in [0, T] x fl, we have (8.1). Setting X('`) 9 X("ks), we obtain (2.8) from the bounded convergence theorem. 2.18. By assumption, we have

E = E

d<M>, < co.

Uniform integrability and the existence of a last element for IM(X) follow from Problem 1.5.24, as does uniform integrability of (1m(X))2; similarly for /N( Y). Applying Proposition 2.14 with X., Y. replaced by X.1{.). 71, Y.1{, T}, respectively, we obtain X. Ysd <M, N>,

(f

T+I

d<M>.

f

T+t

1/2

Y.2

ds)

,


228 whence

XsYsd<M,N>s
s XS

T

f

Ys2

ds)112

a.s. P. As T -0 co, the right-hand side of this inequality converges to zero; therefore, Y)>, =

s

converges as t -0 co and is bounded by the integrable random variable

(f: X d<M>.

0

a.s. P. The dominated convergence theorem gives then

lim ER' (X)INY)] = lim E, = Es, fro.

=E

f

X sYsd <M, N>s.

We also have

E[e,f(X)C,(Y)] = E[(1:,4, (X) - r (X))(IVY) -Ii (Y))]

+ E[IP4(X)(IVY) -Ii (Y))] + E[I7(Y)(e,o4(X) - ir(x))]

+ E[Ir(X)17(Y)]. We have just shown that the fourth term on the right-hand side converges to (X), IN (Y)>. as t -0 co. The other three terms converge to zero because of Holder's inequality and the uniform integrability of Om( X))2 and ON Y))2. E

2.27. With X E 9"(M), we construct the sequence of bounded stopping times {Tn}:=1 in (2.32). In the notation of (2.33), each )0") is in 2*(M(")) and therefore can be approximated by a sequence of simple processes PC"'''));,°_, g z in the sense T

lim E k-co

j.

- )0")I2 d<M>, = 0, V T < co

o

(Proposition 2.8). Let us fix a positive number T < co and consider n > T; we have P 01, I )0") - X,I2 d<M>, > 0] = > 0] n1 P[T, < T] = P[S < T or 11-,X,2 d<M>, > N], and the last quantity converges to zero as n -0 oo, by assumption. Now, given any E > 0, we have

PH. T In") - X,I2 d<M>, > Ei

P[f

T

In") - n)I2 d<M>, > -8] 2

o P[fT

IX} ")- X,I2d <M >,

+

>0

< -2 E j. T I Xl"'k) - n')I2 d <M > t + P[T < T] E

by the ebygev inequality.

0


n

229

For any given ö > 0 we can select 115> T so that P[T,, < T] < 312 for every n0, and we can find an integer kr,, 1 so that E J.T XP4.k..4) -)0 )12 d I

ei5

, < 4

o

It follows that for e = S = j; j > 1, there exist integers n, and k" such that, with Yul A X('0"), we have

- X,I2 d <Mx >

P[f

j

Then according o Proposition 2.26, both sequences of random variables T

YtLI) - X,12

sup 11,(YllI) - /,(X)I

d<M>,, 0

s.

It follows from (3.9) that 1

vo.

= .10 vodxv + -E

2 J.1

32

t

aXiax;

f(Xs) d <Ma), Mo)>s,

and now (3.10) reduces to the HO rule applied to f(X i).

3.15. Let X and Y have the decomposition (3.7). The sum in question is m-1

-

E

1=o

The first term is ro

1 'n-1 X11)

dMs +

+

E (}

2 1=0

- i,)(x+, - X ).

dBs, where

rn -1 1 =0

and the continuity of Y implies the convergence in probability of Po (Ys' Y)2 d<M> s and Po [Ysn - Ys] dBs to zero. It follows from Proposition 2.26 that rn-1

E

i=0

The other term is

- x)

rsdxs. o


E

L i=c1

(M1.

- MOlArt,,

231 m-i

-

+-E

"

2 i=o

m-i

-

1 m-i

+ -2 Xo (N,, -

- Be.) +

(B4.

'

)(C.,

"

- C,)

-

-

which converges in probability to l<M, N>, because of Problem 1.5.14 and the bounded variation of B and C on [0, t].

3.29. For x; > 0, i = 1,

xr +

d, we have

d(x +

+

dm+1(xr +

+ xd)m

+ x,r).

Therefore 104,112m

(8.2)

[E (VT]

dm E 00)12,. i=i

and

E 014(i)>7 < d(E

(8.3)

i=1

T)

= dAT.

Taking maxima in (8.2), expectations in the resulting inequality and in (8.3), and applying the right-hand side of (3.37) to each M('), we obtain EOM 114;12 m < dm E Eumolti2m

dm E KmEum(oyn

i=i

Kmdm+1E(A7)

1=1

A similar proof can be given for the lower bound on EU M 4.5.

(i) The nondecreasing character of T is obvious. Thus, for right-continuity, we need only show that lim045 T(0) < T(s), for 0 s < S. Set t = T(s). The definition of T(s) implies that for each E > 0, we have A(t + c) > s, and for s < 0 < A(t + E), we have T(0) < t + E. Therefore, lim,4s T(9) < t. (ii) The identity is trivial for s > S; if s < S, set t = T(s) and choose E > 0. We

have A(t + e) > s, and letting e l 0, we see from the continuity of A that A(T(s)) > s. If t = T(s) = 0, we are done. If t > 0, then for 0 < e < t, the definition of T(s) implies A(t - e) < s. Letting E l 0, we obtain A(T(s)) < s. (iii) This follows immediately from the definition of T(). (iv) By (iii), T(A(t)) = t if and only if A(T) = A(t), in which case 49(t) = q9(t). Note

that if S < cc and A(t) = A(cc) for some t < co, then T(A(t)) = co and (i9 is constant and equal to 41(t) on [t, co); hence, (p(co) = lira, (Mu) exists and equals (p(t).

(v) This is a direct consequence of the definition of T and the continuity of A. (vi) For a 5 t1 < t2 5 b, let G(t) = 11,,12,(t). According to (v), t1 5 T(s) < t2 if and only if A(t,) s < A(t2), so A(b)

G(t)dA(t) = A(t2) - A(t1) =

G(T(s))ds. A(a)

The linearity of the integral and the monotone convergence theorem imply that the collection of sets C e .41([a,b]) for which (8.4)

1,(t) dA(t)

f

A(b)

A(o)

1,(T(s))ds

232

3. Stochastic Integration forms a Dynkin system. Since it contains all intervals of the form [t1, t2) c [a, b], and these are closed under finite intersection and generate 4([a, b]), we have (8.4) for every C e M([a, b]) (Dynkin System Theorem 2.1.3). The proof of (vi) is now straightforward.

4.7. Again as before, every <M>, (resp., T(s)) is a stopping time of {s} (resp., {F,}), and the same is true of S <M>, (Lemma 1.2.11). The local martingale /II has quadratic variation CAI>, < <M>,(s2)= S n s2 < s2 < co (Problem 4.5 (ii)),

so again both a, a2 - are uniformly integrable martingales, and by optional sampling: E [AlT(s2)

MT(Si)1 T(si )] = 0,

EP-4($2) - ICIT(.0)21,77(.)] = EE<M>T(s2) - <M>T001,71501; a.s. P.

It follows that MoT A {MTN, Ws; 0 < s < co} is a martingale with <M 0 T >s =

<M>T(s), and by Problem 4.5 (iv), M 0 T has continuous paths. Now if { Ws, Ws; 0 < s < co} is an independent Brownian motion, the process

BsAWs-W s A S +M- T(s),4- s, 0_sT(s) = s, i.e., a Brownian motion. For this process, (4.17) is established by using Problem 4.5 (iv). 4.11. Let (p be a deterministic, strictly increasing function mapping [0, co) onto [0,1), and define M E ir 1°' by M, --4

f

ow Xs dWs,

so <M >1 =

fow XS ds;

o

0 < t < co,

o

and lim, <M>, = co, a.s. on E. According to Problem 4.7, there is a Brownian motion B such that Jt

X s dWs = Bp(,),

where p(t) A <M>,-i(i).

We have limgti p(t) = oo a.s. on E, and so, by the law of the iterated logarithm for Brownian motion,

lm B,1) = -lim Bp(1) = +co, a.s. on E.

it

it 1

4.12. (Adapted from Watanabe (1984).) From Problem 4.7 we have the representation (4.17) for a suitable Brownian motion B. Taking n > 3 max(' xi, pT) and denoting R = inf ft _>.: 0;141 n/31, we have the inclusions

{max I X,' (:)1,5T

n}

g { max 141 051.sT

.1= {<M>T. 3

R,,}

{PT

Rn},

which lead, via (2.6.1), (2.6.2), and (2.9.20), to m ax P [CIT

I Xr1

n]

P[R 5 pT]

2R J

The conclusion follows.

2P[T13 '- PT] = 4P BpT dz
1

of this interval. Choosing a modification of Po 1()( Ws) dWs which is continuous in a (cf. (6.21)), we see that the Lebesgue (and Riemann) integral on the left-hand side of (6.24) is approximated by the sum

2"-I b

E -h(b17))( f i(bno")dw) =

k=0 2"

F(Ws)dWs,

where the uniformly bounded sequence of functions

2"-I b

E -hoe)lor,,,,(x);

F(x)

k=0 2n

n

1

converges uniformly, as n -* co, to the Lebesgue (and Riemann) integral F(x)

Therefore, the sequence of stochastic integrals {it F(Ws)dWs}`°_i converges in L2 to the stochastic integral Po F(Ws)dWs, which is the right-hand side of (6.24). 6.13.

(i) Under any P, B(a) is a continuous, square-integrable martingale with quadratic variation process , =

f [sgn(Ws - a)]2 ds = t;

0 5 t < co, a.s. P .

o

According to Theorem 3.16, B(a) is a Brownian motion. (ii) For w in the set Q* of Definition 6.3, we have (6.2) (Remark 6.5), and from this we see immediately that L o(a, w) = 0 and L,(a, a)) is nondecreasing in t. For each z E R, there is a set n e.57" with p(n) = 1 such that 2°,(a) is closed for all CO E n. For wenn Ce, the complement of .2°,0(a) is the countable

union of open intervals U. NI I. To prove (6.26), it suffices to show that Si. dL,(co) = 0 for each a e NI. Fix an index a and let 1 = (u, v). Since W(w) -a

has no zero in (u, v), we know that I W(a)) - al is bounded away from the origin on [u + (1/n), v - (1/n)], where n > 2/(v - u). Thus, for all sufficiently small e > 0, 1

meas { 0< s < u + -; I Ws - al 5 e} = meas {0 < s

v

--1; I Ws - aI 5 E},

co). It follows that jr,,,(I,,,),-(1/0] dL,(a,w) = whence 4,(1/)(a, co) = 0, and letting n oo we obtain the desired result. = -B,(0) + 24(0); 0 .5 t < oo, a.s. P°. (iii) Set z = a = 0 in (6.25) to obtain I The left-hand side of this relation is nonnegative; B,(0) changes sign infinitely often in any interval [0, E], e > 0 (Problem 2.7.18). It follows that L,(0) cannot remain zero in any such interval.

(iv) It suffices to show that for any two rational numbers 0 5 q < r < co, if < W,(co) = a for some t e (q, r) then Lq(a, w), P-a.e. w. Let T(w) inf.{ t q; W(w) = a}. Applying (iii) to the Brownian motion {Ws ., - a; 0 5 s < co} we conclude that


234 Li-00(a, a)) < LT(0,),(a, w)

for all s > 0, Pz-a.e. w,

by the additive functional property of local time (Definition 6.1 and Remark 6.5). For every we T < r} we may take s = r - T(w) above, and this yields Lq(a, w) < L,(a, a)).

6.19. From (6.38) we obtain lim),4f (y) 5 f(x), limyt f(y) 5 f(z) and f(y) 5 lim,ty f(x), f(y) 5_ lin- ir.ty f(z). This establishes the continuity of f on R. For E l fixed and 0 < h, < h2, we have from (6.38), with x = y = u + h1,

z= + h2:

of(; h1) A.N;h2) On the other hand, applying (6.38) with x = - h2, y = - hi, and z = yields

(8.5)

AN; -h2) 5 4f(; -hi).

(8.6)

Finally, with x = - e, y =

z = + 6, we have

Af(; -E)

(8.7)

AN; 6); E, 6 > a

Relations (8.5)-(8.7) establish the requisite monotonicity in h of the difference quotient (6.39), and hence the existence and finiteness of the limits in (6.40). In particular, (8.7) gives D- f(x) 5 Liff (x) upon letting E 10, 610, which establishes the second inequality in (6.41). On the other hand, we obtain easily from (8.5) and (8.6) the bounds (8.8)

(y - x) to+f(x) 5 f(y) - f(x) 5 (y - x) Df(y); x < y,

which establish (6.41).

For the right-continuity of the function Dtf(), we begin by observing the inequality D+ f (x) 5 limy 4,, D+f(y); x e R, which is a consequence of (6.41). In the

opposite direction, we employ the continuity of f, as well as (8.8), to obtain for x < z:

f(z) - f(x) = lim f (z) -f (y)

z -X

y ,i,

z -y

lim Go+ f (y). y 4,.

Upon letting z 1 x, we obtain D+ f(x)_ limy 4 x D+f(y). The left-continuity of IT f(-) is proved similarly. From (8.8) we observe that, for any function (p: R - l satisfying D- f(x) < (p(x) S D+f(x);

(8.9)

x E ER,

we have for fixed y E R, (8.10)

f(x)

Gy(x)

f(y) + (x - y)(p(y);

x e R.

The function Gy) is called a line of support for the convex function fe). It is immediate from (8.10) that f(x) = supyERGy(x); the point of (6.42) is that f() can be expressed as the supremum of countably many lines of support. Indeed, let E be a countable, dense subset of R. For any XE R, take a sequence { y }'.1 of numbers in E, converging to x. Because this sequence is bounded, so are the sequences {D±f(yO} 1 (by monotonicity and finiteness of the functions D±)) and {q)(y)},T=1 (by (8.9)). Therefore, lim, Gyn(x) = f(x), which implies that f(x) = supy e EGy(x).


235

6.20. (iii) For any x < y < z, we have (8.11)

(1)(y) - (1)(x)

1

y-x

y-x

1

z

y

f

co(u)du

9(y)

co(u)du -(1)(z) - 4)(Y) y

9(z)

x

This gives (D(Y)

z -y y -x 4)(x) + z -x z -x 4:0(z),

which verifies convexity in the form (6.38). Now let x j y, z 1 y in (8.11), to obtain 9-(y) < D (D(Y) < 9(.1) < D+O(Y) 5- 9+(y);

Ye

At every continuity point x of 9, we have 9±(x) = 9(x) = D± (1)(x). The left- (respectively, right-) continuity of 9_ and D- co (respectively, and WO) implies tp__(y) = D -1(y) (respectively, 9.,(y) = D+4)(y)) for all ye R. (iv) Letting x 1 y (respectively, x T y) in (6.44), we obtain Ir.f(Y) < (PAY) 5 9(Y) 5 9+(Y) 5- D+.1(Y);

Y e R.

But now from (6.41) one gets 50+(x)

D+ f(x) 5- D.1(Y) -5 9-01 5 OA x
0, and that {B, 4=' Ws+t, - x, 0 < s < co} is a Brownian motion under P°. Now, for every we {2T < t}: 2T (m)

f(x + 13,(a)))ds o

LA.)

f(W(a)))du

f f(Wx(co))du < oo,

t} g {fp f(x + BOds < co}, a.s. P°. We conclude that this latter event has positive probability under P°, and (vi) whence {2 Tx

follows upon taking S = Tx. Lemma 6.26 gives, for each x e K, the existence of an open neighborhood U(x) of x with fu(x)f(y)dy < co. Now (iii) follows from the compactness of K. For fixed x e R, define gx(y) = f(x + y) and apply the known implication (iii) (ii) to the function gx. 7.3. We may write f as the difference of the convex functions


236

f,(x)

f(0) + xf'(0) +

[f"(z)]+ dz dy, o

f2(x) -4

o

f1 x

)

o

0

[f"(z)]- dz dy,

and apply (7.4). In this case, Adx) = f "(x) dx, and (7.3) shows that I% A,(a)m(da) =

ifof "(Xs)d <M>s.

7.6. Let 1%,(w) denote the total variation of V(w) on [0, t]. For P-a.e. we a we have I7,(w) < cc; 0 < t < co. Consequently, for a < b, 131(a) - .1(b)1

-

+ IJ,(a) - Jr(b)I < I V - VzI +

and these last expressions converge to zero a.s. as r

1(am(XS)dVs, 0

t and b La. Furthermore,

the exceptional set of CO E S2 for which convergence fails does not depend on t or a. Relation (7.20) is proved similarly.

7.7. The solution is a slight modification of Solution 6.12, where now we use Lemma 7.5 to establish the continuity in a of the integrand on the left-hand side.

3.9. Notes Section 3.2: The concept of the stochastic integral with respect to Brownian motion was introduced by Paley, Wiener & Zygmund (1933) for nonrandom integrands, and by K. Ito (1942a, 1944) in the generality of the present section. ItO's motivation was to achieve a rigorous treatment of the stochastic differential equation which governs the diffusion processes of A. N. Kolmogorov (1931). Doob (1953) was the first to study the stochastic integral as a martingale, and to suggest a unified treatment of stochastic integration as a chapter of martingale theory. This task was accomplished by Courrege (1962/1963), Fisk (1963), Kunita & Watanabe (1967), Meyer (1967), Millar (1968), DoleansDade & Meyer (1970). Much of this theory has become standard and has

received monograph treatment; we mention in this respect the books by McKean (1969), Gihman & Skorohod (1972), Arnold (1973), Friedman (1975), Liptser & Shiryaev (1977), Stroock & Varadhan (1979), and Ikeda & Watanabe (1981) and the monographs by Skorohod (1965) and Chung & Williams (1983).

Our presentation draws on most of these sources, but is closest in spirit to Ikeda & Watanabe (1981) and Liptser & Shiryaev (1977). The approach suggested by Lemma 2.4 and Problem 2.5 is due to Doob (1953). A major recent development has been the extension of this theory by the "French school" to include integration of left-continuous, or more generally, "predictable," processes with respect to discontinuous martingales. The fundamental reference for this material is Meyer (1976), supplemented by Dellacherie & Meyer (1975/1980); other accounts can be found in Metivier & Pellaumail (1980), Metivier (1982), Kopp (1984), Kussmaul (1977), and Elliott (1982). Section 3.3: Theorem 3.16 was discovered by P. Levy (1948: p. 78); a different

3.9. Notes

237

proof appears on p. 384 of Doob (1953). Theorem 3.28 extends the Burkholder-

Davis-Gundy inequalities of discrete-parameter martingale theory; see the excellent expository article by Burkholder (1973). The approach that we follow

was suggested by M. Yor (personal communication). For more information on the approximations of stochastic integrals as in Problem 3.15, see Yor (1977).

Section 3.4: The idea of extending the probability space in order to accommodate the Brownian motion W in the representation of Theorem 4.2 is due to Doob (1953; pp. 449-451) for the case d = 1. Problem 4.11 is essentially from McKean (1969; p. 31). Chapters II of Ikeda & Watanabe (1981) and XII of Elliott (1982) are good sources for further reading on the subject matter of Sections 3.3 and 3.4. For a different proof and further extensions of the F. B.

Knight theorem, see Cocozza & Yor (1980) and Pitman & Yor (1986) (Theorems B.2, B.4), respectively. Section 3.5: The celebrated Theorem 5.1 was proved by Cameron & Martin

(1944) for nonrandom integrands X, and by Girsanov (1960) in the present generality. Our treatment was inspired by the lecture notes of S. Orey (1974). Girsanov's work was presaged by that of Maruyama (1954), (1955). Kazamaki (1977) (see also Kazamaki & Sekiguchi (1979)) provides a condition different

from the Novikov condition (5.18): if exp(14) is a submartingale, then 4 = exp(M, - l<M>1) is a martingale. The same is true if E[exp(1111,)] < co (Kazamaki (1978)). Proposition 5.4 is due to Van Schuppen & Wong (1974). Section 3.6: Brownian local time is the creation of P. Levy (1948), although

the first rigorous proof of its existence was given by Trotter (1958). Our approach to Theorem 6.11 follows that of Ikeda & Watanabe (1981) and McKean (1969). One can study the local time of a nonrandom function divorced from probability theory, and the general pattern that develops is that regular local times correspond to irregular functions; for instance, for the highly irregular Brownian paths we obtained Holder-continuous local times (relation (6.22)). See Geman & Horowitz (1980) for more information on this topic. On the other hand, Yor (1986) shows directly that the occupation time B

r; (B, co) of (6.6) has a density.

The Skorohod problem of Lemma 6.14, for RCLL trajectories y, was treated by Chaleyat-Maurel, El Karoui & Marchal (1980). The generalized Ito rule (Theorem 6.22) is due to Meyer (1976) and Wang (1977). There is a converse to Corollary 6.23: if f( W) is a continuous semimartingale, then f is the difference of convex functions (Wang (1977), cinlar, Jacod, Protter & Sharpe (1980)). A multidimensional version of Theorem 6.22, in which convex functions are replaced by potentials, has been proved by Brosamler (1970). Tanaka's formula (6.11) provides a representation of the form f(147,) f(Wo) + f o g(W)dWs for the continuous additive functional L,(a), with a e fixed. In fact, any continuous additive functional has such a representation, where f may be chosen to be continuous; see Ventsel (1962), Tanaka (1963).

238


We follow Ikeda & Watanabe (1981) in our exposition of Theorem 6.17. For more information on the subject matter of Problem 6.30, the reader is referred to Papanicolaou, Stroock & Varadhan (1977). Section 3.7: Local time for semimartingales is discussed in the volume edited by Azema & Yor (1978); see in particular the articles by Azema & Yor (pp. 3-16) and Yor (pp. 23-36). Local time for Markov processes is treated by Blumenthal & Getoor (1968). Yor (1978) proved that local time AM for

a continuous semimartingale is jointly continuous in t and RCLL in a. His proof assumes the existence of local time, whereas ours is a step in the proof of existence. Exercise 7.13 comes from Azema & Yor (1979); see also Jeulin & Yor (1980)

for applications of these martingales in the study of distributions of random variables associated with local time. Exercise 7.14 is taken from Yor (1979).

CHAPTER 4

Brownian Motion and Partial Differential Equations

4.1. Introduction There is a rich interplay between probability theory and analysis, the study of which goes back at least to Kolmogorov (1931). It is not possible in a few sections to develop this subject systematically; we instead confine our attention to a few illustrative cases of this interplay. Recent monographs on this subject are those of Doob (1984) and Durrett (1984). The solutions to many problems of elliptic and parabolic partial differential equations can be represented as expectations of stochastic functionals. Such representations allow one to infer properties of these solutions and, conversely, to determine the distributions of various functionals of stochastic processes by solving related partial differential equation problems. In the next section, we treat the Dirichlet problem of finding a function which is harmonic in a given region and assumes specified boundary values. One can use Brownian motion to characterize those Dirichlet problems for which a solution exists, to construct a solution, and to prove uniqueness. We shall also derive Poisson integral formulas and see how they are related to exit distributions for Brownian motion. The Laplacian appearing in the Dirichlet problem is the simplest elliptic operator; the simplest parabolic operator is that appearing in the heat equation. Section 3 is devoted to a study of the connections between Brownian motion and the one-dimensional heat equation, and, again, we give probabilistic proofs of existence and uniqueness theorems and probabilistic interpretations of solutions. Exploiting the connections in the opposite direction, we show how solutions to the heat equation enable us to compute boundary crossing probabilities for Brownian motion. Section 4 takes up the study of more complicated elliptic and parabolic

4. Brownian Motion and Partial Differential Equations

240

equations based on the Laplacian. Here we develop formulas necessary for the treatment of Brownian functionals which are more complex than those appearing in Section 2.8. The connections established in this chapter between Brownian motion and elliptic and parabolic differential equations based on the Laplacian are a foreshadowing of a more general relationship between diffusion processes and second-order elliptic and parabolic differential equations. A good deal of the more general theory appears in Section 5.7, but it is never so elegant and surprisingly powerful as in the simple cases of the Laplace and heat equations developed here. In particular, in the more general setting, one must rely on existence theorems from the theory of partial differential equations, whereas in this chapter we can give probabilistic proofs of the existence of solutions to the relevant partial differential equations.

4.2. Harmonic Functions and the Dirichlet Problem The connection between Brownian motion and harmonic functions is profound, yet simply explained. For this reason, we take this connection as our first illustration of the interplay between probability theory and analysis. Recall that a function u mapping an open subset D of Rd into R is called harmonic in D if u is of class C2 and Au (52 u/a4) = 0 in D. As we shall prove shortly, a harmonic function is necessarily of class C' and has the mean-value property. It is this mean-value property which introduces Brownian motion in a natural way into the study of harmonic functions. Throughout this section, {147 A; 0 < t < co}, (S2, {F"}ER, is a ddimensional Brownian family and {A} satisfies the usual conditions. We denote by D an open set in Rd and introduce the stopping time (Problem 1.2.7) (2.1)

TD

inf It

0; W, e De 1,

the time of first exit from D. The boundary of D will be denoted by ap, and D = D u ap is the closure of D. Recall (Theorem 2.9.23) that each component of W is almost surely unbounded, so (2.2)

Px[TD < co] = 1; V x eD c Rd, D bounded.

Let Br A {x E Rd; Ilxll < r} be the open ball of radius r centered at the origin. The volume of this ball is 2r d d/2

(2.3)

d F (d)' 2

and its surface area is (2.4)

2rd-1 nal2 Sr

F(d/2)

- dr

4.2. Harmonic Functions and the Dirichlet Problem

241

We define a probability measure t, on 8B, by

r > 0.

po(dx) = P°[W,Bre dx];

(2.5)

A. The Mean-Value Property Because of the rotational invariance of Brownian motion (Problem 3.3.18), the measure u, is also rotationally invariant and thus proportional to surface measure on BB,. In particular, the Lebesgue integral of a function f over B, can be written in iterated form as

f(x)dx =

(2.6)

Sp

f(x))ip(dx)dp.

.1 aeo

eo

2.1 Definition. We say that the function u: D -4 IR has the mean-value property

if, for every a e D and 0 < r < oo such that a + B, c D, we have

u(a) = J

u(a + x)Eir(dx). aBr

With the help of (2.6) one can derive the consequence

u(a) = -1 Vr

u(a

x)dx

Br

of the mean-value property, which asserts that the mean integral value of u over a ball is equal to the value at the center. Using the divergence theorem one can prove analytically (cf. Gilbarg & Trudinger (1977), p. 14) that a harmonic function possesses the mean-value property. A very simple probabilistic proof can be based on It6's rule. 2.2 Proposition. If u is harmonic in D, then it has the mean-value property there.

PROOF. With a e D and 0 < r < oo such that a + B, c D, we have from Ito's rule d

U(Wt

E i=1

) = U(WO)

0

uxi

(Ws)dKi)

A to*,

1

+-

2

to ra,Br

f

Au(W.$)ds;

0 < t < oo.

0

Because u is harmonic, the last (Lebesgue) integral vanishes, and since (8u /ax;);

1 < i < d, are bounded functions on a + B the expectations under P° of the stochastic integrals are all equal to zero. After taking these expectations on both sides and letting t oo, we use (2.2) to obtain u(a) = E° u(W,..r ) =

f ()Br

u(a + x)yr(dx).

242


2.3 Corollary (Maximum Principle). Suppose that u is harmonic in the open, connected domain D. If u achieves its supremum over D at some point in D, then u is identically constant.

PROOF. Let M = supxe, u(x), and let DM = {x e D; u(x) = M }. We assume that DM is nonempty and show that DM = D. Since u is continuous, DM is closed relative to D. But for a e Dm and 0 < r < co such that a + Br D, we have

the mean value property:

M = u(a) =

1

V,

u(a + x)dx,

J Br

which shows that u = M on a + Br. Therefore, DM is open. Because D is connected, either DM or D \DM must be empty.

2.4 Exercise. Suppose D is bounded and connected, u is defined and continuous on D, and u is harmonic in D. Then u attains its maximum over D on D. If v is another function, harmonic in D and continuous on D, and v = u on (3D, then v = u on D as well.

For the sake of completeness, we state and prove the converse of Proposition 2.2. Our proof, which uses no probability, is taken from Dynkin & Yushkevich (1969).

2.5 Proposition. If u maps D into IR and has the mean value property, then u is of class C° and harmonic.

PROOF. We first prove that u is of class C. For e > 0, let ge:

[0, co) be

the C' function c(e)exp [11x112 g e(x) =

- 2 ];

0;

114 < 114

E,

where c(e) is chosen so that (because of (2.6))

g,(x)dx = c(e)

(2.7) 18,

f 0

s exp (p 2

1

E2

dp

= 1.

For e > 0 and a e D such that a + Be c D, define u(a + x)ge(x)dx =

ue(a) A J B,

f

u(y)g e(y - a) dy.

ad

From the second representation, it is clear that u, is of class C' on the open subset of D where it is defined. Furthermore, for every a e D there exists e > 0 so that a + Be D; from (2.6), (2.7), and the mean-value property of u, we may then write


243

u,(a) = J u(a + x)g,(x)dx

f

= c(e)

j'

= c(E)

u(a + x)exp(

Sp,

P

Jae

'

1

2

-

2)1.2,(dx)dp E

1

Spu(a)exp(

P, - E, dP = u(a), and conclude that u is also of class C. o

In order to show that Au = 0 in D, we choose a ED and expand a la Taylor

in the neighborhood a + B

au

d

(2.8)

u(a + y) = u(a) + E 3,1

()n

1

d

02

d

(a) + E E yiy; Xi i=1 J=1 2

(a) WC;UXJ

YEk,

+ 0(II312);

where again e > 0 is chosen so that a + BE c D. Odd symmetry gives us Jas,

YtiMdY) = 0,

Jas,

Yiblit(dY) = 0;

i

j,

so upon integrating in (2.8) over OA and using the mean-value property we obtain (2.9)

u(a) =

fas, u(a + y)ue(dy) 1

= u(a) + 2

d

02

E 04 -(a)

pc(dy) + o(0) as,

But E2

= 711

as,

as,

Y lit(dY) =

d

and so (2.9) becomes p2

2d

+ o(e2) = 0.

Dividing by E2 and letting e 0, we see that Au(a) = 0.

B. The Dirichlet Problem We take up now the Dirichlet problem (D,f): with D an open subset of Rd and f: OD R a given continuous function, find a continuous function u: such that u is harmonic in D and takes on boundary values specified by f; i.e., u is of class C2 (D) and

244


(2.10)

Au = 0;

(2.11)

u = f; on aD.

in D,

Such a function, when it exists, will be called a solution to the Dirichlet problem

(D, f). One may interpret u(x) as the steady-state temperature at x e D when the boundary temperatures of D are specified by f. The power of the probabilistic method is demonstrated by the fact that we can immediately write down a very likely solution to (D, f), namely x

u(x) A Exf(W,.);

(2.12)

provided of course that Exlf(W,D)1 < co;

(2.13)

V x e D.

By the definition of TD, u satisfies (2.11). Furthermore, for a a D and Br chosen

so that a + Br c D, we have from the strong Markov property: u(a) = Eaf(N)) = Ea {Ea ER wz,,) 1,,,*)}

= Ea lu(W,± )1 =

f

u(a + x)y,.(dx).

aar

Therefore, u has the mean-value property, and so it must satisfy (2.10). The only unresolved issue is whether u is continuous up to and including OD. It turns out that this depends on the regularity of aD, as we shall see later. We summarize our discussion so far and establish a uniqueness result for (D,f) which strengthens Exercise 2.4. 2.6 Proposition. If (2.13) holds, then u defined by (2.12) is harmonic in D.

2.7 Proposition. If f is bounded and

P`Tri, < cc] = 1; V a eD,

(2.14)

then any bounded solution to (D, f) has the representation (2.12).

PROOF. Let u be any bounded solution to (D,f), and let D A Ix e D; infy cap fix

- yll > 1 /n }. From ItO's rule we have d

u(WtAnAtDn) = u(Wo) + E

i=i

-(WO dWe; 0 < t < oo, n > 1. ax,

Since (au/axi) is bounded in B n D, we may take expectations and conclude that u(a) = Ea 14(Wi A TB., TD);

As t

co, n

0 < t < co,

n

1,

a e D.

cc, (2.14) implies that u(Wt A 28 A 28 ) converges to f(W,D), a.s. P°.

The representation (2.12) follows from the bOunCled convergence theorem.

4.2. Harmonic Functio.is and the Dirichlet Problem

245

2.8 Exercise. With D = {(x,, x2); x2 > 0} and f(x,, 0) = 0; x, e R, show by example that (D, f) can have unbounded solutions not given by (2.12). In the light of Propositions 2.6 and 2.7, the existence of a solution to the Dirichlet problem boils down to the question of the continuity of u defined by (2.12) at the boundary of D. We therefore undertake to characterize those points a e aD for which (2.15)

lim Exf(W,D) = f(a) x-a xeD

holds for every bounded, measurable function f: aD -9 at the point a.

which is continuous

2.9 Definition. Consider the stopping time of the right-continuous filtration {,,,} given by aD 9 inf ft > 0; 14/,e Del (contrast with the definition of TD in (2.1)). We say that a point a e aD is regular for D if Pa[crp = 0] = 1; i.e., a Brownian path started at a does not immediately return to D and remain there for a nonempty time interval.

2.10 Remark. A point a e aD is called irregular if /3/up = 0] < 1; however, the event {o- = 0} belongs to Fo","_, and so the Blumenthal zero-one law (Theorem 2.7.17) gives for an irregular point a: P °[aD = 0] = 0. 2.11 Remark. It is evident that regularity is a local condition; i.e., a e aD is regular for D if and only if a is regular for (a + 13,) n D, for some r > 0. In the one-dimensional case every point of aD is regular (Problem 2.7.18) and the Dirichlet problem is always solvable, the solution being piecewiselinear. When d > 2, more interesting behavior can occur. In particular, if D = {x e Rd; 0 < jlx II < 1} is a punctured ball, then for any x e D the Brownian

motion starting at x exits from D on its outer boundary, not at the origin (Proposition 3.3.22). This means that u defined by (2.12) is determined solely

by the values off along the outer boundary of D and, except at the origin, this u will agree with the harmonic function a(x) o Exf(141,B) =-- Exf(W,D);

x e B,.

Now u(0) f(0), so u is continuous at the origin if and only if f(0) = When d > 3, it is even possible for aD to be connected but contain irregular points (Example 2.17). 2.12 Theorem. Assume that d > 2 and fix a e aD. The following are equivalent:

(i) equation (2.15) holds for every bounded, measurable function f: aD which is continuous at a; (ii) a is regular for D;

246


(iii) for all e > 0, we have lim Px[TD > e] = 0.

(2.16)

x-qi

xeD

PROOF. We assume without loss of generality that a = 0, and begin by proving

the implication (i)

(ii) by contradiction. If the origin is irregular, then

P ° [6D = 0] = 0 (Remark 2.10). Since a Brownian motion of dimension d > 2 never returns to its starting point (Proposition 3.3.22), we have

lim r[WD e Br] = P° [W,D = 0] = 0.

40

Fix r > 0 for which P°[W,,, e B,] < (1/4), and choose a sequence On In°., for which 0 < 6 < r for all n and ön .1. 0. With ; A inf {t z 0; II Wtli b.}, we have P° [Tni. 0] = 1, and thus, limn, P° ['s . < CD] = 1. Furthermore, on the event Itn < aDI we have Wz. e D. For n large enough so that r [rn < o-D] > (1/2), we may write

-1

4

> r[W,DeB,] > PqW,, e B Tn < api = E°(1{, 0 and n > 1] < E 3-11 < 1. n=1

If cusplike behavior is avoided, then the boundary points of D are regular, regardless of dimension. To make this statement precise, let us define for

250 ye


\ {0} and 0 5 8 5

1C,

the cone C(y, 0) with direction y and aperture 0 by

C(y, 0) = {x e Rd; (x,

cos 0}.

II

2.18 Definition. We say that the point a e OD satisfies Zaremba's cone con-

0 and 0 < 0 < it such that the translated cone

dition if there exists y

a + C(y, 0) is contained in Rd \ D.

2.19 Theorem. If a point a E ap satisfies Zaremba's cone condition, then it is regular.

PROOF. We assume without loss of generality that a is the origin and C(y, 0) c d \ D, where y 0 and 0 < 0 < n. Because the change of variable z = (x/.1i) maps C(y, 0) onto itself, we have for any t > 0,

[ = c(m) (2ittr2 exp 1

P°[ Wt CU,

=

f

1

exp [

11x1121dx

2t

I 21 dz 2

q > 0,

where q is independent of t. Now P° [6D < t] > P° [W, E C(y, 0)] = q, and letting t j 0 we conclude that Pqo-D = 0] > 0. Regularity follows from the Blumenthal zero-one law (Remark 2.10).

2.20 Remark. If, for a e ar) and some r > 0, the point a satisfies Zaremba's cone condition for the set (a + 13,) r D, then a is regular for D (Remark 2.11).


251

D. Integral Formulas of Poissont We now have a complete solution to the Dirichlet problem for a large class of open sets D and bounded, continuous boundary data functions f: OD R. Indeed, if every boundary point of D is regular and D satisfies (2.14), then the unique bounded solution to (D,f) is given by (2.12) (Propositions 2.6, 2.7 and Theorem 2.12). In some cases, we can actually compute the right-hand side of (2.12) and thereby obtain Poisson integral formulas. 2.21 Theorem (Poisson Integral Formula for a Half-Space). With d 2, D = {(x1,..., x); x, > 0} and f: OD bounded and continuous, the unique bounded solution to the Dirichlet problem (D,f) is given by O

u(x) =

(2.17)

f(/2) d

xdf (Y)

ird/2

d

dy;

xeD.

2.22 Problem. Prove Theorem 2.21.

The Poisson integral formula for a d-dimensional sphere can be obtained from Theorem 2.21 via the Kelvin transformation. Let go: Rd VOI

Rd \ {0} be

defined by q;o(x) = (x/11.42). Note that cp is its own inverse. We simplify notation by writing x* instead of co(x). For r > 0, let B = {x e Rd: llx - Cul < r}, where c = red and e1 is the unit vector with a one in the i-th position. Suppose f: aB R is continuous (and hence bounded), so there exists a unique function u which solves the Dirichlet problem (B, f). The reader may easily verify that cp(B) = H

fx* e IRd; (x*, c) > 11

and (p(OB \ {O}) = aH = {x* e Rd; (x*, c) = -I}. We define u*: H -+ R, the Kelvin transform of u, by 1

u*(x*) = oc*Ild-2 u(x).

(2.18)

A tedious but straightforward calculation shows that Au*(x*) = Au(x), so u* is a bounded solution to the Dirichlet problem (H, f*) where (2.19)

f *(x*) =

x*II d -2 fix);

Because H = (1/2r)ed + D, where D is as in Theorem 2.21, we may apply (2.17)

to obtain The results of this subsection will not be used later in the text.


252

1

(2.20)

u*(x*) =

(d/2) f

xd

lid/2

r)f*(Y*) Y* - x*

dy*;

x* e H.

Formulas (2.18)-(2.20) provide us with the unique solution to the Dirichlet problem (B, f). These formulas are, however, a bit unwieldy, a problem which can be remedied by the change of variable y = q)(y*) in the integral of (2.20). This change maps the hyperplane OH into the sphere 5B. The surface element on aB is Sra,(dy - c) (recall (2.4), (2.5)). A little bit of algebra and calculus on manifolds (Spivak (1965), p. 126) shows that the proposed change of variable in (2.20) involves

dy* =

(2.21)

Srar(dy

- c)

1131112(")

(The reader familiar with calculus on manifolds may wish to verify (2.21) first for the case y* = e, + (1/2r)ed and then observe that the general case may be reduced to this one by a rotation. The reader unfamiliar with calculus on manifolds can content himself with the verification when d = 2, or can refer to Gilbarg & Trudinger (1977), p. 20, for a proof of Theorem 2.23 which uses the divergence theorem but avoids formula (2.21).)

On the other hand, (2.22)

(2.23)

ILV* - X*112 =

Ilx - y112 1142

r2 - Ilx - cii 2 = 114 2[2(c, x*) - 1] = 2r Ike (x:

- 2r). 1

Using (2.18), (2.19), and (2.21)-(2.23) to change the variable of integration in (2.20), we obtain (2.24)

u(x) = rd-2(r -

cii 2)

I f011tr(dY as

x e B.

I1Y -

Translating this formula to a sphere centered at the origin, we obtain the following classical result.

2.23 Theorem (Poisson Integral Formula for a Sphere). With d > 2, Br = {xe 141d; 114 < r}, and f: aB, continuous, the unique solution to the Dirichlet problem (B,, f) is given by f(Y)fc(dY)

(2.25)

1142) foe,

Y-

;

x e B,..

2.24 Exercise. Show that for x E Br, we have the exit distribution (2.26)

rd-2(r2 Px[WtBe dY1 =

11

x

X112"1

-

Y Ild

(dY);

= r-


253

E. Supplementary Exercises 2.25 Problem. Consider as given an open, bounded subset D of Rd and the bounded, continuous functions g: D -, R and f: 8D -, R. Assume that u: D -gcR is continuous, of class OD), and solves the Poisson equation 1

2Au = -g;

in D

subject to the boundary condition u=f;

on aD.

Then establish the representation ,1)

(2.27)

u(x) = Ex[ f(W,D) + .1

g(W) dt];

x e D.

o

In particular, the expected exit time from a ball is given by r2

(2.28)

Ex TB,.

=

i

ixil 2

d

;

X E Br.

(Hint: Show that the process {M, gL- u(W, AO + Po^ `D g(W) ds, Ft; 0 < t < col

is a uniformly integrable martingale.) 2.26 Exercise. Suppose we remove condition (2.14) in Proposition 2.7. Show

that v(x) A Px [-up = co] is harmonic in D, and if aeaD is regular, then limx.a v(x) = 0. In particular, if every point of al) is regular, then with xe D

u(x) = Ex[f(41/01{,D,)}], the function u + Av is a bounded solution to the Dirichlet problem (D, f) for any A e R. (It is possible to show that every bounded solution to (D, f) is of this form; see Port & Stone (1978), Theorem 4.2.12.)

2.27 Exercise. Let D be bounded with every boundary point regular. Prove that every boundary point has a barrier. 2.28 Exercise. A complex-valued Brownian motion is defined to be a process W = {IV) + ilgt(2), .Ft; 0 < t < co}, where W = {(W('0, W(2)), A; 0 < t < co} is a two-dimensional Brownian motion and i = .\/- 1:

(i) Use Theorem 3.4.13 to show that if W is a complex-valued Brownian motion and f: C --0 C is analytic and nonconstant, then (under an appropriate condition) f(W) is a complex-valued Brownian motion with a random time-change (P. Levy (1948)).

(ii) With e C \ {0}, show that M, A e''',, 0 < t < co is a time-changed, complex-valued Brownian motion. (Hint: Use Problem 3.6.30.) (iii) Use the result in (ii) to provide a new proof of Proposition 3.3.22. For additional information see B. Davis (1979).

254


4.3. The One-Dimensional Heat Equation In this section we establish stochastic representations for the temperatures in infinite, semi-infinite, and finite rods. We then show how such representations allow one to compute boundary-crossing probabilities for Brownian motion. Consider an infinite rod, insulated and extended along the x-axis of the (t, x) plane, and let f(x) denote the temperature of the rod at time t = 0 and location x. If u(t, x) is the temperature of the rod at time t > 0 and position x e R, then, with appropriate choice of units, u will satisfy the heat equation 314

(3.1)

102u

at = 2 0x2'

with initial condition u(0, x) = f(x); x E R. The starting point of our probabilistic treatment of (3.1) is furnished by the observation that the transition density p(t; x, y)

1

y

Px[W,edy] =

1

e-(x-y)212t;

t > 0,

x, ye R,

of the one-dimensional Brownian family satisfies the partial differential equation (3.2)

Op

102p

at

2 0x2

Suppose then that f: R -+ R is a Borel-measurable function satisfying the condition

f_. e-`"2 I f (x)I dx < co

(3.3)

for some a > 0. It is well known (see Problem 3.1) that u(t, x) A Exf(Wi) =

(3.4)

f(y)p(t; x, y) dy

is defined for 0 < t < (1/2a) and x E R, has derivatives of all orders, and satisfies the heat equation (3.1).

3.1 Problem. Show that for any nonnegative integers n and m, under the assumption (3.3), we have ani-m

(3.5)

Otnaxm

ani-m

oo

u(t, x) =

f(y)

OtnOxm

p(t; x, y) dy;

0 0, x E R,

solves the heat equation (3.1) on every strip of the form (0, x R; furthermore, it satisfies condition (3.8) for every 0 < a < (1/2T), as well as (3.7) for every x 0. However, the limit in (3.7) fails to exist for x = 0, although we do have lima° h(t, 0) = 0.

B. Nonnegative Solutions of the Heat Equation If the initial temperature f is nonnegative, as it always is if measured on the absolute scale, then the temperature should remain nonnegative for all t > 0; this is evident from the representation (3.4). Is it possible to characterize the nonnegative solutions of the heat equation? This was done by Widder (1944),

4.3. The One-Dimensicnal Heat Equation

257

who showed that such functions u have a representation CO

u(t, x) = J

p(t; x, y) dF(y); x e

where F: is nondecreasing. Corollary 3.7 (i)', (ii)' is a precise statement of Widder's result. We extend Widder's work by providing probabilistic characterizations of nonnegative solutions to the heat equation; these appear as Corollary 3.7 (iii)', (iv)'. 3.6 Theorem. Let v(t, x) be a nonnegative function defined on a strip (0, T) x where 0 < T < cc. The following four conditions are equivalent:

(i) for some nondecreasing function F: CO

(3.11)

p(T - t; x, y)dF(y); 0 < t < T,

v(t, x) = J

X E fR;

-co

(ii) v is of class C1'2 on (0, T) x IR and satisfies the "backward" heat equation

av

-at + -21 02axev = 0

(3.12)

on this strip; (iii) for a Brownian family {W. ..,;; 0 < s < op}, (S2,

ikPxxe

fixed t e (0, T), x e 0:2, the process {v(t + s, Ws), .mss; 0 tingale on (SI, Px);

(iv) for a Brownian family {Ws, .; 0 (3.13)

v(t, x)

s

s < co},

aindeaacrh_

T-}

{Px}xefra we have

0 < t < t + s < T, xeEJ.

Ex v(t + s,VV,);

PROOF. Since (a/at)p(T - t; x, y) + (1/2)(02/ex2)p(T - t; x, y) = 0, the impli-

cation (i)

(ii) can be proved by showing that the partial derivatives of

v can be computed by differentiating under the integral in (3.11). For a > 1/2T we have

dF(y) = j_ooe-aY2

-v (T -

< cc.

This condition is analogous to (3.3) and allows us to proceed as in Solution 3.1.

For the implications (ii)(iii) and (ii)

(iv), we begin by applying Ito's

rule to v(t + s, Ws); 0 < s < T - t. With a < x < b, we consider the passage times To and Tb as in (2.6.1) and obtain: v(t +(s n Ta n Tb), Ws A T. A

= V(t, Wo)

sAToTb

sATanTh a -04 ± 0-, Wcy)dHfc,

ax

a

1a2

-2

x2

±

W(,) do-.

258


Under assumption (ii) the Lebesgue integral vanishes, as does the expectation of the stochastic integral because of the boundedness of (0/ax)v(t + o, y) when a < y ._ B.

It follows that Ex[v(t + T6, b)1{Tb<s}] db = .133 fs

v(t + a,b)h(a,b - x)da db

0

B

s

ly

fs

J o

v(t + o-,b)p(r; x, b) db doB

-113- I Ex v(t + a,W,)doo

r is 0

where the next to last inequality is a consequence of (3.15). This proves (3.13)

for x e fR, 0 1, of a nondecreasing function F: R -, R such that (3.22) holds on (0, n) x R. For t > n, we have from (3.23): L

4.3. The One-Dimensional Heat Equation

u(t, x) = Exu(--n2,

-/2)) =

261

cc)

01

z

c0

CO

PG; z,y)p (t =

f

14

Z)p(t -12

2'

- 2; x,

dz

x,

z) dz

(y)

p(t; x, y) dF(y).

3.8 Exercise (Widder's Uniqueness Theorem).

defined on the strip (i) Let u(t, x) be a nonnegative function of class (0, T) x R, where 0 < T < co, and assume that u satisfies (3.1) on this strip and lim u(t, y) = 0;

x e R.

110

Show that u = 0 on (0, T) x R. (Hint: Establish the uniform integrability of the martingale u(t - s, Ws); 0 5 s < t.) (ii) Let u be as in (i), except now assume that lima° u(t, y) = f(x); x e R. Show that CO

u(t, x) =

f-co p(t; x, y)f(y)dy;

0 < t < T, x e R.

Can we represent nonnegative solutions v(t, x) of the backward heat equation (3.12) on the entire half-plane (0, co) x R, just as we did in Corollary 3.7

for nonnegative solutions u(t, x) of the heat equation (3.1)? Certainly this cannot be achieved by a simple time-reversal on the results of Corollary 3.7. Instead, we can relate the functions u and v by the formula (3.24)

v(t, x) =

j2ir exp (x2)u(1- x

0 < t < co , x e R. "

The reader can readily verify that v satisfies (3.12) on (0, co) x R if and only if u satisfies (3.1) there. The change of variables implicit in (3.24) allows us to deduce the following proposition from Corollary 3.7.

3.9 Proposition (Robbins & Siegmund (1973)). Let v(t,x) be a nonnegative function defined on the half-plane (0, co) x R. With T= co, conditions (ii), (iii), and (iv) of Theorem 3.6 are equivalent to one another, and to (i) ":

(i)" for some nondecreasing function F: R (3.25)

v(t, x) = J

exp

1

R,

yx - y2 t) dF(y);

0 < t < oo, x e R.


262

PROOF. The equivalence of (ii), (iii), and (iv) for T = co follows from their equivalence for all finite T If v is given by (3.25), then differentiation under the integral can be justified as in Theorem 3.6, and it results in (3.12). If v satisfies (ii), then u given by (3.24) satisfies (ii)', and hence (i)', of Corollary 3.7. But (3.24) and (3.22) reduce to (3.25). 0

C. Boundary-Crossing Probabilities for Brownian Motion The representation (3.25) has rather unexpected consequences in the computation of boundary-crossing probabilities for Brownian motion. Let us consider a positive function v(t, x) which is defined and of class C`2 on (0, co) x IR, and satisfies the backward heat equation. Then v admits the representation (3.25) for some F, and differentiating under the integral we see that

- v(t,x)< 0; a

(3.26)

0 < t < oo, x E R

at

and that v(t, ) is convex for each t > 0. In particular, lim,4,0 v(t, 0) exists. We assume that this limit is finite, and, without loss of generality (by scaling, if necessary), that lim v(t, 0) = 1.

(3.27)

ti.o

We also assume that lim v(t, 0) = 0,

(3.28)

r00

lim v(t, x) = co;

(3.29)

0 < t < co,

x-co

lim v(t, x) = 0,

(3.30)

0 < t < co.

It is easily seen that (3.27)-(3.30) are satisfied if and only if F is a probability distribution function with F(0+) = 0. We impose this condition, so that (3.25) becomes (3.31)

v(t, x) =

1

c°

1

exp yx - 2y2 t)dF(y); 0 < t < oo, X E R,

0+

where F(co) = 1, F 0 + ) = 0. This representation shows that v(t, .) is strictly increasing, so for each t > 0 and b > 0 there is a unique number A(t, b) such that (3.32)

v(t, A(t, b)) = b.

It is not hard to verify that the function A(- , b) is continuous and strictly increasing (cf. (3.26)). We may define A(0, b) = lima° A(t, b). We shall show how one can compute the probability that a Brownian path W, starting at the origin, will eventually cross the curve A(.,b). The problem of


263

computing the probability that a Brownian motion crosses a given, timedependent continuous boundary {/i(t); 0 < t < co} is thereby reduced to finding a solution v to the backward heat equation which also satisfies (3.27)-(3.30) and v(t, 0)) = b; 0 < t < co, for some b > 0. In this generality both problems are quite difficult; our point is that the probabilistic problem can be traded for a partial differential equation problem. We shall provide an explicit solution to both of them when the boundary is linear. Let { W gt; 0 < t < co}, (CI, .x), {Px}. pl be a Brownian family, and define Z, = v(t, Wt);

0 < t < oo.

For 0 < s < t, we have from the Markov property and condition (iv) of Proposition 3.9:

EqZ,1";] = f(14/5) = v(s, Ws) = Z a.s. P °, where f(y) A EY v(t, Wi_s). In other words, {Z Ft; 0 < t < co} is a continuous,

nonnegative martingale on (0, P°). Let {tn} be a sequence of positive numbers with tn.j. 0, and set Z0 = Z. This limit exists, P ° -a.s., and is independent of the particular sequence {t.} chosen; see the proof of Proposition 1.3.14(i). Being "O",1-measurable, Zo must be a.s. constant (Theorem 2.7.17).

3.10 Lemma. The extended process Z A {Z gi; 0 < t < co} is a continuous, nonnegative martingale under P° and satisfies Z0 = 1, 4 = 0, P°-a.s. PROOF. Let {t.} be a sequence of positive numbers with t l O. The sequence {4},D=1 is uniformly integrable (Problem 1.3.11, Remark 1.3.12), so by the Markov property for W, we have for all t > 0:

E°[Z,IF0] = E°Z, = lim E°Zi. = E°Z0 = Z0. n-,co

This establishes that {Z .a.07,; 0 < t < co} is a martingale. Zi exists P°-a.s. (Problem 1.3.16), as does Z0 Since

lima° 4,

it suffices to show that lima° 4 = 1 and lim, Z, = 0 in P °- probability. For every finite c > 0, we shall show that lim sup

(3.33)

1v(t,x) - 11 = 0.

Indeed, for t > 0, 1x1 0, we can find tc, depending on c and a, such that


264

l - E < v(t, x) < 1 ± E;

I XI < C-s/t,

0 < t < tc,,

Consequently, for 0 < t < tc, P°[14 - 11> e] = P°[ 1v((, Wt) - 11> E]

P°[1Wt1> c-s/i] = 2[1 - (I)(e)],

where (1)(x) 1=--`

i' x

,1

-.

N/27r

e-2212 dz.

Letting first t 10 and then c cc, we conclude that Z, t 10. A similar argument shows that

1 in probability as

lim sup v(t, x) = 0,

(3.35)

too ixi .c.,./i

and, using (3.35) instead of (3.33), one can also show that Z, -0 in probability

as t - co.

111

It is now a fairly straightforward matter to apply Problem 1.3.28 to the martingale Z and obtain the probability that the Brownian path {W,(w); 0 < t < co} ever crosses the boundary {A(t, b); 0 < t < co}. 3.11 Problem. Suppose that v: (0, cc) x FR - (0, cc) is of class C' '2 and satisfies

(3.12) and (3.27)-(3.30). For fixed b > 0, let A( , b): [0, co) -, R be the continuous function satisfying (3.32). Then, for any s > 0 and Lebesgue-almost every a e R with v(s, a) < b, we have (3.36)

(3.37)

P° [Wt > A(t, b), for some t > sl Ws = a] =

P° [W

v(s, a)

b'

A(t, b), for some t > s]

= 1 -(I)(

A(s,b)) .,/s

+

1 f (A(s,b) (I) u

o+

Is

y Nfi)dF (y),

where F is the probability distribution function in (3.31).

3.12 Example. With it > 0, let v(t, x) = exp(px - p2 t /2), so A(t,b) = fit + y, where 13 = (it/2), y = (1/12)log b. Then F(y) = 11,,,00(y), and so for any s > 0, 13 > 0, y e R, and Lebesgue-almost every a < y + 13s: (3.38)

P° [W, > fit + y, for some t > sI W., = a] = e-"('''' I's),

and for any s > 0, iq > 0, and y e R: (3.39)

P°[W,

fit + y, for some t > s] = 1 -(I) (

+ /3is -s

+

e-211Y(13(

,

.,/s

I3A.


265

The observation that the time-inverted process Y of Lemma 2.9.4 is a Brownian motion allows one to cast (3.38) with y = 0 into the following formula for the maximum of the so-called "tied-down" Brownian motion or "Brownian bridge": (3.40)

P° [ max W > /3 (:1

]

1,177.

e-2p(p-a)/T

(57.

for T > 0, > 0, a.e. a < 13, and (3.39) into a boundary-crossing probability on the bounded interval [0, T]: (3.41)

P °[W > /3 + yt, for some t e [0, Ti]

= 1 - (1)(7.,j+ 13,) + e-2137,1)(VT

,/ T

.\/ T

> 0,y e R.

3.13 Exercise. Show that P° [W, > ft + y, for some t > 0] = e-211Y, for 13 > 0 and y > 0 (recall Exercise 3.5.9).

D. Mixed Initial/Boundary Value Problems We now discuss briefly the concept of temperatures in a semi-infinite rod and the relation of this concept to Brownian motion absorbed at the origin. Suppose that f: (0, oo) R is a Borel-measurable function satisfying co° e-"21f(x)Idx < oo

(3.42)

for some a > 0. We define (3.43)

0 0.

The reflection principle gives us the formula (2.8.9)

Px [ W, e dy, To > t] = p At; x, y)dy A [p(t; x, y) - p(t; x, -y)] dy

for t > 0, x > 0, y > 0, and so o

(3.44)

f(y)p(t; x, y)dy -

ui(t, x) = I 0

j

AP(t; x, AdY,

which gives us a definition for t. valid on the whole strip (0, 1/2a) x R. This representation is of the form 9.4), where the initial datum f satisfies f(y) = -f( -y); y > 0. It is clear then that u, has derivatives of all orders, satisfies the heat equation (3.1), satisfies (3.6) at all continuity points of f, and lim ui (t, 0) = 0; x10

1

0 < t < -2a .


266

We may regard u (t, x); 0 < t < (1/2a), x > 0, as the temperature in a semiinfinite rod along the nonnegative x-axis, when the end x = 0 is held at a constant temperature (equal to zero) and the initial temperature at y > 0 is

f(v)

Suppose now that the initial temperature in a semi-infinite rod is identically

zero, but the temperature at the endpoint x = 0 at time t is g(t), where g: (0, 1/2a)

R is bounded and continuous. The Abel transform of g, namely

(3.45)

u2(t, x) -4 Ex [g(t -To)1{,0,}]

=

J

=

.1

r

g(t - r)h(r, x) dr g(s)h(t - s, x)ds;

0 < t < -1 , x > 2a

o

0

with h given by (3.10), is a solution to (3.1) because h is, and h(0, x) = 0 for x > 0. We may rewrite this formula as

u2(t, x) = E° [g(t -7'x)1{,,,,}]; 0 < t
0,

and then the bounded convergence theorem shows that lim u2(s, x) = g(t); s-t lim u2(t, y) = 0;

tlo

0 < t < -1 , 2a

0 < x < co.

y -.x

We may add ul and u2 to obtain a solution to the problem with initial datum f and time-dependent boundary condition g(t) at x = 0.

3.14 Exercise (Neumann Boundary Condition). Suppose that f: (0, co) is a Borel-measurable function satisfying (3.42), and define u(t, x)

R

1

Exf(11411);

0 < t < - , x > 0. 2a

Show that u is of class C1'2, satisfies (3.1) on (0, 1/2a) x (0, co) and (3.6) at all continuity points of f, as well as lim

0

s_.t uX

u(s, x) = 0;

0 0, x e Rd, we may compute formally (4.3)

1 2

-1 f° Cat Au dt = (a + k)z,, -f. 2

0

The stochastic representation for the solution z of the elliptic equation (4.3) is known as the Kac formula; in the second subsection we illustrate its use when d = 1 by computing the distributions of occupation times for Brownian motion. The second subsection may be read independently of the first one.


268

Throughout this section, {W, .; 0 s t

0.t5T

d

2Kead"2 E {px[wp) > n]

+ Px[ -wp)

n]


270

where we have used (2.6.2). But by (2.9.20), eadn, px[+ W- ) > n]

T eadn2

1

e(n -T- x()))212T

2n n -T

which converges to zero as n oo, because 0 < a < (1/2Td). Again by the dominated convergence theorem, the third term is shown to converge to Ex [v(T, W -i) exP -g-k(Ws)dsl] as n co and rT T - t. The FeynmanKac formula (4.7) follows. 4.5 Corollary. Assume that f: Rd -> R, k: Rd -> [0, oo), and g: [0, oo) x Rd -> R are continuous, and that the continuous function u: [0, co) x Rd -R is of class C''2 on (0, co) x Rd and satisfies (4.1) and (4.2). If for each finite T > 0 there

exist constants K > 0 and 0 < a < (1/2Td) such that max lu(t,x)1 + max Ig(t,x)I < Kea11x112;

OStST

V x e R,

0

,

o

0,

eR

/ 2a enables us to replace (4.16) by the equivalent condition OD

(4.16)'

f e-alf1WtHdt < Vco, x e R.

Ex

PROOF OF THEOREM 4.9. For piecewise-continuous functions g which satisfy condition (4.16), we introduce the resolvent operator G. given by OD

e.' a g(W)dt =

(G.g)(x) A Ex

1 Jo

=

1

-03

x

1

03

+I

e(Y-x)

.,./Tot

' g(y) dy

e-Iv -xl

I

.,./2a

-cc

g(y) dy1;

e(x-Y

x e R.

x

Differentiating, we obtain (Gag)'(x) = (4.18)

J

e(x-v1 Nr2xg(y)dy - J

(Gag)" (x) = - 2g(x) + 2a(Gag)(x);

x

(y)dy;

e(Y-x1

xe

x e R,

Dg.

It will be shown later that

G7(kz) = Gaf -z

(4.19)

and G2(1kz1)(x) < co;

(4.20)

V x e R.

If we then write (4.18) successively with g = f and g = kz and subtract, we obtain the desired equation (4.17) for x e R\(Df u DkZ), thanks to (4.19). One can easily check via the dominated convergence theorem that z is continuous, so DkZ g Dk. Integration of (4.17) yields the continuity of z'. In order to verify (4.19), we start with the observation

0
b.

;

y

Matching the values of z() and z'() across the points x = 0 and x = b, we obtain the values of the four constants A, B, C, and D. In particular, z(0) = A is given by

sinh b.,/2a

a

2

-1

a + y [-cosh b

,/2a

L

(,./2ot + \/2(a + y)) cosh 62a + (.12ot

1

,/2(a + y) ± Y) +

sinh bN/2a

whence 2

lim z(0) =

yt

(cosh b,./2(ot +

- 1)

N/2(oz + 13) cosh b.,/2(a + /3) + ,./2a sinh b.\/2(« + fl)

and lim lim z(0) = a4,0 ytco

1

1

/3

cosh b

-[1

The result (4.23) now follows from (4.24).

4.5. Solutions to Selected Problems 2.16. Assume without loss of generality that a = O. Choose 0 < r < fl b II A 1. It suffices to show that a is regular for Br c D (Remark 2.11). But there is a simple arc C in

Old\D connecting a = 0 to 6, and in 13,\C, a single-valued, analytic branch of log(x, + ix2) can be defined because winding about the origin is not possible. Regularity of a = 0 is an immediate consequence of Example 2.14 and Proposition 2.15.

276


2.22. Every boundary point of D satisfies Zaremba's cone condition, and so is regular. It remains only to evaluate (2.12). Whenever Y and Z are independent random variables taking values in measurable spaces (G, W) and (H, le), respectively, and f: G x H R is bounded and measurable, then

Ef(Y,Z) =

fHfG f(y,z)P[Y dy]P[Z dz].

We apply this identity to the independent random variables TD = inf {t

0;

Wt(d) = 01 and {(W,"), W(d-1)); 0 < t < oo 1, the latter taking values in C[0, cold'. This results in the evaluation (see (2.8.5)) cc

E'fiwt,,) =

J la

ad-t

/(Y1,

, Yd-1, 0)P[(14/t(1),

,

Wt(d-1))E (dY I,

,dYd-i)] P"[TDE dt]

ao

xdfol

- x112]dt t(2rztr2exP

2t

dy,

and it remains only to verify that 2,42 r (do)

Az

1

(5.1)

ed+2)/2 exP

[

A > 0.

2t dt =

Ad

For d = 1, equation (5.1) follows from Remark 2.8.3. For d = 2, (5.1) can be verified by direct integration. The cases of d > 3 can be reduced to one of these two cases by successive application of the integration by parts identity

r" Jo

e-''212` dt =

r1+

2(ot

1)

"1

dt, a > 1.

ta e

Az

1

2.25. Consider an increasing sequence {D};,°_, of open sets with /5 c D; V n 1 and U,T=, D = D, so that the stopping times zn = inf {t > 0; IV, D} satisfy limn-. tn = TD, a.s. Px. It is seen from ItO's rule that

tnr Mr)

0 0, E > 0, 0 < to < t, s (1/2(a + E)), and set B = {(t, x); to < t < tI,Ixj < fi}. For (t, JOE B, ye R, we have ay

(x

Y)2

2t

1

ay2 2t- -, (Ix1 - ly1)2 aye --2ty2 + - IxIlyI , t, 1

1

E

R2

#

E

Y2 - 2- Y2 + -t,IYI 2

+

2

2Et?

For any nonnegative integers n and m, there is a constant C(n, m) such that on+m

Ot"Ox 'n

p(t; x, y)

(x - y)2} C(n, m)(1 + 1Y12"-"") exP

2t

;

(t, x)e B, ye

and so an+m

(5.3)

et"Ox'"

5- IBA

p(t; x, y)

m)(1 + ly12"-"")expl-(a + -2)3/2 + -2SE2t1}

D(n,m)If(y)le-aY2; (t, x)e B, y E R,

where D(n, m) is a constant independent of t, x, and y. It follows from (3.3) that thA integral in (3.5) converges uniformly for (t, x) e B, and is thus a continuous function of (t, x) on (0,1/2a) x We prove (3.5) for the case n = 0, m = 1; the general case is easily established by induction. For (t, x) E B, (t, x + h) e B, we have 1

-[u(t, x + h) - u(t, x)] = h

J

,

-[p(t; x + h, y) - p(t, x, yflf(y)dy h

a

=

-pct; oho, 31, YV(Y)dy, Ox

where, according to the mean-value theorem, O(t, y) lies between x and x + h. We now let h 0, using the bound (5.3) and the dominated convergence theorem, to obtain (3.5).

3.2. (Widder (1944)): We suppose f is continuous at xo and assume without loss of generality that f(xo) = 0. For each E > 0, there exists (5 > 0 such that If(y)I < E for Iy - xo 15 S. We have for X E [x0 - (5/2), -X0

(6/2)],


278

(5.4)

I u(t, x)I

f

so+6

so -6

I fly)1P(t; x, y)dy

I f( APO; -; A dy +

so-6

fco

+ ..+6

I f(Y)IP(t; x, Ady.

The middle integral is bounded above by e; we show that the other two converge to zero, as t 10. For the third integral, we have the upper bound

(5.5)

r

(y - xo by1

z

e'Y If(y)1 exp ay 2

2t

27cJt J so+6

dy.

For t sufficiently small, exp[ay2 - (y - xo - 6/2)2 /2t] is a decreasing function of y for y xo + 6 (it has its maximum at y = (x0 + 6/2)/(1 - 2at)). Therefore, the expression in (5.5) is bounded above by 6

p t; 0, -2) exp[a(x,

.,

+ 6)2] j.

e "2 If(Al dY

xo+ 6

which approaches zero as t10. The first integral in (5.4) is treated similarly.

4.6. Notes Section 4.2: The Dirichlet problem has a long and venerable history (see, e.g., Poincare (1899) and Kellogg (1929)). Zaremba (1911) was the first to observe that the problem was not always solvable, citing the example of a punctured region. Lebesgue (1924) subsequently pointed out that in three or more dimensions, if D has a sufficiently sharp, inward-pointing cusp, then the problem can fail to have a solution (our Example 2.17). Poincare (1899) used barriers to show that if every point on ap lies on the surface of a sphere which does not otherwise intersect D, then the Dirichlet problem can be solved in D. Zaremba (1909) replaced the sphere in Poincare's sufficient condition by a cone. Wiener (1924b) has given a necessary and sufficient condition involving the capacity of a set.

The beautiful connection between the Dirichlet problem and Brownian motion was made by Kakutani (1944a, b), and his pioneering work laid the foundation for the probabilistic exposition we have given here. Hunt (1957/1958) studied the links between potential theory and a large class of transient Markov processes. These matters are explored in greater depth in Ito & McKean (1974), Sections 7.10-7.12; Port & Stone (1978); and Doob (1984).

Section 4.3: The representation (3.4) for the solution of the heat equation is usually attributed to Poisson (1835, p. 140), although it was known to both Fourier (1822, p. 454) and Laplace (1809, p. 241). The heat equation for the semi-infinite rod was studied by Widder (1953), who established uniqueness

4.6. Notes

279

and representation results similar to Theorems 3.3 and 3.6. Hartman & Wintner (1950) considered the rod of finite length. For more examples and further information on the subject matter of Subsection C, including applications to the theory of statistical tests of power one, the reader is referred to Robbins & Siegmund (1973), Novikov (1981), and the references therein. Section 4.4: Theorem 4.2 was first established by M. Kac (1949) for d = 1; his work was influenced by the derivation of the Schrodinger equation achieved by R. P. Feynman in his doctoral dissertation. Kac's results were strengthened and extended to the multidimensional case by M. Rosenblatt (1951), who also provided Holder continuity conditions on the potential k in order to guarantee a C1.2 solution. Proposition 4.12 is taken from ItO & McKean (1974).

k(x) = co; then the

Let k: l -4 [0, co) be continuous and satisfy eigenvalue problem (k(x) - A)tli(x) = - tr(x); 2

xe

with e L2(R), has a discrete spectrum Al < A2 < ... and corresponding eigenfunctions ti/J;17_1 g L2(R). Kac (1951) derived the stochastic representation 1

(6.1)

k(14(,) ds}]

- lirn - log Ex[exp

Al

,_) t

for the principal eigenvalue, by combining the Feynman-Kac expression

u(t,x) = Ex[exp { -Jo k(141s)ds}] for the solution of the Cauchy problem Ou

ot

(6.2)

1

+ k u = -Au;

(0, c3o) x

u(0, x)

xel

2

1;

(Corollary 4.5), with the formal eigenfunction expansion

u(t,x)

00

E cie-Aittp,(x) .i=1

for the solution of (6.2). Recall also Exercise 4.6, and see Karatzas (1980) for a control-theoretic interpretation (and derivation) of this result. Sweeping generalizations of (6.1), as well as an explanation of its connection with the classical variational expression (6.3)

A, =

inf

k(x)tly 2 (X) dx + 2

J.

(

/(x))2dx } ,

_co

if°, 14,2(x) dx=1.

are provided in the context of the theory of large deviations of Donsker &

280


Varadhan (1975), (1976). This theory constitutes an important recent development in probability theory, and is overviewed succinctly in the monographs by Stroock (1984) and Varadhan (1984).

The reader interested in the relations of the results in this section with quantum physics is referred to Simon (1979).

Alternative approaches to the arc-sine law for fi.(t) can be found in Exercise 6.3.8 and Remark 6.3.12.

CHAPTER 5

Stochastic Differential Equations

5.1. Introduction We explore in this chapter questions of existence and uniqueness for solutions to stochastic differential equations and offer a study of their properties. This endeavor is really a study of diffusion processes. Loosely speaking, the term diffusion is attributed to a Markov process which has continuous sample paths and can be characterized in terms of its infinitesimal generator. In order to fix ideas, let us consider a d-dimensional Markov family X =

{X A; 0 < t < co}, (52, ,F), Px 1 xe Rd, and assume that X has continuous paths. We suppose, further, that the relation (1.1)

1

- [ef(X,) -f (x)] = (cif )(x); dxePd rlo t lim

holds for every f in a suitable subclass of the space C2(Rd) of real-valued, twice continuously differentiable functions on Rd; the operator s/f in (1.1) is given by I

(1.2)

(sif )(x)

d

d

.E1 kE1 a ik(x)

a2f( x)

d

ofi x)

+

2

for suitable Borel-measurable functions aik: Rd R, 1 5 i, k S d. The lefthand side of (1.1) is the infinitesimal generator of the Markov family, applied to the test function f. On the other hand, the operator in (1.2) is called the secondorder differential operator associated with the drift vector b = (1)1, . , bd) and the diffusion matrix a = {aik} 0. The monograph by Nelson (1967) can be consulted for a detailed study of the kinematics and dynamics of such random motions.

1.1 Definition. Let X = IX ,57;,; 0 < t < co }, dimensional Markov family, such that

(12, g;),

{1'1. Esd

be a d-

(i) X has continuous sample paths; (ii) relation (1.1) holds for every f e C2(Rd) which is bounded and has bounded first- and second-order derivatives; (iii) relations (1.3), (1.4) hold for every x e Rd; and

(iv) the tenets (a)-(d) of Definition 2.6.3 are satisfied, but only for stopping times S.

Then X is called a (Kolmogorov-Feller) diffusion process.

There are several approaches to the study of diffusions, ranging from the purely analytical to the purely probabilistic. In order to illustrate the traditional analytical approach, let us suppose that the Markov family of Definition 1.1 has a transition probability density function Px [X, e dy] = F (t; x, y)dy;

(1.5)

t > 0.

Vxe

Various heuristic arguments, with (1.1) as their starting point, can then be employed to suggest that r(t; x, y) should satisfy the forward Kolmogorov equation, for every fixed x e Rd: (1.6)

0

Ot

f(t; x, y) = ..il*F(t; x, y);

(t, y)E(0, 00) x Rd,

and the backward Kolmogorov equation, for every fixed y e Rd: (1.7)

- I-(t; x, y) = sir(t; x, y); 0

at

(t, x) e (0, co) x Rd.

The operator d* in (1.6) is given by (1.8)

(.91*.f)(Y)

i=i k=1

uy (4, [aik(Y)f(Y)]

1

=1

Ebt(Y)f(Y)1,

the formal adjoint of .4 in (1.2), provided of course that the coefficients k, ask

possess the smoothness requisite in (1.8). The early work of Kolmogorov (1931) and Feller (1936) used tools from the theory of partial differential

5.1. Introduction

283

equations to establish, under suitable and rather restrictive conditions, the existence of a solution C(t; x,y) to (1.6), (1.7). Existence of a continuous Markov process X satisfying (1.5) can then be shown via the consistency Theorem 2.2.2 and the entsov-Kolmogorov Theorem 2.2.8, very much in the spirit of our approach in Section 2.2. A modern account of this methodology is contained in Chapters 2 and 3 of Stroock & Varadhan (1979).

The methodology of stochastic differential equations was suggested by P. Levy as an "alternative," probabilistic approach to diffusions and was carried out in a masterly way by K. ItO (1942a, 1946, 1951). Suppose that we have a continuous, adapted d-dimensional process X = {X 3,7,; 0 < t < cc) which satisfies, for every x e Rd, the stochastic integral equation (1.9)

AT)

= xi +

t

r

't

o

J =1

0

f bi(Xs)ds + E

on a probability space KZ

0-1;(xjawch; 0 < t < co, I R; 1 < i < d, 1 < j < r

are Borel-measurable. Then it is reasonable to expect that, under certain conditions, (1.1)-(1.4) will indeed be valid, with (1.10)

au,(x)

Er aii(90"ki(X). i=1

We leave the verification of this fact as an exercise for the reader. 1.2 Problem. Assume that the coefficients bi, au are bounded and continuous, and the Or-valued process X satisfies (1.9). Show that (1.3), (1.4) hold for every x e Or, and that (1.1) holds for every f e C2(Rd) which is bounded and has bounded first- and second-order derivatives.

Ito's theory is developed in Section 2 under the rubric of strong solutions. A strong solution of (1.9) is constructed on a given probability space, with respect to a given filtration and a given Brownian motion W. In Section 3 we take up the idea of weak solutions, a notion in which the probability space, the filtration, and the driving Brownian motion are part of the solution rather than the statement of the problem. The reformulation of a stochastic differential

equation as a martingale problem is presented in Section 4. The solution of this problem is equivalent to constructing a weak solution. Employing martingale methods, we establish a version of the strong Markov propertycorresponding to (iv) of Definition 1.1-for these solutions; they thereby earn the right to be called diffusions.

The stochastic differential equation approach to diffusions provides a powerful methodology and the useful representation (1.9) for a very large class of such processes. Indeed, the only important strong Markov processes with continuous sample paths which are not directly included in such a development are those which exhibit "anomalous" boundary behavior (e.g., reflection, absorption, or killing on a boundary).

284

5. Stochastic Differential Equations

Certain aspects of the one-dimensional case are discussed at some length in Section 5; a state-space transformation leads from the general equation

to one without drift, and the latter is studied by the method of random time-change. The notion and properties of local time from Sections 3.6, 3.7 play an important role here, as do the new concepts of scale function, speed measure, and explosions. Section 6 studies linear equations; Section 7 takes up the connections with partial differential equations, in the spirit of Chapter 4 but not in the same detail. We devote Section 8 to applications of stochastic calculus and differential equations in mathematical economics. The related option pricing and consumption/investment problems are discussed in some detail, providing concrete illustrations of the power and usefulness of our methodology. In particular, the second of these problems echoes the more general themes of stochastic control theory. The field of stochastic differential equations is now vast, both in theory and in applications; we attempt in the notes (Section 10) a brief survey, but we abandon any claim to completeness.

5.2. Strong Solutions In this section we introduce the concept of a stochastic differential equation with respect to Brownian motion and its solution in the so-called strong sense. We discuss the questions of existence and uniqueness of such solutions, as well as some of their elementary properties. Let us start with Borel-measurable functions Mt, x), x); 1 < i < d, 1 < j < r, from [0, co) x Rd into R, and define the (d x 1) drift vector b(t, x) = {13,(t, x)} and the (d x r) dispersion matrix o-(t, x) = {6;;(t, x) }, The intent is to assign a meaning to the stochastic differential equation (2.1)

sjSr

dX, = b(t, X ,) dt + o(t, X,)dW

written componentwise as (2.1)'

= bat, X,)dt + E

xt) dwto;

1 < i < d,

.i=i

where W = {W; 0 < t < oo } is an r-dimensional Brownian motion and X = {X,; 0 < t < co} is a suitable stochastic process with continuous sample paths and values in Rd, the "solution" of the equation. The drift vector b(t, x) and the dispersion matrix o-(t, x) are the coefficients of this equation; the (d x d) matrix a(t, x) A x)o-T(t, x) with elements (2.2)

aik(t, x) A E 0-,;(t,x)o-kp,x); 1=1

will be called the diffusion matrix.

1 < i, k < d

5.2. Strong Solutions

285

A. Definitions In order to develop the concept of strong solution, we choose a probability space (0, P) as well as an r-dimensional Brownian motion W = .317,w; 0 < t < co} on it. We assume also that this space is rich enough to accommodate a random vector taking values in Rd, independent of 'cow, and with given distribution

ell; re ARd)-

u(r) =

We consider the left-continuous filtration

a (0 v

= cr(, Ws; 0

s

t);

0 < t < op,

as well as the collection of null sets P-4 {Ns O.; 3 G e W., with N

G and P(G)= 0},

and create the augmented filtration a (g, u .41),

(2.3)

0 < t < oo;

(u t

0

by analogy with the construction of Definition 2.7.2. Obviously, { W A; o < t < co} is an r-dimensional Brownian motion, and then so is { W

0 < t < co} (cf. Theorem 2.7.9). It follows also, just as in the proof of Proposition 2.7.7, that the filtration {",} satisfies the usual conditions. 2.1 Definition. A strong solution of the stochastic differential equation (2.1), on the given probability space (S2, F, P) and with respect to the fixed Brownian motion W and initial condition is a process X = {X,; 0 .5 t < oo} with continuous sample paths and with the following properties: (i) X is adapted to the filtration {..F,} of (2.3),

(ii) P[X0 = ] = 1, Xs)1 + Xs)} ds < co] = 1 holds for every 1 < i < d, r and 0 t < co, and (iv) the integral version of (2.1)

(iii) P[Po 1

(2.4)

j

X, = X0 +

J b(s, X s) ds +

ft o-(s, Xs) dWs;

0 1, namely Xt -.E. 0; however, for 0 < a < 1, all functions of the form

0 < t < s, s < t < oo,

;

with # = 1/(1 - a) and arbitrary 0 < s < co, solve (2.7). It seems then reasonable to attempt developing a theory for stochastic differential equations by imposing Lipschitz-type conditions, and investigating what kind of existence and/or uniqueness results one can obtain this way. Such a program was first carried out by K. ItO (1942a, 1946). 25 Theorem. Suppose that the coefficients b(t, x), o-(t, x) are locally Lipschitzcontinuous in the space variable; i.e., for every integer n 1 there exists a

constant K > 0 such that for every t (2.8)

b(t,

- Mt,

0, 114

+ 110'(t, x) -

n and Ilyll

n:

Y)11 < KnIlx - .3111.

Then strong uniqueness holds for equation (2.1).

2.6 Remark on Notation. For every (d x r) matrix a, we write d

(2.9)

r

116112 -4 E E 6. i =1 j =1

Before proceeding with the proof, let us recall the useful Gronwall inequality.

2.7 Problem. Suppose that the continuous function g(t) satisfies

288


0 < g(t) < a(t) + # J r g(s) ds; 0 < t < T,

(2.10)

0

with fl > 0 and a: [0, T]

integrable. Then

f

g(t) < a(t) + 13

(2.11)

a(s)e"-s) ds; 0 < t < T

0

PROOF OF THEOREM 2.5. Let us suppose that X and 2 are both strong solutions,

defined for all t > 0, of (2.1) relative to the same Brownian motion W and the same initial condition on some (SI, P). We define the stopping times rn = inf ft > 0; IX, II > n} for n > 1, as well as their tilded counterparts, and

we set S T A in. Clearly lim, S = co, a.s. P, and t A Su

Xt A S

Xt A S

b(u, gu)} du

{NU, Xu)

+

f

tAsu

Using the vector inequality Ilv, + + vkIIZ k2(Ilvi 112 + 1114112), the Holder inequality for Lebesgue integrals, the basic property (3.2.27) of stochastic integrals, and (2.8), we may write for 0 < t E 11X

A

- jet A sull2

[f

T:

tAS,,

Ilb(u, X) - b(u, gu)II du12

o

r

[E f

+ 4E

tAS

Xu) -

u)) d W utn 12

o

i=1 fI A

Ilb(u, Xu) - b(u, g)112 du

4tE

0 t A Su

+ 4E .1 4(T + 1)K,;

Xu) - a(u, 5tu)Il2 du

ft E X

uA

- luAsull2 du.

We now apply Problem 2.7 with g(t) A E II X t A s, 21A 5112 to conclude that {X, A S; 0 < t < 00 } and {X n 5; 0 < t < co} are modifications of one another, oo, we see that the same is true for and thus are indistinguishable. Letting n t < co }. {X,; 0 _< t < co } and {)-1,; 0

2.8 Remark. It is worth noting that even for ordinary differential equations, a local Lipschitz condition is not sufficient to guarantee global existence of a solution. For example, the unique (because of Theorem 2.5) solution to the equation


289

X, = 1 + f t X1 ds o

is X, = 1/(1 - t), which "explodes" as t T 1. We thus impose stronger conditions in order to obtain an existence result.

2.9 Theorem. Suppose that the coefficients b(t, x), a(t, x) satisfy the global Lipschitz and linear growth conditions 116(t, x) - b(t, y)II + lio(t, x) - alt, AII Lc. K 11x - yil,

(2.12)

(2.13)

ilb(t, x)O2 + ki(t, X)Il 2 < K2(1 + 0x112),

for every 0 < t < oo, x e Rd, y e Rd, where K is a positive constant. On some probability space (0, ..F, P), let be an Rd-valued random vector, independent of the r-dimensional Brownian motion W = {Hit, ,F,w ; 0 < t < co}, and with finite second moment: (2.14)

Ell92 < 00.

Let {A} be as in (2.3). Then there exists a continuous, adapted process X = {X .57;,; 0 < t < col which is a strong solution of equation (2.1) relative to W, with initial condition . Moreover, this process is square-integrable: for every T > 0, there exists a constant C, depending only on K and T, such that

E Vie

(2.15)

C(1 + E g 112 )ect;

0

t

T

The idea of the proof is to mimic the deterministic situation and to construct recursively, by analogy with (2.6), a sequence of successive approximations by setting Xr --- and

Xr1) A-

(2.16)

+

f

t

t

b(s, XP)ds +

o

5

o(s, XP)dWs; 0 < t < co,

o

for k > 0. These processes are obviously continuous and adapted to the filtration {,,}. The hope is that the sequence {X(k)}1°_, will converge to a solution of equation (2.1). Let us start with the observation which will ultimately lead to (2.15).

2.10 Problem. For every T > 0, there exists a positive constant C depending only on K and T, such that for the iterations in (2.16) we have E Il V)II2

(2.17)

C (1 + E g 112)ect; OT,k 0.

PROOF' OF THEOREM 2.9. We have Xr1) - V) = B, + M, from (2.16), where B, -4

t

o

{b(s V)) - b(s, Xlk-1))1 ds,

M, A 5 z fa(s, X!k)) - Os, Xlk-1))1 dWs. o

Thanks to the inequalities (2.13) and (2.17), the process {M, = (MP ), ... , Mr),


290

..97t; 0 < t < co

is seen to be a vector of square-integrable martingales, for

which Problem 3.3.29 and Remark 3.3.30 give

E[max

0 <s 1. For each n > 1, there exists a continuous function p on fJ with support in (a, a_,) so that 0 < p(x) < (2/nh2(x)) holds for every x > 0, and 117-' p(x) dx = 1. Then the function p,,(u)dudy;

4 /(x)

(2.26)

o

is

xeR

o

even and twice continuous y differentiable, with tir,;(x)1 < 1 and 0(x) = Ixi for x e rt. Furthermore, the sequence {tp}-_, is

nondecreasing. Now let us suppose that there are two strong solutions x(') and X(2) of (2.1)

with n) = Xo a.s. It suffices to prove the indistinguishability of xo) and X(2) under the assumption (2.27)

E

XV)12 ds < oo;

0

t < co, i = 1, 2;

292


otherwise, we may use condition (iii) of Definition 2.1 and a localization argument to reduce the situation to one in which (2.27) holds. We have

A, A- XP ) - V ) = ft {b(s, X.P)) - b(s, V))} ds o

ft

+

{Os, X11)) - Os, V))1 dWs,

o

and by the ItO rule, (2.28)

*At) = fo tk(As)[b(s, XS')) - b(s, X12))] ds + -1 I t ti,;(As)[a(s, Vs1)) - o-(s, X12))]2 ds 2

+

0

it0 IMA.,)[a(s, X,P)) - o(s, X,12))] dWs.

The expectation of the stochastic integral in (2.28) is zero because of assumption

(2.27), whereas the expectation of the second integral in (2.28) is bounded above by ES ip"(As)h2(14,1) ds < 2t/n. We conclude that (2.29)

Eik(A,)

E

ft ii/;,(A.,)[b(s, XV)) - b(s, X52))] ds + -tn o

_.K

o

t

.1EI.A.,lds +-t ; n

0, n

1.

A passage to the limit as n -> co yields EIA,1 < K Po E IN ds; t z 0, and the conclusion now follows from the Gronwall inequality and sample path

0

continuity.

2.15 Example (Girsanov (1962)). From what we have just proved, it follows that strong uniqueness holds for the one-dimensional stochastic equation (2.30)

X,=

1 t I Xsix dWs;

0

t < co,

o

as long as a > (1/2), and it is obvious that the unique solution is the trivial one X, -a 0. This is also a solution when 0 < a < (1/2), but it is no longer the only solution. We shall in fact see in Remark 5.6 that not only does strong uniqueness fail when 0 < a < (1/2), but we do not even have uniqueness in the weaker sense developed in the next section. 2.16 Remark. Yamada & Watanabe (1971) actually establish Proposition 2.13 under a condition on b(t, x) weaker than (2.23), namely, (2.23y

I b(t, x) - b(t, Al

K(1 x - YI);

0

t < co, x e R, y e R,


293

where K: [0, co) -+ [0, oo) is strictly increasing and concave with K(0) = 0 and f(0.0(du/K(u)) = co for every s > 0.

2.17 Exercise (Ito & Watanabe (1978)). The stochastic equation

X, = 3 fX:I3 t

ds + 3

dW,

has uncountably many strong solutions of the form

:") =

t
W, = 01. Note that the function o(x) = 3x213 satisfies condition (2.24), but the function b(x) = 3x" fails to satisfy the condition of Remark 2.16.

The methodology employed in the proof of Proposition 2.13 can be used to great advantage in establishing comparison results for solutions of onedimensional stochastic differential equations. Such results amount to a certain kind of "monotonicity" of the solution process X with respect to the drift coefficient b(t, x), and they are useful in a variety of situations, including the study of certain simple stochastic control problems. We develop some comparison results in the following proposition and problem. 2.18 Proposition. Suppose that on a certain probability space (1-2, .F, P) equipped

with a filtration {,,} which satisfies the usual conditions, we have a standard, one-dimensional Brownian motion {W 37;,; 0 < t < oo} and two continuous, adapted processes Xth; j = 1, 2, such that (2.31)

Xli) = XVI +

ft bf(s, Xn ds + o

i

o- (s, XV)) dW.,;

0 < t < co

o

holds a.s. for j = 1, 2. We assume that

(i) the coefficients o-(t, x), k(t,x) are continuous, real-valued functions on [0, co) x R, (ii) the dispersion matrix o-(t, x) satisfies condition (2.24), where h is as described in Proposition 2.13,

(iii) XV) < XV) a.s., (iv) bi(t, x) < b2(t, x), V 0 < t < oo, x e R, and (v) either b1(t, x) or b2(t, x) satisfies condition (2.23). Then (2.32)

P[XP) < XP), V 0 < t < co] = 1.

PROOF. For concreteness, let us suppose that (2.23) is satisfied by b1(t,x). Proceeding as in the proof of Proposition 2.13, we assume without loss of

294


generality that (2.27) holds. We recall the functions kfr(x) of (2.26) and create a new sequence of auxiliary functions by setting ton(x) = tfr(x) 1,0,)(x); x e R,

n > 1. With A, = XP) - V), the analogue of relation (2.29) is

Eq),,(60 -n 5 E .1

(p'(As)[bi(s, x!") - b2(s, X12)11 ds

o

=E

ii co;,(As)[bi(s, X11)) - bi(s, X?))] ds 0

it

+E

cp',,(As)[bi(s, X?)) - b2(s, V))] ds < K ft WO ds,

o

o

by virtue of (iv) and (2.23) for 61(t, x). Now we can let n -' co to obtain E(A,+) < K Po E(45-1-)ds; 0 < t < co, and by the Gronwall inequality (Problem 2.7), we have E(A,+) = 0; i.e., XP) < XP) a.s. P.

2.19 Exercise. Suppose that in Proposition 2.18 we drop condition (v) but strengthen condition (iv) to (iv)'

bi(t, x) < b2(t, x);

0 < t < co, x e 41.

Then the conclusion (2.32) still holds. (Hint: For each integer m > 3, construct a Lipschitz-continuous function b,,,(t, x) such that (2.33)

bi (t, x) 5 b,(t, x) < b2(t, x);

0 < ( < m, ixi < m).

It should be noted that for the equation (2.34)

X,= + f i b(s, X s) ds + Wr;

0 5 t < co,

/3

with unit dispersion coefficient and drift b(t, x) satisfying the conditions of Theorem 2.9, the proof of that theorem can be simplified considerably. Indeed, since there is no stochastic integral in (2.34), we may fix an arbitrary w e SI

and regard (2.34) as a deterministic integral equation with forcing function { W,(co); 0 5 t < co}. For the iterations defined by (2.16), and with k = 1, 2, ..., Er)(w) A max IIV)(w) - k-i

osst

)(OE

we have the bound ak)(co) < K Pc, Dr1)(co)ds; 0 < t < co, valid for every w e e. The latter can be iterated to prove convergence of the scheme (2.16), path by path, to a continuous, adapted process X which obeys (2.34) surely. This is the standard Picard-Lindeliif proof from ordinary differential equations and makes no use of probabilistic tools such as the martingale inequality or the Borel-Cantelli lemma.

Lamperti (1964) has observed that, under appropriate conditions on the coefficients b and o-, the general, one-dimensional integral equation


(2.4)"

295

X, = + f

b(Xs)ds +

f

6(X0dWs;

0 :5_ t < cc,

can be reduced by a change of scale to one of the form (2.34); see the following exercise.

2.20 Exercise. Suppose that the coefficients a: IR - (0, oo) and b: R R are of class C2 and C1, respectively; that b' - (1/2)66" - (bof/o-) is bounded; and that (1/a) is not integrable at either +oo. Then (2.4)" has a unique, strong solution

X. (Hint: Consider the function f(x) = 1,I(du/o-(u)) and apply Ito's rule to

f(x,)) A second important class of equations that can be solved by first fixing the

Brownian path and then solving a deterministic differential equation was discovered by Doss (1977); see Proposition 2.21.

D. Approximations of Stochastic Differential Equations Stochastic differential equations have been widely applied to the study of the

effect of adding random perturbations (noise) to deterministic differential systems. Brownian motion offers an idealized model for this noise, but in many applications the actual noise process is of bounded variation and non-Markov. Then the following modeling issue arises.

Suppose that { V}°°_, is such a sequence of stochastic processes which converges, in an appropriate strong sense, to the Brownian motion W = {W .57;,; 0 < t < co}. Suppose, furthermore, that {X},7_, is a corresponding sequence of solutions to the stochastic integral equations (2.35)

)0") = +

J

b(Xr))ds +

ft o-(X!"))dVs(");

n > 1,

where the second integral is to be understood in the Lebesgue-Stieltjes sense. As n -- co, will {X(")},7_, converge to a process X, and if so, what kind of integral equation will X satisfy? It turns out that under fairly general conditions, the proper equation for X is (2.36)

X, = + f

b(Xs)ds +

f

o-(X )odWs,

where the second integral is in the Fisk-Stratonovich sense. Our proof of this depends on the following result by Doss (1977).

2.21 Proposition. Suppose that a is of class C2(R) with bounded first and second derivatives, and that b is Lipschitz-continuous. Then the one-dimensional stochastic differential equation

296

(2.36)'


2"

X, = + f fb(Xs) + -o-(X )cr'(X )} ds + f 1

cr(Xs)dly,

has a unique, strong solution; this can be written in the form X t(w) = u(W(w), Yt(co));

0 < t < oo, co E

R and a process Y which solves an for a suitable, continuous function u: R2 ordinary differential equation, for every w el/

2.22 Remark. Under the conditions of Proposition 2.21, the process {cr(X,), .97E; 0 < t < co} is a continuous semimartingale with decomposition 1

o-(X,) = cr(0 +

+

f

J

1

[b(Xs)er'(Xs) + 2cr(X5)(cr'(X5))2 + - o-"(Xs)cr2(X5)1ds 2

a(X3)0=(X,)dll7s,

r

and so, according to Definition 3.3.13,

fo cr(XJ0 dW, = -1 ft o-(Xs)o-'(Xs)ds + 2

0

f

o-(Xs)dW.s.

In other words, equations (2.36) and (2.36)' are equivalent. PROOF OF PROPOSITION 2.21. Let u(x, y): R2 - R be the solution of the ordinary

differential equation Ou

-= o-(u),

(2.37)

u(0, y) = y;

Ox

such a solution exists globally, thanks to our assumptions. We have then

-

02

2

(2.38)

Ox2

= a(u)o-'(u),

x0y

aU

= ce(u)-0y,

a y) = 1, yu(0,

which give (2.39)

0

-oyu(x, y) = exp

1

cr'(u(z, y)) dz} A

p(x, Y).

Let A > 0 be a bound on o' and a". Then e-Alx1 < p(x, y) < eAlx1, and (2.39) implies the Lipschitz condition Iu(x,y1) - ti(x,Y2)1 < eAl'IlYi - y21.

If L is a Lipschitz constant for b, then lb(u(x, Yi)) - b(u(x,Y2))1 < LeAlx1

1y1

Y21

and consequently, for fixed x, b(u(x, y)) is Lipschitz-continuous and exhibits linear growth in y. Using the inequality le l - ez21 < (ez' v ez2)lz - z21, we


297

may write lx1

I p(x, yi) - p(x, y2)I

[P(x, Yi) v P(x,Y 2)] is

Id(u(z, Y in - a' (u(z, Y2))1 dz

0

eAlx1 f 'xi

A lu(ztY 1) - u(zt Y Adz

03

< Alx1e2A1'11Yi- Y21.

For fixed x, p(x, y) is thus Lipschitz-continuous and bounded in y. It follows that the product f(x, y) A p(x, y) b(u(x, y)) satisfies Lipschitz and growth conditions of the form (2.40)

I f (x, y 1) -f (x, Y 2)1 0 consider the stochastic differential system

XV =

d.)02) = V') dt;

dY,(1) = 016(t, X``))dt - ca,(') dt + a dW; ye = n, where , ri are a.s. finite random variables, jointly independent of the Brownian motion W.

(i) This system admits a unique, strong solution for every value of a E (0, CO). (ii) For every fixed, finite T > 0, we have lim

sup 1,02) - XtI = 0, a.s.,

ago 0.T

where X is the unique, strong solution to (2.34). 2.27 Exercise. Solve explicitly the one-dimensional equation

dX, = (,/1 + X,2 + PC,)dt + .,/1 + X, MT,. 2.28 Exercise. (i) Suppose that there exists an Rd-valued function u(t, y) = (1410,11,

of class C "2([0, co) x Rd), such that

,ud(t,17))

300


au;

at

(t, y) = bi(t,u(t, y)),

aui y,

(t, y) = o- ;At, u(t, y));

1

j

d

hold on [0, co) x Rd, where each bi(t, x) is continuous and each o-,;(t, x) is of class C1'2 on [0, co) x Rd. Show then that the process X,

u(t, W);

0 < t < oo,

where W is a d-dimensional Brownian motion, solves the Fisk-Stratonovich

equation (2.36)"

dX, = b(t, X,) dt + cr(t, X,) 0 dW,.

(ii) Use the above result to find the unique, strong solution of the onedimensional Ito equation d X --

[ 1 2+ t X,

a(1 + t)21dt + a(1 + t)2 dW; 0 < t < oo.

5.3. Weak Solutions Our intent in this section is to discuss a notion of solvability for the stochastic differential equation (2.1) which, although weaker than the one introduced in the preceding section, is yet extremely useful and fruitful in both theory and applications. In particular, one can prove existence of solutions under assumptions on the drift term b(t, x) much weaker than those of the previous section, and the notion of uniqueness attached to this new mode of solvability will lead naturally to the strong Markov property of the solution process (Theorem 4.20).

3.1 Definition. A weak solution of equation (2.1) is a triple (X, W), {A}, where

P),

P) is a probability space, and {F,} is a filtration of sub-a-fields of .97; satisfying the usual conditions,

(i) (f2,

(ii) X = IX A; 0 < t < col is a continuous, adapted Rd-valued process, W={W 0 < t < oo} is an r-dimensional Brownian motion, and (iii), (iv) of Definition 2.1 are satisfied.

The probability measure p(T) A P[X0 e 11, F e 4(1?) is called the initial distribution of the solution. The filtration {.57,} in Definition 3.1 is not necessarily the augmentation of the filtration = o-() v 0 < t < oo, generated by the "driving Brownian motion" and by the "initial condition" = X0. Thus, the value of the solution X,(w) at time t is not necessarily given by a measurable functional of the Brownian path { Ws(w); 0 < s < t} and the initial condition (w). On the other

5.3. Weak Solutions

301

hand, because W is a Brownian motion relative to {A}, the solution X,(w) at time t cannot anticipate the future of the Brownian motion; besides { Ws(a)); 0 < s < t} and (co), whatever extra information is required to compute Xt(w) must be independent of { Wo(w) - W(w); t < 0 < col.

One consequence of this arrangement is that the existence of a weak solution (X, W), (fl, P), } does not guarantee, for a given Brownian motion { l , A; 0 < t < cc } on a (possibly different) probability space (n,,,P), the existence of a process )7 such that the triple (51, 1,P), (n, F),

{A} is again a weak solution. It is clear, however, that strong solvability implies weak solvability.

A. Two Notions of Uniqueness There are two reasonable concepts of uniqueness which can be associated with weak solutions. The first is a straightforward generalization of strong uniqueness as set forth in Definition 2.3; the second, uniqueness in distribution, is better suited to the concept of weak solutions.

3.2 Definition. Suppose that whenever (X, W), .F, P), {A}, and (X, W), (SI, y, P), {A}, are weak solutions to (2.1) with common Brownian motion W (relative to possibly different filtrations) on a common probability space P) and with common initial value, i.e., P[X0 = 5C0] = 1, the two processes X and X are indistinguishable: P[X, = V0 < t < co] = 1. We say then that pathwise uniqueness holds for equation (2.1).

3.3 Remark. All the strong uniqueness results of Section 2 are also valid for pathwise uniqueness; indeed, none of the proofs given there takes advantage of the special form of the filtration for a strong solution. 3.4 Definition. We say that uniqueness in the sense of probability law holds for equation (2.1) if, for any two weak solutions (X, W), P), {A}, and (g, 11), c, F), with the same initial distribution, i.e., P[X 0 e

= i3 [g0 e 1]; V I- e R(Rd),

the two processes X, g have the same law.

Existence of a weak solution does not imply that of a strong solution, and uniqueness in the sense of probability law does not imply pathwise uniqueness. The following example illustrates these points amply. However, pathwise uniqueness does imply uniqueness in the sense of probability law (see Proposition 3.20).

35 Example. (H. Tanaka (e.g., Zvonkin (1974))). Consider the one-dimensional equation

302


X, =

(3.1)

f sgn(Xs) d Ws;

0 < t < co,

where

sgn(x) =

x > 0,

1;

1 -1; x , = jo sgn2(Xs) ds = t. Therefore, X is a Brownian

motion (Theorem 3.3.16), and uniqueness in the sense of probability law holds. P), {A} is also a weak solution, so once On the other hand, (- X, W), (SI,

we establish existence of a weak solution, we shall also have shown that pathwise uniqueness cannot hold for equation (3.1).

Now start with a probability space (0, .

P) and a one-dimensional

Brownian motion X = {X .519;ix; 0 < t < co} on it; we assume P[X0 = 0] = 1 and denote by IR the augmentation of the filtration } under P. The same argument as before shows that sgn(Xs)dXs;

0 < t < co

0

is a Brownian motion adapted to {.3,' }. Corollary 3.2.20 shows that (X, W), (1, P), { x} is a weak solution to (3.1). With {#,w} denoting the augmen-

tation of {Fr}, this construction gives

Ax, which is the opposite

inclusion from that required for a strong solution! Let us now show that equation (3.1) does not admit a strong solution. Assume the contrary; i.e., let X satisfy (3.1) on a given (0, P) with respect to a given Brownian motion W, and assume ..9%x Ary for every t > 0. Then X is

necessarily a Brownian motion, and from Tanaka's formula (3.6.13) with a = 0, we have sgn(Xs)dXs = I X ,I - 2Lx, (0)

W, = 0

=

1

- lim - meas{0 40 2e

s

t;

I Xsj

e},

0 < t < co , a.s. P,

where a(0) is the local time at the origin for X. Consequently, Aw c and thus also Ax Axl holds for every t > 0. But this last inclusion is absurd.

B. Weak Solutions by Means of the Girsanov Theorem The principal method for creating weak solutions to stochastic differential equations is transformation of drift via the Girsanov theorem. The proof of the next proposition illustrates this approach.

5.3. Weak Solutions

303

3.6 Proposition. Consider the stochastic differential equation (3.2)

d X, = b(t, X,) dt + dW,;

0 < t < T,

where T is a fixed positive number, W is a d-dimensional Brownian motion, and b(t, x) is a Borel-measurable, Rd-valued function on [0, T] x O which satisfies (3.3)

K(1 + DM; 0 .t,T,xe9;Rd

ilb(t,x)11

for some positive constant K. For any probability measure u on (Rd,.4(Rd)), equation (3.2) has a weak solution with initial distribution II.

PROOF. We begin with a d-dimensional Brownian family X = {X 1, let us introduce the o--fields

(3.19)

ajcp, con A o(z(s); 0 < s < t) =

(a(cp, co)'"))

for 0 < t < cc, where tpt: C[0, oor is the truncation mapping (x))'" (ptz)(s) A z(t A s); z e C[0, oor, 0 5 s < co. As in Problem 2.4.2, it is shown that A(C[0, corn) = oM), where (6, is the countable collection of all finitedimensional cylinder sets of the form

C = {ze C[0, co)'"; (z(ti),..., z(t,,))e AI

with n > 1, tie [0, t] n Q, for 1 5 i < n, and A e AR") equal to the product of intervals with rational endpoints. The continuity of ze C[0, corn allows us to conclude that a set of the form C is in a(%), even if the points t, are not necessarily rational. It follows that a t(c [0, cor) is countably generated. On the other hand, the generating class (6; is closed under finite intersections, so any two probability measures on A(C[0, con which agree on (6, must also agree on ai(c 0, con,

by the Dynkin System Theorem 2.1.3. It follows that A(c[o, con is also countably determined.

More generally, Theorem 2.1.3 shows that if a 6 -field .97 is generated by a countable collection of sets (6' which happens to be closed under pairwise intersection, then ,9-", is also countably determined. In a topological space with a countable base (e.g., a separable metric space), we may take this to be the collection of all finite intersections of complements of these basic open sets. 3.18 Theorem. Suppose that SI is a complete, separable metric space, and denote

the Borel 6 field y = WI). Let P be a probability measure on (0,,), and let be a sub-o--field of F. Then a regular conditional probability Q for F given exists and is unique. Furthermore, if dr is a countably determined sub-o--field of then there exists a null set N e such that (iv)

Q(o); A) = Wu)); A e

, w e SI\N.

In particular, if X is a -measurable random variable taking values in another complete, separable metric space, then with ° denoting the a-field generated by X, (iv) implies (iv)'

Q(w; {w' e

X(&) = X(01) = 1; P-a.e.

When the a-field is generated by a random variable, we may recast the assertions of Theorem 3.18 as follows. P) be as in Theorem 3.18, and let X be a measurable 3.19 Theorem. Let (SI, mapping from this space into a measurable space (S, 91, on which it induces the


308

distribution P X-1 (B) -A- P [w e O.; X (co) e B], B e 9'. There exists then a function Q(x; A): S x 07", [0, 1], called a regular conditional probability for y given

X, such that

(i) for each xeS, Q(x; ) is a probability measure on (52, g"), (ii) for each A e ", the mapping xi- Q(x; A) is Y-measurable, and (iii) f o r each A e 9-;, Q(x; A) = P[Al X = x], P X -1-a.e. xeS.

If Q'(x; A) is another function with these properties, then there exists a set Neff with PX-1(N) = 0 such that Q(x; A) = Q'(x; A) for all A e .f; and xe S\N. Furthermore, if S is also a complete, separable metric space and = M(S), then N can be chosen so that we have the additional property: B e ff, x e S\N.

Q(x; Ico e O.; X (co)e BI) = 1,(x);

(iv)

In particular, Q(x; 1w e O.; X(w) = x }) = 1;

(iv)'

PX-1-a.e. xeS.

D. Results of Yamada and Watanabe on Weak and Strong Solutions Returning to our initial question about the relation between pathwise uniqueness and uniqueness in the sense of probability law, let us consider two weak }; j = 1, 2, of equation (2.1) with solutions (X-', W-'), (51i, vi), y(B) A vi[n) e 13] = v2[X,1321e 13];

(3.20)

We set Ic(f) =

B e M(f?).

- X,(3j); 0 < t < co, and we regard the j-th solution as

consisting of three parts: Xe, W(i), and 17"). This triple induces a measure Pi on

(0, .4(0)) A Old x C[0, cc)' x C[0, co)d, M(Rd) © .4(C[0, cor) 0 .4(C [0, co)d))

according to the prescription (3.21)

pi(A) A vi[(XW), W"), Yu) e A];

A e .4(13), j = 1, 2.

We denote by B = (x, w, y) the generic element of O. The marginal of each Pi on the x-coordinate of B is the marginal on the w-coordinate is Wiener

measure P,, and the distribution of the (x, w) pair is the product measure x P1 because X,(3-(*) is .F1)-measurable and W") is independent of .9-1) (Problem 2.5.5). Furthermore, under Pi, the initial value of the y-coordinate is zero, almost surely. The two weak solutions (X"), W")) and (X(2), W(2)) are defined on (possibly) different sample spaces. Our first task is to bring them together on the same, canonical space, while preserving their joint distributions. Toward this end, we note that on (0, .4(0), Pi) there exists a regular conditional probability for .4(0) given (x, w). We shall be interested only in conditional probabilities of

5.3. Weak Solutions

309

sets in MO) of the form Rd x C[0, coy x F, where F e 4(C[0, °or). Thus, with a slight abuse of terminology, we speak of Qi(x, w; F):

x C[0, coy x .4(C[0, cor)

[0,1]

as the regular conditional probability for M(C[0, oor) given (x, w). According

to Theorem 3.19, this regular conditional probability enjoys the following properties: (3.22) (i) for each x e Rd, w e Cr[0, co), Q3(x, w; ) is a probability measure on

(C[0, cc)", MC [0, cor)), (3.22) (ii) for each F e [0, cor), the mapping (x,w)i--. Q;(x,w; F) is MO) AC [0, con-measurable, and (3.22) (iii) PJ(G x F) = $G Qi(X w; F)p.(dx)P,,,(dw); F e M(C [0, cor), G e .4(01d) ®M(C[0, coy).

Finally, we consider the measurable space (0,F), where SI = O x C[0, co)d and F is the completion of the o--field Af(0) 0 AC[0, cor) by the collection ..)1( of null sets under the probability measure P(dco) A Qi(x, w; dyi )Q2(x, w;dy2)p(dx)P,(dw).

(3.23)

We have denoted by w = (x, w4142) a generic element of a In order to endow (SI,

P) with a filtration that satisfies the usual conditions, we take A -4 Q { (x, w(s), y (s), y2(s)); 0 < s < t} ,

,A

,+;

u

0 < t < co.

It is evident from (3.21), (3.22) (iii), and (3.23) that (3.21)'

P [co e SI; (x, w, yj) e

and so the distribution of (x

= vi[(4-1), Wu), ri))e A]; A G.4(:3), j = 1, 2, w) under P is the same as the distribution of

(Xu), WLD) under vi. In particular, the w-coordinate process {w(t), A; 0 < t < co}

P), and it is then not difficult is an r-dimensional Brownian motion on (SI, to see that the same is true for {w(t), Ft; 0 < t < co}. 3.20 Proposition (Yamada & Watanabe (1971)). Pathwise uniqueness implies uniqueness in the sense of probability law. {..FP)}; PROOF. We started with two weak solutions (Xth, Wu)), (C/J, j = 1, 2, of equation (2.1), with (3.20) satisfied. We have created two weak solutions (x + yj, w), j = 1, 2, on a single probability space (II, F, P), {",}, such that (X), WU)) under v.; has the same law as (x + yj, w) under P. Pathwise uniqueness implies P[x + y1 (t) = x + y2(t), V 0 < t < co] = 1, or equivalently,

(3.24)

no) = (x5w41,Y2)E CI; yi = y2] = 1.

It develops from (3.21)', (3.24) that


310

viun),14/0), Y(1))e A] = P[co ei); (x, w, yi )e A] = P[to e 0; (x, w, y2) e in = v2[(X,(32), W(2), Y(2)) e A];

A e R(D),

and this is uniqueness in the sense of probability law.

CI

Proposition 3.20 has the remarkable corollary that weak existence and pathwise uniqueness imply strong existence. We develop this result.

3.21 Problem. For every fixed t > 0 and F e .4,(C[0, co)d), the mapping (x, w)i- Qi(x, w; F) is 4,-measurable, where IA 1 is the augmentation of the filtration 1.4(Rd) ® at(c [0, con} by the null sets of p(dx)P,(dw).

(Hint: Consider the regular conditional probabilities Z(x, w; F): Rd x C[0, co)' x .4,(C[0, co)d) -* [0, 1] for RAC [0, co)d), given (x, cp,w). These

enjoy properties analogous to those of Qi(x,w; F); in particular, for every F e .4,(C[0, co)d), the mapping (x, w) r- Ql(x, w; F) is .4(Rd) 0 .4,(C [0, conmeasurable, and

Pj(G x F) = 1 Vi(x,w; F) p(dx)P,(dw)

(3.25)

G

for every G e .4(Rd) 0 .4,(C[0, con. If (3.25) is shown to be valid for every G e .4(Rd) 0 .4(C[0, con, then comparison of (3.25) with (3.22) (iii) shows that Qi(x, w; F) = Q;(x,w; F) for it x P,-a.e. (x, w), and the conclusion follows. Establish (3.25), first for sets of the form (3.26)

G = G1 x (pt.' G2 n o-,-1 G3);

G1 e .4(Rd),

G2, G, e .4(C [0, co)'),

where (o-,w)(s) A w(t + s) - w(t); s > 0, and then in the generality required.)

3.22 Problem. In the context of Proposition 3.20, there exists a function k: Rd x C[0, co)'

C[0, co)d such that, for it x P,-a.e. (x, w) e Rd x C[0, co)",

we have (3.27)

Q 1 (x, w; {k(x, w)} ) = Q2(x, w; {k(x, w)} ) = 1.

This function k is .4(Rd) 0 .4(C [0, con/ .4(C [0, co)d)-measurable and, for each

0 < t < co, it is also A/.4,(C[0, co)d)-measurable (see Problem 3.21 for the definition of A). We have, in addition, (3.28)

P[o) = (x, w, yi, Y2)6 n; Y1 = Y2 = k(X, W)] = 1.

3.23 Corollary. Suppose that the stochastic differential equation (2.1) has a weak

solution (X, W), (0, .F, P), 1.0 with initial distribution 12, and suppose that pathwise uniqueness holds for (2.1). Then there exists a .4(Rd) 0 .4(C [0, con/ .4(C[0, co)d)-measurable function h: Rd x C[0, co)' C[0, co)d, which is also A/at(c [o, oo)d)-measurable for every fixed 0 < t < co, such that (3.29)

X. = h(X0, W), a.s. P.

5.4. The Martingale Problem of Stroock and Varadhan

311

Moreover, given any probability space P) rich enough to support an Rd-valued random variable with distribution p and an independent Brownian

motion {ll, (3.30)

; 0 < t < ool, the process X.

h(, W.)

is a strong solution of equation (2.1) with initial condition

PROOF. Let h(x, w) = x + k(x, w), where k is as in Problem 3.22. From (3.28) and (3.21)' we see that (3.29) holds. For c and 14, as described, both (X0, W) and ITO induce the same measure p x P,, on Rd x [0, con, and since W) satisfies (2.1), so does (X. = Pr.). The process )7 is (X. = h(X 0, adapted to {A} given by (2.3), because h is / MAC [0, co)d)- measurable. The functional relations (3.29), (3.30) provide a very satisfactory formulation of the principle of causality articulated in Remark 2.2.

5.4. The Martingale Problem of Stroock and Varadhan We have seen that when the drift and dispersion coefficients of a stochastic differential equation satisfy the Lipschitz and linear growth conditions of Theorem 2.9, then the equation possesses a unique strong solution. For more general coefficients, though, a strong solution to the stochastic differential equation might not exist (Example 3.5); then the questions of existence and uniqueness, as well as the properties of a solution, have to be discussed in a different setting. One possibility is indicated by Definitions 3.1 and 3.4: one attempts to solve the stochastic differential equation in the "weak" sense of finding a process with the right law (finite-dimensional distributions), and to do so uniquely. A variation on this approach, developed by Stroock & Varadhan (1969), formulates the search for the law of a diffusion process with given drift and dispersion coefficients in terms of a martingale problem. The latter is equivalent to solving the related stochastic differential equation in the weak sense, but does not involve the equation explicitly. This formulation has the advantage of being particularly well suited for the continuity and weak convergence arguments which yield existence results (Theorem 4.22) and "invariance principles", i.e., the convergence of Markov chains to diffusion processes (Stroock & Varadhan (1969), Section 10). Furthermore, it casts the

question of uniqueness in terms of the solvability of a certain parabolic equation (Theorem 4.28), for which sufficient conditions are well known. This section is organized as follows. First, the martingale problem is formulated and its equivalence with the problem of finding a weak solution to the corresponding stochastic differential equation is established. Using this martingale formulation and the optional sampling theorem, we next establish the strong Markov property for these solution processes. Finally, conditions

312


for existence and uniqueness of solutions to the martingale problem are provided. These conditions are different from, and not comparable to, those given in the previous section for existence and uniqueness of weak solutions to stochastic differential equations.

4.1 Remark on Notation. We shall follow the accepted practice of denoting by Ck(E) the collection of all continuous functions f: E gl which have continuous derivatives of every order up to k; here, E is an open subset of some

gl is a continuous function, we

Euclidean space Rd. If f(t, x): [0, T) x E

write f e C([0, T) x E), and if the partial derivatives (afia(),(afiaxi),(a2fiaxiaxi); 1 < i,j < d, exist and are continuous on (0, T) x E, we write f e C"2((0, T) x E). The notation f e C" 2 ( [0, T) x E) means that f e C" 2 ((0, T) x E) and the

indicated partial derivatives have continuous extensions to [0, T) x E. We shall denote by Ct(E), Ct(E), the subsets of Ck(E) of bounded functions and functions having compact support, respectively. In particular, a function in Ct(E) has bounded partial derivatives up to order k; this might not be true for a function in ct(E).

A. Some Fundamental Martingales In order to provide motivation for the martingale problem, let us suppose P), {.??7,} is a weak solution to the stochastic differential that (X, W), equation (2.1). For every t > 0, we introduce the second-order differential operator a2foo 1 d d af(x) (4.1)

(.91, f )(x) A

-

2 i=i k=1

+

atk(t, x)

NOXi 10Xk

b.(t, x)

;

f e C2 (Rd),

a Xi

1=1

where aik(t, x) are the components of the diffusion matrix (2.2). If, as in the next proposition, f is a function of t e [0, oo) and x e gld, then (.91,f)(t, x), is obtained by applying sr', to f(t, -). 4.2 Proposition. For every continuous function f(t, x): [0, oo) x gld -, gl which belongs to C"2((0, oo) x Rd), the process Mf = {Mir , .F;; 0 < t < op} given by (4.2)

Ng A f(t, Xt) - f(0, X0) -

t (a-f + as

s)ds

sisf)(s, X

Jo

is a continuous, local martingale; i.e., Mf je, loc. If g is another member of C([0, co) x Rd) C. , 2 ( (05 oo) x glg), then Mg e .41'''" and d d 't a a (4.3)

<Mf, Mg>, = 1 1

i=1 k=1

afk(s, Xs)

0

ax;

f(s, Xs)

ax,,

g(s, XS) ds.

Furthermore, if f e C0([0, oo) x Rd) and the coefficients r; 1 5 i s d, 1 s j 5 r, are bounded on the support of f, then Mf e JP',


313

PROOF. The Ito rule expresses M1 as a sum of stochastic integrals: r

d

(4.4)

Mir = E E gi,J), with MI" n r aii(s, Xs)f(s, X s)dWo. i=1 j=1

ax,

Introducing the stopping times

S

inf

> 0; X,11 > n or

a5(s, Xs)

n for some (i,j)}

and recalling that a weak solution must satisfy condition (iii) of Definition 2.1,

we see that limn, S = co a.s. The processes Ali f(n)

tAS

r

d

(4.5)

AlfAs = E E

i=1 j=1

J

0 a ii(S, X s)- f(s XS) d Wsti); Oxi

n > 1,

are continuous martingales, and so M1 e dr'. The cross-variation in (4.3) follows readily from (4.5). If f has compact support on which each au is bounded, then the integrand in the expression for M" in (4.4) is bounded, so Mf With the exception of the last assertion, a completely analogous result is valid for functional stochastic differential equations (Definition 3.14). We elaborate in the following problem.

4.3 Problem. Let bi(t, y) and c(t, y); 1 < i < d, 1 < j < r, be progressively measurable functionals from [0, co) x C[0, oo)d into R. By analogy with (2.2), we define the diffusion matrix a(t, y) with components (4.6)

ak(t, y) A E aiy,y)o-,;(t,y); 0 < t < co, y e C[0, co )d. .J=1

{;}, is a weak solution to the functional Suppose that (X, W), stochastic differential equation (3.15), and set 1

(sift'u)(Y) =

d

Ou(y(t)) 02u(y(t)) + bi(t, y) i.1 oxi Oxiax,

d

E E aik1t, 2 i=i k=1

0 < t < oo, u e C2(Rd), y e C[0, oo)d. Then, for any functions f, g e C([°, co) x Rd) cr. ,2(0, co. x Rd), the process (4.2)'

mif 4 f(t, Xt) - f(0, X0) -

ft o

+ .4.11(s, X) ds, Ft; 0 < t < 00 os

is in .11`.1", and d

(4.3)'

d

t

<M1, M9 >, = E E

i=1 k=1

aik(s, X) o

a

oXi

f(s, X,)-g(s, XS) ds. Oyck

Furthermore, if f e Co([0, co) x Rd) and for each 0 < T < co we have

314 (4.7)

5. Stochastic Differential Equations KT,

Haft,

t

0

T,

}le CLO, 00)d,

where K T is a constant depending on T, then f e ./11`2.

The simplest case in Proposition 4.2 is that of a d-dimensional Brownian motion, which corresponds to k(t, x) 0 and o-u(t, x) Su; 1 j < d. Then the operator in (4.1) becomes d

82f

1

1

2

2 i=1 ax?;

sif = -Af = E

fe C2(g1°).

4.4 Problem. A continuous, adapted process W = {W, ,Ft; 0 < t < co} is a d-dimensional Brownian motion if and only if

f(w) - fiwo) - -2

0

Ai(Ws) ds, 0. Problem 1.2.2 shows that T(y) = T(pT(y)(y)) holds for every y e C[0, oo)d and so, with A e ,IT and t A T(y), we have PROOF.

ye A

ye[A n

5 t1]-4* cpty e [A n

5 tn.:* (Aye A.

The second of these equivalences is a consequence of the facts An{ T< t} e M and y(s) = (cp,(y))(s); 0 < s < t. We conclude that A = {ye C[0, oo)d; ch(y)(y)e Al =

e C[0, oo)d; y( n T)e Al.

For the next lemma, we recall the discussion of regular conditional probabilities in Subsection 3.C, as well as the formula (2.5.15) for the shift operators: (Osy)(t) = y(s + t); 0 < t < oo for s > 0 and y e C[0, oo)d.


321

4.19 Lemma. Let T be a bounded stopping time of {.4,} and 5 a countably determined sub-a-field of MT such that y(T) is 5-measurable. Suppose that b and Q are bounded on compact subsets of Rd, and that the probability measure P on (SI, = (C[0, co)°, .4(C [0, cx)r), solves the time-homogeneous martingale problem of Definition 4.15. We denote by Q,(F) = Q(a); F): S2 x P4 -+ [0,1] the regular conditional probability for .4 given 1. There exists then a P-null event N e 5 such that, for every w N, the probability measure P. 4 Q,° 13i.1 solves the martingale problem (4.21), (4.22) with x = w(T(w)).

PROOF. We notice first that, thanks to the assumptions imposed on /, Theorem 3.18 (iv)' implies the existence of a P-null event N e 5, such that Q(0); {yen; Y(T(Y)) = w(T(w)) }) = 1,

and therefore also PAY e f); .Y(0) = (0(7.(0))] = Q.Ey e n; Y(T(Y)) = w(T(w))] = 1 hold for every w 0 N. Thus (4.22) is satisfied with x = w(T(w)).

In order to establish (4.21), we choose 0 < s < t < co, Gel, Fe.4 f e q(Rd); define

f(Y(t)) - f(Y(s)) - I (sin1Y1un du; ye C[0, co)d,

Z(Y)

and observe that (4.23)

Z(y)1F(Y)13.(dY) =

Z(07-0).01F(07.61Y)0(0;dY) 1-1

= E[loiF(Z 0 OT)ig](a) = E[E(Z 0 oTiaT+3). lei.,F11] (03)

= 0,

P-a.e. co.

We have used in the last step the martingale property (4.21) for P and the optional sampling theorem (Problem 1.3.23 (i)).

Let us observe that, because of our assumptions, the random variable Z is bounded; relation (4.23) shows that the /-measurable random variable w

Z(y)P,(dy) is zero except on a P-null event depending on s, t, f, and F.

Consider a countable subcollection g of as and a P-null event N(s, t, f) e , such that Z(y)Pc,(dy) = 0;

V co

N(s, t, f),

V F e g.


322

agree on 6', and since is countably determined, the subcollection g can be chosen so as to permit the conclusion Then the finite measures v,±-(F) A IF Z± (y)P,(dy); F e

f,

Z(y)P,(dy) = 0;

We may set now N(f) = lj Li e Q 0

V ru

N(s,t, f),

VFe

N(s, t, f), and use the boundedness and

s 1,

fRa

there exists a weak solution of (4.25).

PROOF. For integers j > 0, n > 1 we consider the dyadic rationals 4") = jr" and introduce the functions i(i(t) = t11); t e [tj"), 01_1). We define the new coefficients (4.26)

b(n)(t,

b(Y(0.(t))),

(7(")(t, Y) g 6(Y(IP.(0);

0 < t < co, y e C[0, oo)d,

which are progressively measurable functionals. Now let us consider on some probability space (0,F, P) an r-dimensional

Brownian motion W = {W, Fr; 0 < t < col and an independent random vector with the given initial distribution

and let us construct the filtration

1,,1 as in (2.3). For each n > 1, we define the continuous process V") = Wm),

0 < t < co} by setting X,(,,") = and then recursively:

X") = 41!) + b(X(n)))(t - t(p) + o-PC:!))(1,V, - K.));

j

0, tj") < t
._ 0; 1X,1

n},

we have

,AS

(5.3)

P[f {1b(Xs)I + e(Xs)} ds < col = 1; V0co

t < co] = 1


330

as the explosion time for X. The assumption of continuity of X in the extended real numbers implies that (5.6)

and

S = inf{t > 0: X, st RI

Xs = +co a.s. on {S < co}.

We stipulate that X, = Xs; S < t < co. The assumption of finiteness of X0 gives P[S > 0] = 1. We do not assume that lim, X, exists on {S = col, so Xs may not be defined on this event. If P[S = co] = 1, then Definition 5.1 reduces to Definition 3.1 for a weak solution to (5.1). We begin with a discussion of the time-change which will be employed in both the existence and uniqueness proofs.

A. The Method of Time-Change Suppose we have defined on a probability space a standard, one-dimensional

Brownian motion B = {Bs, FB; 0 < s < co} and an independent random variable with distribution p. Let {s) be a filtration satisfying the usual conditions, relative to which B is still a Brownian motion, and such that is q0-measurable (a filtration with these properties was constructed for Definition 2.1). We introduce TA

(5.7)

du 0 U 2g fs+

Bu);

< < CC,

a nondecreasing, extended real-valued process which is continuous in the topology of [0, co] except for a possible jump to infinity at a finite time s. From Problem 3.6.30 we have To A lim T, = co

(5.8)

a.s.

s t co

We define the "inverse" of (5.9)

Ts

by

0 < t < co

A, -4 inf{s > 0; T > t};

and

A3,

lim A,.

Whether or not Ts reaches infinity in finite time, we have a.s. (5.10)

Ao = 0,

A, < co; 0

t < co

and

Acc, = inf {s

0: T, = co}.

Because Ts is continuous and strictly increasing on [0, IL), A, is also continuous on [0, co) and strictly increasing on [0, TA,,_). Note, however, that if T, jumps

to infinity (at s = A), then (5.11)

A, = Az°, V t > TA._ ;

if not, then TA._ = TA.= co, and (5.11) is vacuously valid. The identities (5.12)

TA, = t;

0 5 t < T/c_,

5.5. A Study of the One-Dimensional Case (5.13)

331

0 < s < Aco,

AT= = s;

and

hold almost surely. From these considerations we deduce that 0; A, > sl;

'Ts = inflt

(5.14)

0 < s < oo, a.s.

In other words, A, and Ts are related as in Problem 3.4.5. Let us consider now the closed set dy

E

/(a) = { x E R; f_e

(5.15)

= co, V e > 01,

define

R A inf{s z 0;

(5.16)

+ Bs e 40)}.

5.2 Lemma. We have R = Acc, a.s. In particular, c R+

(5.17)

du

v2( + B.)

= co,

a.s.

PROOF. Define a sequence of stopping times

0; p( + B 1(a))

R,, A inf s

1

-1; n > 1,

where

p(x, 1(a)) A inf{ x - yl; ye/(a)}.

(5.18)

Because 1(a) is closed, we have limn, R = R, a.s. (recall Solution 1.2.7). For n > 1, set (x);

p(x,1(a)) z

a(x) = p(x, 1(a))
1, so <M>, = A,; 0 < t < co, a.s.

T - At A T =

(5.22)

that the process of (5.9) is given by

As the next step, we

A, =

(5.23)

t

62 (X )dv;

0 < t < co, a.s.

0

Toward this end, fix w e {R = 4,0}. For s < A,(w), (5.7) and (5.10) show that the function 7'(w) restricted to u e [0, s] is absolutely continuous. The change of variable v = 7;,(w) is equivalent to Ay(w) = u (see (5.12), (5.13)) and leads to the formula At(m)

(5.24)

A,(w) =

62(((o) + B.(0)dr,,((o) =

a2 (X (w)) dv,

valid as long as A,(w) < 21,0(w), i.e., t < T(w), where (5.25)

T

TA._ = inf { u > 0; A = A. }.

If t(w) = co, we are done. If not, letting t T t(w) in (5.24) and using the continuity of A(w), we obtain (5.23) for 0 < t < t(o)). On the interval [-r(w), co],

A00(co) = A.(w) = R(w). If R(w) < co, then

A,(w)

c(w) + BR(W)(w) E 1(a)

A',40(co)

Z(o-),

and so o-(X,(w)) = o-(X too(w)) = 0;

t(w) < t < co.

Thus, for t > t(co), equation (5.23) holds with both sides equal to 240.)(co). From (5.22), (5.23), and the finiteness of A, (see (5.10)), we conclude that

<M> is a.s. absolutely continuous. Theorem 3.4.2 asserts the existence of a Brownian motion 4V = 1147 0 < t < col and a measurable, adapted

process p = Ip At; 0 < t < col on a possibly extended probability space F), such that M, =

f

dITT,

<M>, =

f

py2 dv;

0

t < co , P a.s.

In particular, P[pi2 = 62(Xj) for Lebesgue a.e. t > 0] = 1. We may set

W=

ft

0

sgn(p09-(XJ)dIR;

0 < t < co;


334

observe that W is itself a Brownian motion (Theorem 3.3.16); and write

X,= + M, = + f t a(X)dW; 0 < t < co, P-a.s. 0

Thus, (X, W) is a weak solution to (5.19) with initial distribution ft. To prove the necessity of (E), we suppose that for every x e 64E, (5.19) has

a nonexploding weak solution (X, W) with X0 = x a.s. Here W = {W, ",; 0 < t < co} is a Brownian motion with W0 = 0 a.s. Then

0 < t < co

M, = X, - X0, ..17;,;

(5.26)

is in de.'" and <M>, =

(5.27)

I: a2(Xv)dv < co;

0 < t < oo, a.s.

o

According to Problem 3.4.7, there is a Brownian motion B = 14, W's; 0 s < co} on a possibly extended probability space, such that

M, = Boot; 0 < t < co, a.s.

(5.28)

Let Ts = inf{t _. 0; <M>, > s }. Then s A <M>co = <M> T., (Problem 3.4.5 (ii)),

so using the change of variable u = <M> (ibid. (vi)) and the fact that d<M> assigns zero measure to the set {v > 0; a2(X) = 0 }, we may write (5.29)

du

5 A <MX°

1<m)T.

j0

a2(X0 + B.)

=

f

du

a2(X0 + B.)

T,

ii; d<mx, j0 a2(X,,)

d<M>v l {,,,2(x d>, 0}

(x

T,

=

J.0

1 {2(x 0") dv _co > 0] > 0, and thus for sufficiently small positive s, P[T, < co, <MX° > 0] > 0. Now apply Lemma 3.6.26 with T(w

A <M>.(w); s;

if Ts(co) < oo, <M>.(w) > 0, otherwise.

We conclude from this lemma and (5.29) that x e 1 (o)`. It follows that /(u) g Z(a). 5.6 Remark. The solution (5.21) constructed in the proof of Theorem 5.4 is nonconstant up until the time (5.30)

t A inf.{ t > 0; X, e /(a)}.

(This definition agrees with (5.25).) In particular, if X0 = x e 1(o)` , then X is

5.5. A Study of the One-Dimensional Case

335

not identically constant. On the other hand, if x e Z(a), then Y = x also solves (5.19). Thus there can be no uniqueness in the sense of probability law if Z(a) is strictly larger than 1(a). This is the case in the Girsanov Example 2.15;

if a(x) = jxIa and 0 < a < (1/2), then 1(a) = 0 and Z(a) = {0}. Engelbert & Schmidt (1985) show that if a: IR -* IR is any Borel-measurable function satisfying 1(a) = 0, then all solutions of (5.19) can be obtained from the one constructed in Theorem 5.4 by "delaying" it when it is in Z(a). An

identically constant solution corresponds to infinite delay, but other, less drastic, delays are also possible.

5.7 Theorem. (Engelbert & Schmidt (1984)). For every initial distribution p, the stochastic differential equation (5.19) has a solution which is unique in the sense of probability law if and only if

(E + U)

1(a) = Z(cr).

PROOF. The inclusion 1(a) g Z(a), i.e., condition (E), is necessary for existence

(Theorem 5.4); in the presence of (E), the reverse inclusion is necessary for uniqueness (Remark 5.6). Condition (E) is also sufficient for existence (Theorem 5.4), so it remains only to show that the equality 1(a) = Z(a) is sufficient for uniqueness. We shall show, in fact, that the inclusion 1(a) Z(0) is sufficient for uniqueness. Assume 1(a) Z(a) and let (X, W), (n,F,p), {,,} be a weak solution

of (5.19) with initial distribution It. We define M as in (5.26) and obtain (5.27)-(5.29), where B is a standard Brownian motion. We also introduce via (5.30). By assumption,

a2(X,) > 0;

(5.31)

0

t < T, a.s.,

so <M> is strictly increasing on [0, t]. In particular, (5.32)

0; X0 + Bse 1(a)} = <M>t,

R A inf {s

a.s.

We set (5.33)

T, A int.{ t

0; <M>, >

so T is nondecreasing, right-continuous, and (5.34)

{TS

,}.

We claim that (5.35)

du JS

a2 (X0 + Bu)

-Ts ;

0 < s < <MX°, a.s.

In light of (5.29), (5.31), to verify this claim it suffices to show (5.36)

R

<M>., a.s.

Indeed, on the set {R < <M>,,,} we have by the definition (5.33) that TR < CO;

336


letting sIR in (5.29), we obtain then TR+ o

du

6.2(X0 + B)

on {R < <M> 1.

, we have Ts < co and so (5.37) follows immediately from (5.35). For s = <M> = R, it is clear from (5.33) that Ts = oo, and (5.37) follows from (5.17). It is then apparent that (5.37) also holds for s > <M> with both sides equal to infinity.

Comparing (5.37) and (5.7), we conclude that the discussion preceding Lemma 5.2 can be brought to bear. The process A defined in (5.9) coincides then with <M>, which is thus seen to be continuous and real-valued and to satisfy s+

<M>, = inf Is

du

> t } ; 0 t < co. fo u2(X0 + B) We now prove uniqueness in the sense of probability law by showing that the distribution induced by the solution process X on the canonical space (C[0, co), ,l(C[0, co))) is completely determined by the distribution it of Xo. The distribution induced on this space by the process X0 + B is completely (5.38)

0;

determined by it because B is a standard Brownian motion independent of X0 (Remark 3.4.10). For CO E C [0, Go), define 9.(w) to be the right-continuous process tp,(w) A inf Is

0;

iso+

du o-2(w(u))

> t}; 0 < t < co

with values in [0, co]. Because of its right-continuity, 9 is 4( [0, co)) ® .4(C[0, co))-measurable (Remark 1.1.14). Let 7C: [0, CO) x C [0, CO) -+ R be the measurable projection mapping tr,(co) = w(t). Consider the P. co)) ® M(C [0, co))-measurable process Co)) A co(0) + tr ,,,,(,,)(a)). We have (5.39)

X, = X0 + Boo, = 11/,(X0 + B.),

0

t < co,

and so the law of X is completely determined by that of X0 + B..

5.8 Remark. It is apparent from the proof of Theorem 5.7 that under the assumption 1(a) 2 Z(r), any solution to (5.19) remains constant after arriving in 1(o). See, in particular, the representation (5.39) and recall that <M>, is constant for t < t < CO.


337

Using local time for continuous semimartingales, it is possible to give a simple but powerful sufficient condition for pathwise uniqueness of the solution to equation (5.19).

5.9 Theorem. Suppose that there exist functions f: L -> [0, co) and h: [0, co) -> [0, co) such that: (i)

at every x el(a)t, the quotient (f/a)2 is locally integrable; i.e., there exists e > 0 (depending on x) such that

(f(Y))2 dy 0 such that

+ y) - a(x)I

f(x)klYD;

Vxe

y e [ - a, a].

Then pathwise uniqueness holds for the equation (5.19).

PROOF. Observe first of all that if a(x) = 0, then (ii) and (iii) imply that fe

dy

h'(y)dy = oo,

> 2f -2(x)

cf2(x +

V e > O.

0

Thus, Z(a) 1(a). Suppose now that X(`) = {XI°, .F.,; 0 < t < co}; i = 1, 2, are solutions to

0 < t < co} on (5.19) relative to the same Brownian motion W = {W, some probability space (SI, .F, P), with P[XP = XVI = 1. Defining -c(l) = inf{t > 0; XP) e /(a)}, we recall from Remark 5.8 that XP) = 'CPL.; 0 < t < co, a.s., so it suffices to prove that xp) )02); 0 < t < T(1), (5.40)

For each integer k > (1/a), there exists sk > 0 We set A, = XF) ) d = k. Using the continuity of the local time for A

for which 1,11c h-2

(Remark 3.7.8), (3.7.3), and assumption (iii), we may write for every random time r:

1 fs 1

(5.41)

lik

A°(0) = lim k

a0

h-2(x)M(x)dx

k

= lim k

t

0

1 ft

< lim -,, k-,,,, PC

0

h-2(A)1

f 2 (XV)) ds,

(As )(a(X(2))

- a(XP)))2 ds

a.s. P.

Now M, A XI" - 4 ) is in jr.loc, and so M admits the representation M, = B<m>, where B is a Brownian motion (Problem 3.4.7). Furthermore,


338

<M>, = focr2(X!,1))dv, and 62(X,")) > 0 for 0 < v < -r"), so a change of variable results in the relation <m>,

f

t

f 2 (XV)) dv =

o

f2041)

B.)0.-2(A-,i) o+

B,Jdu; 0 < t < -r").

o

We take T. A inf It > 0; p(X:1),I(o-))

-1

n},

R A <M>,. = inf { s > 0; p(X0 + B5,1(0)) < ;11}, where p is given by (5.18). We alter f and a near 1(6) by setting 1

f(x) = o-(x) = 1;

if p(x, /(o-))
1, a.s. Since t T -c" ) as n co, we conclude from the continuity of ANO) that A'.:`0)(0) = 0, a.s. P. Remark 3.7.8 shows that E I X).,,i, - X:1,),i)1 = 0; 0 < t < oo, and (5.40) follows.

We recall from Corollary 3.23 that the existence of a weak solution and pathwise uniqueness imply strong existence. This leads to the following result.

5.10 Corollary. Under condition (E) of Theorem 5.4 and conditions (i)-(iii) of Theorem 5.9, the equation (5.19) possesses a unique strong solution for every initial distribution p. 5.11 Remark. If the function f is locally bounded, condition (i) of Theorem 5.9 follows directly from the definition of 1(6). We may take h(y) = y"; a > (1/2),

and (iii) becomes the condition that a be locally Holder-continuous with exponent at least (1/2); see Examples 2.14 and 2.15.


339

B. The Method of Removal of Drift It is time to return to the stochastic differential equation (5.1), in which a drift term appears. We transform this equation so as to remove the drift and thereby reduce it to the case already studied. This reduction requires assumptions of nondegeneracy and local integrability: (ND) (LI)

a2 (x) > 0;

Vx

> 0 such that

V x e R, 3

e R,

x+(AI ' b dy

< co.

Under these assumptions, we fix a number c e R and define the scale function

exp { -2 14 b() 61} g x e R. a2g) The function p has a continuous, strictly positive derivative, and p" exists (5.42)

P(x)

x

almost everywhere and satisfies (5.43)

p"(x) -

2b(x)

0.2(x) p'(x).

Henceforth, whenever we write p", we shall mean the (locally integrable) function defined by (5.43) on the entire of R; this definition is possible because of (ND). The function p maps R onto (p( -co), p(co)) and has a continuously differ-

entiable inverse q: (p( -co), p(oo)) -> R. This latter function has derivative qf(y)= (1/ p'(q(y))), which is actually absolutely continuous, with (5.44)

P(q(Y)) "

y))2

p'(q(Y))

2b(q(y)) a2(q(y))

1

(q( y))) 2

a.e. in (p( -co), p(oo)).

We extend p to [-co, co] and q to [p(- co), p(co)] so that the resulting functions are continuous in the topology on the extended real number system.

5.12 Problem. Although not explicitly indicated by the notation, p(x) defined by (5.42) depends on the number c e R. Let us for the moment display this dependence by writing pc(x). Show that (5.45)

Pa(x) = pa(c) + PL(c)Pc(x).

In particular, the finiteness (or nonfiniteness) of pc(+ co) does not depend on the choice of c.

5.13 Proposition. Assume (ND) and (LI). A process X = {X Ft; 0 < t < oo} is a weak (or strong) solution of equation (5.1) if and only if the process

340


Y=Y

p(X,), r#,; 0 < t < col is a weak (or strong) solution of

=

(5.46)

+

f

0 < t < oo,

d(Ys)dWs;

where

p(-00) < Yo < p(00) a.s.,

(5.47) (5.48)

(y) =

p( -co) < y < p(co),

{p'(q(y))6.(q(y));

otherwise.

0;

The process X may explode in finite time, but the process Y does not.

PROOF. Let X satisfy (5.1) up to the explosion time S, define Y, = p(X,), and recall S from (5.2); we obtain from the generalized Ito rule (Problem 3.7.3) and (5.43): t

5,,

LAS = P(X0)

(5.49)

0 < t < co, n > 1.

6(Ys)dWs;

Because X is a continuous, extended-real-valued process defined for all 0 < t < co (Definition 5.1) and p: [-co, co] [p(-oo),p(co)] is continuous, Y is also continuous for all 0 < t < co. Hence, as n co, the left-hand side of (5.49) converges to Y n s, and the right-hand side must also have a limit. In light of Problem 3.4.11, this means that st0A s d2(Ys) ds < co; 0 < t < co, a.s. Because Xs, and hence Ys, is constant for S < s < co, we must in fact have Sto ei2(Ys)ds < co; 0 Lc. t < co, a.s. It follows that Po 6(Ys)d147, is defined for all

0 < t < cc and is a continuous process. From (5.49) we see that (5.46) holds for 0 _5 t < S. This equality must also hold for 0 < t S on the set {S < co}, because of continuity. The integrand a(Ys) vanishes for S < s < co, and so (5.46) in fact holds for all 0 < t < co. The solution of this equation cannot explode in finite time; see Problem 5.3. For the converse, let us begin with a process Y satisfying (5.46), (5.47) which

takes values in [p( -co), p(oo)]. We can always arrange for Y to be constant after reaching one of the endpoints of this interval, because the integrand in (5.46) vanishes when Ys (p( -co), p(oo)). Define X, A q(Y); 0 5 t < co, and let S be given by (5.2). Then the generalized Ito rule of Problem 3.7.3 gives, in conjunction with the properties of the function q (in particular, (5.44)) and (5.48), 1

Xt

S,,

= X0 ±

(107,$)dY.s ±

(1"117.0ds o

tAS

= X°

1

ei(Ys)dW, +

+

PVCs)

f

o

s- b(Xs)

1

472(Xs) (P'1Xs02

tAs,,

tAs,,

= X0 +

tA

b(Xs) ds +

almost surely, for every n

1.

cr(Xs)dWs; 0 0, so P[S = co] = 1. We turn now to the questions of existence of a weak solution, and of pathwise uniqueness, for equation (5.1). The assumptions on b and a imply (ND), (LI), and the local integrability of a', so weak existence follows from Theorem 5.15. According to Proposition 5.13, the pathwise uniqueness of (5.1) -(2b/o-2)p' is locally bounded, is equivalent to that of (5.46). Because p" p' is locally Lipschitz. It follows that ii(y) defined by (5.48) is locally Lipschitz at every point y E(p( -oo), p(oo)). We have shown that any solution X to (5.1) does not explode, so any solution Y to (5.46), (5.47) must remain in the interval (p( -co), p(co)). Under these conditions, the proof given for Theorem 2.5 shows that pathwise uniqueness holds for (5.46). We appeal to Corollary 3.23 in order to conclude the argument.

Proposition 5.13 raises the interesting issue of determining necessary and sufficient conditions for explosion of the solution X to (5.1). Since Y given

by (5.46) does not explode, and Y, = p(X,), it is clear that the condition p(+cc) = +oo guarantees that X is also nonexploding; however, this sufficient condition is unfortunately not necessary (see Remark 5.18). We develop the necessary and sufficient condition known as Feller's test for explosions in Theorem 5.29.

5.18 Remark. Consider the case of b(x) = sgn(x), a(x)

a > 0. The scale

function p of (5.42) is bounded, and according to Proposition 5.17 the equation (5.1) has a nonexploding, unique strong solution for any initial distribution.

5.19 Remark. The linear growth condition I b(x)I + I a(x)I < K(1 + Ix I );

VX E R

is sufficient for P[S = co] = 1; cf. Problem 3.15.

C. Feller's Test for Explosions We begin here a systematic discussion of explosions. Rather than working exclusively with processes taking values on the entire real line, we start with an interval 1 = (1",r);

-oo 0;

V x e /, 3 e> 0

(LI)'

such that

V x e l,

ix", + I b(y)I

J.-.

dy < co.

a2(Y)

We define the scale function p by (5.42), where now the number c must be in I. We also introduce the speed measure m(dx) 4

(5.51)

2 dx p'(x)a 2 (X) '

xe I

and the Green's function (5.52)

Ga,b(x, .Y) =

(p(x A y) - p(a))(p(b) - p(x v y)) ; x, y E [a, b] g_ I. p(b) - p(a)

In terms of these two objects, a solution to the equation (5.53)

b(x)M'(x) + lo-2(x)M"(x) = - 1; a < x < b,

(5.54)

M(a) = M(b) = 0

is given by

i G,,,(x, y)m(dy) to

(5.55) M,(x) A-

a -1.

=

(P(x) - P(.0)m(dY) +P(x)

P(a) f b (PO) - P(A)m(dY)

p(b) - p(a)

a

a

As was the case with p", the second derivative Ma",6 exists except possibly on a set of Lebesgue measure zero; we define M,". at every point in (a, b) by using (5.53). Note that G., be , ), and hence Ma.b(), is nonnegative.

5.20 Definition. A weak solution in the interval I = (1', r) of equation (5.1) is a triple (X, W), (Si, ,F, P), {..-,}, where

(i)' condition (i) of Definition 5.1 holds, (ii)' X = {X ..*--,; 0 < t < co} is a continuous, adapted, [t, r]-valued process with X0 e / a.s., and { W A; 0 < t < col is a standard, one-dimensional Brownian motion, (iii)' with {e}n°°_,, and {r},7,, strictly monotone sequences satisfying t < to < r < r, lim,,,, f = t, iim 00 rn = r, and (5.56)

S A inf{t z 0: X, it (e, r)}; n > 1,

the equations (5.3) and (5.4) hold. We refer to (5.57)

S = inf { t

__

0: X, 0 (e , 0} = tun s n-co

344


as the exit time from I. The assumption X0 E I a.s. guarantees that P[S > 0] = 1. If = -co and r = +co, Definition 5.20 reduces to Definition 5.1, once we

stipulate that Xi = Xs; S < t < cc. Let (X, W) be a weak solution in 1 of equation (5.1) with X0 = X E (a, b) and set

t = inf { t

_

.

o-2(Xs)ds> n

0:

;

I,

n = 1, 2, ...,

o

Tab= inf {t >0;Xt (a, b)};

e < a < < r.

We may apply the generalized Ito rule (Problem 3.7.3) to M,,,b(X,) and obtain t A T A T.,b

Ma,b(Xt A T A T,,,b) = Ma,b(X)

(t A Tr, A 1:7,0 ± J

Nra,b(X.00-(Xs)d o

co, we see that

Taking expectations and then letting n (5.58)

E(t A Tad)) = Ma,b(X)

Eltla,b(XtA T b) 5 M a b(X) < 00,,

and then letting t co we obtain ET,b < Ma.b(x) < co. In other words, X exits from every compact subinterval of ((,r) in finite expected time. Armed with this observation, we may return to (5.58), observe from (5.54) that limi, EM,,,b(Xt A Tab) = 0, and conclude ET,b = Ma,b(x);

(5.59)

a < x < b.

On the other hand, the generalized Ito rule applied in the same way to p(X,) gives p(x) = Ep(X, A Tab), whence (5.60)

p(x) = Ep(XT.,,b)= p(a)P[XTo.b = a] + p(b)P[XT..b = b],

co. The two probabilities in (5.60) add up to one, and thus

upon letting t (5.61)

P[XT = a] = a.13

p(b) - p(x)

P[XT = b] =

p(b)- p(a)'

p(x)- p(a) p(b) - p(a)

These expressions will help us obtain information about the behavior of X near the endpoints of the interval (t,r) from the corresponding behavior of the scale function. Problem 5.12 shows that the expressions on the right-hand sides of the relations in (5.61) do not depend on the choice of c in the definition of p.

5.21 Remark. For Brownian motion Won I = (-co, co), we have (with c = 0 in (5.42)) p(x) = x, m(dx) = 2 dx. For a process Y satisfying

=+

d(K)diVs, 0

we again have p(R) = g, but m(dx) = (2 dife12(50). Now Y is a Brownian motion run "according to a different clock" (Theorem 3.4.6), and the speed measure simply records how this change of clock affects the expected value of


345

exit times: (5.62)

- (5-c v y)) 2 di)

ET'a,s(5e) = f b ((x A

b-a

(5:2

Once drift is introduced, the formulas become a bit more complicated, but the idea remains the same. Indeed, suppose that we begin with X satisfying (5.1), compute the scale function p and speed measure m for X by (5.42) and (5.51), and adopt the notation g = p(x), y = p(y), a = p(a), b = p(b); then (5.55), (5.59) show that ET,,,b(x) is still given by the right-hand side of (5.62), where now if is the dispersion coefficient (5.48) of the process Y, A p(Xi). We say Y, = p(X,)

is in the natural scale because it satisfies a stochastic differential equation without drift and thus has the identity as its scale function. 5.22 Proposition. Assume that (ND)', (LI)' hold, and let X be a weak solution of (5.1) in I, with nonrandom initial condition X0 = x e I. Let p be given by (5.42) and S by (5.57). We distinguish four cases:

(a) pv +) = -co, p(r -) = co. Then

P[S = co] =

P[ sup X, =

=

P[

inf X, =

ot -co, p(r -) = co. Then

P[ sup X, < 7.1= its (c) p(1 +) = -co, p(r-) < co. Then P[lim X, =

P[lim X, =

P[

inf X, >

1.

0 -co, p(r -) < co. Then

P[lim X =

its

1.

(1=1 - P[lim X = r] = p(r -) - p(x)

P(r -) - p(e +).

its

5.23 Remark. In cases (b), (c), and (d), we make no claim concerning the finiteness of S. Even in case (d), we may have P[S = co] = 1, as demonstrated in Remark 5.18. PROOF OF PROPOSITION 5.22. For case (a), we have from (5.61) for 1' < a
0; X, = c }. The continuous process ME

A

T.,. =

e-(t A A Tc)I4(X

t ASA Tc);

2 and c = 1, we have v(x) = [x2 - 1 - 2p(x)]/d; 0 < x < co, and v(0 + ) = v(oo) = oo.

5.31 Exercise. Consider the geometric Brownian motion X, which satisfies the stochastic integral equation (5.79)

X,=x+ttf

Xsds + v

Xsdl/K,

where x > 0. Use Theorem 5.29 and Proposition 5.22 to show that X, e (0, oo) for all t, and (i) (ii) (iii)

if it < v2/2, then X, = 0, suPo < X, < oo, a.s.; if it > v2/2, then info. 0, X, = co, a.s.; if it= v2/2, then info 0.

zT e-tA

n!

n=0

The singularity of M(T) follows from zT M(T)z = 0. Let us now assume that for some T > 0, M(T) is singular and show that rank (C) < d. The singularity of M( T)enables us to find a nonzero vector z E Rd such that

0 = zT M(T)z =

From this we see that f(t)

f

lIzTe-mo-112 dt.

is identically zero on [0, T], and

zT e-"`a

evaluating f(0), f'(0),...,f(d-1)(0) we obtain (6.18). When A and a are constant, equations (6.13) and (6.14) take the simplified form V(t) = AV(t) + V(t)AT

(6.13)' (6.14)'

V(t) =

e" [V(0) +

aaT,

--sAcro.re-sAT ds

ll

etAT.

One could hope that by proper choice of V(0), it would be possible to obtain

a constant solution to these equations. Under the assumption that all the eigenvalues of A have negative real parts, so that the integral VA

(6.19)

esito.o.resAT ds

converges, one can verify that V(t) V does indeed solve (6.13)', (6.14)'. We leave this verification as a problem for the reader. 6.6 Problem. Show that if V(0) in (6.14)' is given by (6.19), then V(t) = V(0). In particular, V of (6.19) satisfies the algebraic matrix equation AV + VAT = --o-o-T.

(6.20)

We have established the following result.

6.7 Theorem. Suppose in the stochastic differential equation (6.1) that a(t) a a, a(t) 0, all the eigenvalues of A(t) A have negative real parts, and the initial random vector = X0 has a d-variate normal distribution with mean m(0) = 0 and covariance V = E(X0.11) as in (6.19). Then the solution X is a stationary, zero-mean Gaussian process, with covariance function (6.21)

p(s,

=

Ve(`-"T; (e("-"V;

0 < t < s < op 0 5 s < t < co.

358


PROOF. We have already seen that V(t) = V; (6.21) follows from (6.14)' and (6.11).

6.8 Example (The Ornstein-Uhlenbeck Process). In the case d = r = 1, a(t) .m0, A(t) = -a < 0, and a(t) a > 0, (6.1) gives the oldest example

dX, = -aXidt + a dW,

(6.22)

of a stochastic differential equation (Uhlenbeck & Ornstein (1930), Doob (1942), Wang & Uhlenbeck (1945)). This corresponds to the Langevin (1908)

equation for the Brownian motion of a particle with friction. According to (6.6), the solution of this equation is X, = X 0e-Ga + o-

ft e-2"-s) dWs;

0 < t < co.

o

If EX,!, < co, the expectation, variance, and covariance functions in (6.7)-(6.9) become

m(t) A- EX, = m(0)e', 0.2

0.2

+ (V(0)

V(t) A Var(X,) =

- -2a) e-2', 0.2

p(s,t) A Cov(X Xt) = [V(0) + - (e2'(` ^ s) - Me-2(f'). 2a

If the initial random variable X0 has a normal distribution with mean zero and variance (a2/2a), then X is a stationary, zero-mean Gaussian process with covariance function p(s, t) = (a2/2a)e-'1".

B. Brownian Bridge Let us consider now the one-dimensional equation (6.23)

dX, =

TT

X, t

dt + dWi;

0 < t < T, and Xo = a,

for given real numbers a, b and T > 0. This is of the form (6.1) with A(t) = -11(T - t), a(t) = b /(T - t) and a(t) ._ 1, whence OW = 1 - (t /T). From (6.6) we have

X,=a(1-

-t) t)

t

+b-T +(T

dig -Of Ts; 0 1) stochastic differential equation (6.30)

dX, =

(t) X, + a(t)] dt +

[S;(t)X, + o-j(t)] i =i

, W,(6), 0 < t < co} is an r-dimensional Brownian motion, and the coefficients A, a, Si, oi are measurable, {. }-

where W = {141, = (Him,

adapted, almost surely locally bounded processes. We set 1

ift

ct

J=1

(6.31)

Zt 4- exp

o

S(u)dW°) "

r

- -2 j=1

S.12(u)du, To

[ft A(u) du +

6.15 Problem. Show that the unique solution of equation (6.30) is

5.6. Linear Equations

(6.32)

361

X, = Z, [X0 +

f

1

- la(u) 0

S;(u)o-,(u)} du + 1=1

.1=1

Wol f af(u)d Zu "

In particular, the solution of the equation

dX, = A(t)X,dt +

(6.33)

S;(t)X,dW,u) J=1

is given by (6.34)

X, = Xoexp[f {A(u) -

,

1

2,

Si (u)} au

.1=

f

Si(u) dW,,u)].

0

In the case of constant coefficients A(t) = A, S;(t) S; with 2A < E;=, s.,2 in (6.34), show that lim X, = 0 a.s., for arbitrary initial condition Xo.

D. Supplementary Exercises 6.16 Exercise. Write down the stochastic differential equation satisfied by Y = X1`; 0 < t < co, with k > 1 arbitrary but fixed and X the solution of equation (5.79). Use your equation to compute E(X,k). 6.17 Exercise. Define the d-dimensional Brownian bridge from a to b on [0, T] (a, b e Rd) to be any almost surely continuous process defined on [0, T], with finite dimensional distributions specified by (6.28), where now iix

p(t; x, y) = (27rt)-(d/2) exp

Yil 2

2t

J

x, y e

t > O.

(i) Prove that the processes X given by (6.26) and BO given by (6.29), where W is a d-dimensional Brownian motion with Wo = 0 a.s., are d-dimensional Brownian bridges from a to b on [0, T]. (ii) Prove that the d-dimensional processes fig'; 0 < t < T} and { Br: ; 0 < t < T} have the same law. (iii) Show that for any bounded, measurable function F: C[0, T]' R, we have (6.35)

EF(a + W.) =

f

EF(B!')p(T; a, b)db.

ud

6.18 Exercise. Let (I): R be of class C2 with bounded second partial derivatives and bounded gradient V(I), and consider the Smoluchowski equation (6.36)

dX, = V(1)(X,)dt + dWi;

0 < t < co,

362


where W is a standard, Rd-valued Brownian motion. According to Theorems 2.5, 2.9 and Problem 2.12, this equation admits a unique strong solution for every initial distribution on Xo. Show that the measure u(dx) = e2G(x)dx

(6.37)

on a(Rd)

is invariant for (6.36); i.e., if X(') is the unique strong solution of (6.36) with initial condition Xr = a e Rd, then (6.38)

µ(A) =

f

P(X:")e A) p(da);

V A e M(0')

holds for every 0 < t < (Hint: From Corollary 3.11 and the Ito rule, we have (6.39)

Ef Ma)) = E[f(a +

exp Istro(a +

- (1)(a)

- 1 f (AO + IIV(1)112)(a +

ds}1

for every f e C43(01°). Now use Exercise 6.17 (ii), (iii) and Problem 4.25.)

6.19 Remark. If d = 1 in Exercise 6.18, the speed measure of the process X is

given by m(dx) = 2 exp{ - 20(c)}tt(dx) and is therefore invariant. Recall Exercise 5.40.

6.20 Exercise (The Brownian Oscillator). Consider the Langevin system d X, = Y, dt

dY, = - X, dt - aY, dt + o- dW where W is a standard, one-dimensional Brownian motion and /3, a, and a are positive constants. (i) Solve this system explicitly.

(ii) Show that if (X0, Yo) has an appropriate Gaussian distribution, then (X 171) is a stationary Gaussian process. (iii) Compute the covariance function of this stationary Gaussian process.

6.21 Exercise. Consider the one-dimensional equation (6.1) with a(t) = 0, a(t) a > 0, A(t) < -a < 0, V 0 t < cc, and = x e R. Show that 0.2

E X,2
O.

= (W,(1), Wt(r)),,t; 0 < t < co} be an rdimensional Brownian motion, and let A(t), S(P)(t); p = 1, . . . , r, be adapted, bounded, (d x d) matrix-valued processes on [0, T]. Then the matrix stochas-

6.22 Exercise. Let W =

5.7. Connections with Partial Differential Equations

363

tic integral equation (6.40)

X(t) = I +

f A(s) X (s)ds +

J p=1

t

S(P)(s)X (s)dW,(P)

o

has a unique, strong solution (Theorems 2.5, 2.9). The componentwise formulation of (6.40) is d

Xki(t) = oki + E

t

d

snsp,;(s)dws(P).

A ke(s)X o(s) ds + E E p=11=1

C =1

0

Show that X(t) has an inverse, which satisfies (6.41)

X-1(t) = I + I X -1(s)[

-E P =1

(S(P)(s))2 - A(s)ids P -1

o

f x-i(s)s(P)(s) d HIP) 0

5.7. Connections with Partial Differential Equations The connections between Brownian motion on one hand, and the Dirichlet and Cauchy problems (for the Poisson and heat equations, respectively) on the other, were explored at some length in Chapter 4. In this section we document analogous connections between solutions of stochastic differential equations, and the Dirichlet and Cauchy problems for the associated, more general elliptic and parabolic equations. Such connections have already been presaged in Section 4 of this chapter, in the prominent role played there by the differential operators a', and (a /at) + si as well as in the relevance of the Cauchy problem to the question of uniqueness in the martingale problem (Theorem 4.28).

In Chapter 4 we employed probabilistic arguments to establish the existence and uniqueness of solutions to the Dirichlet and Cauchy problems considered there. The stochastic representations of solutions, which were so useful for uniqueness, will carry over to the generality of this section. As far as existence is concerned, however, the mean-value property for harmonic

functions and the explicit form of the fundamental solution for the heat equation will no longer be available to us. We shall content ourselves, therefore, with representation and uniqueness results, and fall back on standard references in the theory of partial differential equations when an existence result is needed. The reader is referred to the notes for a brief discussion of probabilistic methods for proving existence. Throughout this section, we shall be considering a solution to the stochastic integral equation

364

5. Stochastic Differential Equations s

(7.1)

X!") = x+ f 6(0, XV))d0 +

o-(0, Xl,"))dW,;

t s s
0:

ET A ID) _.._ h(x) -

Ex h(X, A,D) s 2 max ih(y)1 < 00.

yD

Let t -> c0 to obtain (7.7).

El

75 Remark. Condition (7.9) is stronger than ellipticity but weaker than uniform ellipticity in D. Now suppose that in the open bounded domain D, we have that (i) sad is uniformly elliptic, (ii) the coefficients au, b,, k, g are Holder-continuous, and (iii) every point a e OD has the exterior sphere property; i.e., there exists a ball B(a) such that B(a) n D = 0, B(a) n 8D = {a}.

We also retain the assumption that

f is continuous on

OD.

Then there exists


366

a function u of class C(D) n CZ(D) (in fact, with Holder-continuous second partial derivatives in D), which solves the Dirichlet problem (7.5), (7.6); see Gilbarg & Trudinger (1977), p. 101, Friedman (1964), p. 87, or Friedman (1975), p. 134. By virtue of Proposition 7.2, such a function is unique and is given by (7.8).

B. The Cauchy Problem and a Feynman-Kac Representation With an arbitrary but fixed T > 0 and appropriate constants L > 0, 2 > 1, we consider functions f (x): Rd -, R, g(t, x): [0, T] x08° -, 01 and k(t, x): [0, T] x08° -.10, co) which are continuous and satisfy (7.10)

(i)

1./(x)1 < L(1 + 11421)

(ii) f(x) > 0; V xe Rd

Or

as well as (7.11)

L(1 + 11421)

(i) Ig(t,x)I

or (ii) g(t, x) > 0;

V 0 < t < T, x e Rd.

We recall also the operator .21, of (4.1), and formulate the analogue of the Feynman-Kac Theorem 4.4.2:

7.6 Theorem. Under the preceding assumptions and (7.2)-(7.4), suppose that v(t, x): [0, T] x08° - 08° is continuous, is of class CL2([0,T) x08°) (Remark 4.1), and satisfies the Cauchy problem av

(7.12)

--at + kv = At) + g; in [0,T) x08°,

(7.13)

v(T, x) . f(x); x e01°,

as well as the polynomial growth condition (7.14)

max Iv(t, x)1 0

The killed diffusion process is (i.x) A

Txp -); t < s < p("), A;

s y po-x),

370


where A is a cemetery state isolated from Rd. Assume the conditions of Theorem 7.6 and let G(t, x; r, have

be a fundamental solution of (7.21). Then we

(7.30) P[X!" e A] = J G(t, x; r, g; A e .4(GV), 0 < t < r < T, A

and the solution (7.15) of the Cauchy problem (7.12), (7.13) takes the form (7.29).

7.11 Exercise. Suppose that bi(t, x); 1 < i < d, are uniformly bounded on [0, T] x R' and that f(x) and g(t, x) satisfy (7.10) and (7.11), respectively. If v(t, x) is a solution to the Cauchy problem at1

= Av + (b, Vv) + g; in [0, T) x Rd

-

v(T, x) = f(x);

xe

and (7.20) holds, then T-t

v(t, x) -= Ex[f(W7-_,) exp 1.

(b(t + 0, WO, dWo) o

+ 0, W0)112

t

d0}

o

T-t

g(t + s, Ws) exp

-2 -1

where {W .F,; 0 < t < T },

f

Il b(t 0

8

(b(t + 0, Wo), d Wo)

+

Wo)112

d0} ds],

Px 1.E Re is a d-dimensional Brownian

(CI,

family.

7.12 Exercise. Write down the Kolmogorov forward and backward equations with k 0 for one-dimensional Brownian motion with constant drift it, and

verify that the transition probability density of this process satisfies these equations in the appropriate variables. 7.13 Exercise. Let the coefficients b, a in (7.1) be independent of t, and assume that condition (7.9) holds for every open, bounded domain D c Rd. Suppose also that there exists a function f : R° \ {0} R of class C2, which satisfies (7.31)

and is such that F(r)

df(x) 5 0

on Rd\ {0}

is strictly increasing with

(i) Show that we have the recurrence property

F(r) = co.

5.8. Applications to Economics (7.32)

371

Px(T, < co) = 1;

for every r > 0, where B, = {X G Rd; (ii) Verify that (7.32) holds in the case (7.33)

(x, b(x)) +

1>

aii(x)

II

\B,

Vxe

X II

< r} and T,. = inf It

(x, a(x)x);

0; X, E B, 1.

V x E Rd\{0}.

1x112

(iii)

If (7.31) is strengthened to sif(x) < -1; V x e Rd\ {0}, then we have the positive recurrence property (7.34)

ExT, < 00

VXE

\ fir.

5.8. Applications to Economics In this section we apply the theory of stochastic calculus and differential equations to two related problems in financial economics. The first of these is option pricing, where we derive the celebrated Black & Scholes (1973) option pricing formula. The second application is the optimal consumption/investment

problem formulated by Merton (1971). These problems are unified by their reliance on the theory of stochastic differential equations to model the trading of risky securities in continuous time. In the second problem, this theory allows us to characterize the value function and optimal consumption process in a context more general than considered heretofore. We subsequently specialize the model to the case of constant coefficients, so as to illustrate the use of the Hamilton-Jacobi-Bellman equation in stochastic control.

A. Portfolio and Consumption Processes Let us consider a market in which d + 1 assets (or "securities") are traded continuously. We assume throughout this section that there is a fixed time horizon 0 < T < oc. One of the assets, called the bond, has a price Po(t) which evolves according to the differential equation (8.1)

dPo(t) = r(t)P0(t)dt, P0(0) = po;

0 < t < T.

The remaining d assets, called stocks, are "risky"; their prices are modeled by the linear stochastic differential equation for i = 1, .. , d: (8.2)

dPi(t) = bi(t)Pi(t)dt + PM)

Pi(0) = pi;

0 0; 0 < t < T and

= fT are seen to hold almost surely.

1=1

8.13 Exercise. Show that the hedging strategy constructed in the proof of Theorem 8.12 is essentially (in the sense of meas x P-a.e. equivalence) the only

hedging strategy corresponding to initial wealth x = EQ. In particular, the process X of (8.32) gives the unique wealth process corresponding to the fair price; it is called the valuation process of the contingent claim.

8.14 Example (Black & Scholes (1973) option valuation formula). In the setting of Remark 8.8 with d = 1 and constant coefficients r(t) r > 0, 611(t) a- o- > 0, the price of the bond is

Po(t) = poe"; 0 < t < 'T,

and the price of the stock obeys dP, (t) = 61 (t)P, (t) dt + o- Pi(t)(114/, = r P,(t) dt + o-

(t)

For the option to buy one share of the stock at time T at the price q, we have from (8.32) the valuation process (8.33) X, E[e-rci--0(131(T) ig) + 1 Al 0 T.

-

In order to write (8.33) in a more explicit form, let us observe that the function (8.34) v(t, x)

(T - t, x)) - qe-r(T-1)(1)(p_(T - t, x));

0 < t < T, x > 0,

t= T, x > 0

(x - q)+;

with p+ (t, x) =

1

o-.1i

[log x + t (7- +

0-2

)], 2

4:1)(x) =

f ,./27r

x

e -op

12 dz,

5.8. Applications to Economics

379

satisfies the Cauchy problem all

(8.35)

Ot

+ ry =1 a2 x2 2

02

ex

+

rxO-x v ;

x

0,

v(T, x) = (x - q)+ ;

e

on [0, T) x (0, oc,)

as well as the conditions of Theorem 7.6. We conclude from that theorem and the Markov property applied to (8.33) that (8.36)

X, = v(t,P,(t)); 0 < t

T,

a.s.

We thus have an explicit formula for the value of the option at time t in terms of the current stock price P, (t), the time-to-maturity T - t, and the exercise price q.

8.15 Exercise. In the setting of Example 8.14 but with fT = h(P, (T)), where h: [0, co) [0, co) is a convex, piecewise C2 function with h(0) = h'(0) = 0, show that the valuation process for the contingent claim (0,fT) is given by (8.37)

X, = E[cr(T-')h(P,(T))1.97,1 = i

h"(9)1,,,T(t, P, (t)) dq.

We denote here by ty,,,T(t, x) the function of (8.34).

C. Optimal Consumption and Investment (General Theory) In this subsection we pose and solve a stochastic optimal control problem for the economics model of Subsection A. Suppose that, in addition to the data

given there, we have a measurable, adapted, uniformly bounded discount process 13 = {Ns), Fs; 0 < s < T} and a strictly increasing, strictly concave, continuously differentiable utility function U: [0, co) - [0, co) for which

U(0) = 0 and U'( co) --' lime U'(c) = 0. We allow the possibility that [r(0) A limy, 0 U'(c) = co. Given an initial endowment x > 0, an investor wishes to choose an admissible pair (ir, C) of portfolio and consumption processes, so as to maximize T

V,,,e(X) A E

1 e-ilk")"U(COds. o

We define the value function for this problem to be (8.38)

V(x) = sup V,c(x), (n,c)

where the supremum is over all pairs (n, C) admissible for x. From the admissibility condition (8.23) it is clear that V(0) = 0. Recall from Proposition 8.6 that for a given consumption process C, (8.23) is satisfied if and only if there exists a portfolio it such that (n, C) is admissible for x. Let us define 2(x) to be the class of consumption processes C for which

380

5. Stochastic Differential Equations T

.1e-P.'")dsC,dt = x.

(8.39)

E

o

It turns out that in the maximization indicated in (8.38) we may ignore the portfolio process it and we need only consider C e 9(x). 8.16 Proposition. For every x > 0 we have T

V(x) = sup E C e fd(x)

e-f 0 t3(s)ds U(C,) dt. 0

PROOF. Suppose (it, C) is admissible for x > 0, and set

y '' E

i

T

e-J°'(."' Ctdt < x.

0

If y > 0, we may define C, = (x/y)C, so that 0 e g(x). There exists then a portfolio process A such that (A, 0) is admissible for x, and V,c(x) < Vi,e(x).

(8.40)

If y = 0, then: C, = 0; a.e. t e [0, T], almost surely, and we can find a constant

c > 0 such that 0, .--- c satisfies (8.39). Again, (8.40) holds for some it chosen so that (A, 0) is admissible for x.

1=1

Because U': [0, co] '''' [0, U'(0)] is strictly decreasing, it has a strictly decreasing inverse function 1: [0, U'(0)] --4onto [0, co]. We extend I by setting

1(y) = 0 for y > U'(0). Note that /(0) = co and /(x)) = 0. It is easily verified that (8.41)

U(I(y)) - yl(y)

U(c) - yc; 0 < c < op, 0 < y < oo.

Define a function X: [0, co] - [0, co] by E

(8.42)

T

e-1,. r(u) du

I (yZsef°(fl(")-1( "))")ds,

0

and assume that (8.43)

X(y) < co;

0 < y < Do.

We shall have more to say about this assumption in the next subsection, where

we specialize the model to the case of constant coefficients. Let us define y 4- sup{ y

0; .1. is strictly decreasing on [0, y]}.

8.17 Problem. Under condition (8.43), .i" is continuous and strictly decreasing

on [0, y] with X(0) = co and gly) = 0. to : [0, co] -o-4 [0, y] be the inverse of .?C. For a given initial endowment x > 0, define the processes

Let


381

(8.44)

r/: g V(x)zsePo (poo-rom du

(8.45)

C: A join.

The definition of -N implies C* eg(x). We show now that C* is an optimal consumption process.

8.18 Theorem. Let x > 0 be given and assume that (8.43) holds. Then the consumption process given by (8.45) is optimal:

V(x) = E

(8.46)

CP° P(s)ds U(Ct*)dt. 0

PROOF. It suffices to compare C* to an arbitrary C e g(x). For such a C, we have E

e-S°P(s)ds(U(Ct*)

=E

- U(Ct))dt

CS° P(s)'[(U(I(e)) - ri7 Re)) - (U(C,) - ri7Ct)]dt V(X)E

e-f`o r(s)ds (ct)I,

Ct)dt.

The first expectation on the right-hand side is nonnegative because of (8.41); the second vanishes because both C* and C are in (x).

Having thus determined the value function and the optimal consumption process, we appeal to the construction in the proof of Proposition 8.6 for the determination of a corresponding optimal portfolio process 7E*. This does not provide us with a very useful representation for n *, but one can specialize the model in various ways so as to obtain V, C* and ir* more explicitly. We do this in the next subsection.

D. Optimal Consumption and Investment (Constant Coefficients) We consider here a case somewhat more general than that originally studied

by Merton (1971) and reported succinctly by Fleming & Rishel (1975), pp. 160-161. In particular, we shall assume that U is three times continuously differentiable and that the model data are constant: (8.47)

fl(t)

13,

r(t)

r,

b(t)

b,

Q(t) = a,

where b ege and a is a nonsingular, (d x d) matrix. We introduce the linear, second-order partial differential operator given by


382 IAP(t,Y)

- (Pt(t, y) + i3P(t,.11) - 113 - 031(Py(t, y) - 111011 2.Y2(P(t,37),

where B = o-' (b - r1) in accordance with (8.16). Our standing assumption throughout this subsection is that B is different from zero and there exist C' '3 functions G: [0, T] x (0, co) [0, oo ) and S: [0, T] x (0, co) [0, co) such that LG(t, y) = U (1 (y));

(8.48)

y>0

G(T, y) = 0;

(8.49)

0 < t < 'T, y > 0

and

0 < t < T, y > 0

LS(t, y) = y1 (y);

(8.50)

y > O.

S('T, y) = 0;

(8.51)

Here we mean that G,(t, y), Gty(t, y), Gy(t, y), Gyy(t, y), and Gyyy(t, y) exist for all

0 < t < T, y > 0, and these functions are jointly continuous in (t, y). The same is true for S. We assume, furthermore, that G, Gy, S, and Sy all satisfy polynomial growth conditions of the form (8.52)

max H(t, y) < M(1 + y -'' + y A); 1)

y>0

T

for some M > 0 and A > O.

8.19 Problem. Let H: [0, T] x (0, co) -, [0, co) be of class C I 2 on its domain and satisfy (8.52). Let g: [0, T] x (0, co) [0, co) be continuous, and assume that H solves the Cauchy problem LH (t, y) = g(t, y);

H('T, y) = 0;

0 < t < 7; y > 0 y > O.

Then H admits the stochastic representation

I

H(t, y) = E

T

e-fl(s-`)g(s,17,(`'''))ds,

t

where, with t < s < T: (8.53) (8.54)

ys(1,y) A ye(fl -rgs-t) Zt

4 4 exp{ -0T(Ws - Wt) - 1110112(s - t)}.

(Hint: Consider the change of variable e = log y.) From Problem 8.19 we derive the stochastic representation formulas (8.55)

G(t, y) = E

i

T

e-134-`)U (1(17,("))ds,

t

(8.56)

S(t, y) = yE

f t

T

e-r(s-`)Z's I (V' Y)) ds.


383

It is useful to consider the consumption/investment problem with initial times other than zero. Thus, for 0 < t 5 T fixed and x > 0, we define the value function V(t, x) = sup E

(8.57)

0, ,c)

f

fisU (Cs) ds,

r

where (it, C) must be admissible for (t, x), which means that the wealth process determined by the equation (8.58)

(r X - C) du + E

Xs = x + d

f

(b1 - r)tri(u) du

d

+EE

dW,P);

s < T,

t

r

remains nonnegative. Corresponding to a consumption process C, a portfolio process it for which (7r, C) is admissible for (t, x) exists if and only if (cf. Proposition 8.6) E

(8.59)

f

T

e-r(s-`)Zst Cs ds < x.

$

For 0 < t < T, define a function X(t, ): [0, oo] X(t, y) 9 E

(8.60)

J T

[0, co] by

e-r(s-`)41(Y,"Y))ds.

Comparison of (8.56) and (8.60) shows that

y) = S(t, y) < co; 0 < y < oo.

(8.61)

Now y(t) g sup It > 0; X(t, ) is strictly decreasing on [0, y] = co, and we have just as in Problem 8.17 that for 0 < t < T, -) is strictly decreasing on [0, co] with .X(t, 0) = co and (t, co) = 0. We denote by [0, co] the inverse of X(t, ): (8.62)

W(t,

y)) = y;

0

y

co,

): [0, co] -°'

0 0.

(8.66)

Thus, if we can solve the Cauchy problems (8.48), (8.49) and (8.50), (8.51), then we can express V(t, x) in closed form.

8.20 Exercise. Let U(c) = kA

cb,

where 0 < 6 < 1. Show that if

1

1

r6

1-

2

II 0112

1-

is nonzero, then GO,

= -1(1

- e-ku-_,))(yro-t)

S(t, y) = 6G(t, y), e-k( V(t,

If

k = 0,

then

V(t,x) = e-NT -

= e-13`

-6

)5.

k

G(t, y) = (T - t)(1)°1(",

S(t, y) = 45G(t, y),

and

-6 X b.

Although we have the representation (8.66) for the value function in our consumption/investment problem, we have not as yet derived representations for the optimal consumption and portfolio processes in feedback form, i.e., as functions of the optimal wealth process. In order to obtain such representa-

tions, we introduce the Hamilton-Jacobi-Bellman (HJB) equation for this model. This nonlinear, second-order, partial differential equation offers a characterization of the value function and is the usual technique by which stochastic control problems are attacked. Because of its nonlinear nature, this equation is typically quite difficult to solve. In the present problem, we have already seen how to circumvent the HJB equation by solving instead the two linear equations (8.48) and (8.50). 8.21 Lemma (Verification Result for the HJB Equation). Suppose Q: [0, T] x [0, co) -> [0, co) is continuous, is of class C1'2([0, T) x (0, co)), and solves the HJB equation (8.67)

Qt(t,

+ max f[rx -c + (b - r1)T n] Qx(t, c>o

+

1

Mal' 702 Q.(t ,

neRd

+ e-fitU(c)} =0; 0

t

T,

0 < x < co.

Then (8.68)

V(t, x) < Q(t, x); 0 < t < T,

0 5 x < co.


385

PROOF. For any initial condition (t, x) e [0, T) x (0, co) and pair (ir, C) admis-

sible at (t, x), let {Xs; t 5 s < T} denote the wealth process determined by (8.58). With l

//

Tn A T - -

n

A id s e [t, T]; Xs > n or Xs

1

- or

1114412 du = n

,

we have E 1;"Qx(s, X On'. (s)o- OK = 0. Therefore, It6's rule implies, in conjunction with (8.58) and (8.67), 0 < EQ(T., X,,,)

= Q(t,x) + E

1

rn

(Qt(s, Xs) + [rX3 - Cs + (b - rpr ir(s)]Qx(s, Xs)

1

+ - II 0" T n (Ai 2 Qxx(s, Xs)} ds 5 Q(t, x) -E 2

f

r e-PsU(Cs)ds.

Letting n -+ co and using the monotone convergence theorem, we obtain E if CP' U(Cs) ds < Q(t, x). Maximization of the left-hand side over admissible pairs (n, C) gives the desired result.

A solution to the HJB equation may not be unique, even if we specify the boundary conditions (8.69)

Q(t,0) = 0;

0 < t < T and

Q(7; x) = 0;

0 < x < co.

This is because different rates of growth of Q(t, x) are possible as x approaches infinity. One expects the value function to satisfy the HJB equation, and, in light of (8.68), to be distinguished by being the smallest nonnegative solution of this equation.

8.22 Proposition. Under the conditions set forth at the beginning of this subsection, the value function V: [0, T] x [0, co) -* [0, co) is continuous, is of class ,,,,, T) x (0, co)), and satisfies the HJB equation (8.67) as well as the C1,2(IV boundary conditions (8.69).

PROOF. If 0 < y < U'(0), then (8.70)

U(I(y)) = U'(I(y))1'(y) = yr(y);

if y > U'(0), then 1(y) = l'(y) = 0 and (8.70) still holds. Because of our assumption that G and S are of class C, we may differentiate (8.48), (8.50)

with respect to y and observe that (pi (t, A A -YGy(t9Y) and (P2(t, A A -y2(0/0y)(S(t, y)/y) both satisfy 1.4p,(t, y) = - y2 l' (y); Sai(T, Y) = 0;

0 5 t 5 T, y > 0, y > O.


386

In particular, I' is continuous at y = U'(0), i.e., a necessary condition for our assumptions is U"(0) = oo. Problem 8.19 implies cp, = (p2, because both functions have the same stochastic representation. It follows that Gy(t, y) =

(8.71)

y-0

(-1

S(t, y)) = ydy(t, y)

aY

and from (8.66), (8.62) we have (8.72)

Vx(t, x) = CP' V(t, x),

(8.73)

Vt(t, gr(t, y)) = - Vx(t, lit, y)) gr,(t, y).

Finally, (8.50) and (8.61) imply that (8.74)

-1;(t, y) + r.T(t, y) - (f - r + Heir )Y .Ty(t, Y)

- 2 110112 Y2

(t, Y) = 1(y);

0 < y < oo, 0

t < 'T.

We want to check now that the function V(t, x) of (8.66) satisfies the HJB equation (8.67). With Q = V, the left-hand side of this equation becomes CP' times (8.75)

Gt(t, V(t, x)) - /3G(t, V(t, x)) + Gy(t,V(t, x)) Vt(t, x)

+ max[((rx - c) + (b - rpr n)V(t, x) +

ll o-Tn112Vx(t, x) +

U(c)].

c> 0

The maximization over c is accomplished by setting (8.76)

c = I(V(t, x)).

Because of the negativity of Vx, the maximization over it is accomplished by setting (8.77)

n = -(craT)-` (b

r1)V(t,

x)

V ,c(t, x)

Upon substitution of (8.76), (8.77) into (8.75), the latter becomes (8.78)

Gt(t,V(t, x)) - I1G(t,V(t, x)) + Gy(t,V(t, x))V,(t, x) 1

+ rx?J(t, x) - V(t, x)I(V(t, x)) - -2 1102

V2(t, x)

Vt, x)

+ U(I(Mt, x))).

We may change variables in (8.78), taking y = '(t, x) and using (8.71), (8.73), (8.48) to write this expression as

GM, y) - K(t, Y) - A(t, y) + ry.lit, Y) -Y 1(Y) - +110112y2xy(t, y) + U (I (Y)) = A Thirt(t, y) + rajt, y) - (13 - r + 110112)Y.gry(t, Y) - 1110112Y2Aryy(t, Y) - 4.01


387

which vanishes because of (8.74). This completes the proof that V satisfies the

HJB equation (8.67). The boundary conditions (8.69) are satisfied by V by virtue of its definition (8.57) and the admissibility condition (8.59) applied when x = O.

1=1

In conclusion, we have already shown that for fixed but arbitrary (t, x) e [0, T) x (0, oo), there is an optimal pair (n *, C*) of portfolio/consumption processes. Let {X:; t < s S T} denote the corresponding wealth process. If we now repeat the proof of Lemma 8.21, replacing (it, C) by (tr*, C*) and Q by V, we can derive the inequality (8.79) T

0

V(t, x) + E ft {V,(s, X:) + [rX: -C: + (b - r1)T tr*(s)] Vx(s, X:) T

1

+ -211 6 T it* (s)112 Vx(s,

X:)} ds

V(t, x) -E

e- Ps U (Cnds. t

We have used the monotone convergence theorem and the inequality (8.80)

Vi(s, X:`) + [r X: -C: + (b - r DT tr*(s)] Vx(s, X:) + Dar n*(s)112 Vxx(s, X:)

- - CP' [1 (Cn

0;

t < S < T,

which follows from the HJB equation for V. But (8.65) holds, so equality prevails in (8.79) and hence also in the first inequality of (8.80), at least for meas x P-almost every (s, w) in [t, T] x n. Equality in (8.80) occurs if and only if le: and C: maximize the expression

[rX: -c + (b - r1)T it] Vx(s, X:) + ilia TnIi2 vx.(s,xs) + e-Pt um; i.e. (cf. (8.76), (8.77)), (8.81)

C: = .1((s, X:)),

(8.82)

ir:

(0.0.T r 1 0

r1)

(s, X:) ,c(s, X:) '

where again both identities hold for meas x P-almost every (s, w) E [t, T] x a The expressions (8.81), (8.82) provide the optimal consumption and portfolio processes in feedback form.

8.23 Exercise. Show that in the context of Exercise 8.20, the optimal consumption and portfolio processes are linear functions of the wealth process X *. Solve for the latter and show that XI = 0 a.s.

5.9. Solutions to Selected Problems 2.7. We have from (2.10) -d dt

(e-flt f ' o

g(s)ds)=(g(t)-

13

f

g(s)ds) e-flt < a(t)e -flt, o


388

whence f u g(s) ds _5 e°' f u a(s)e - as ds. Substituting this estimate back into (2.10), we obtain (2.11).

2.10. We first check that each V) is defined for all t > 0. In particular, we must show that for k > 0,

0 < t < cc, a.s.

(II b(s, XI') )11 + 110(s, X?))112)ds < oo; Jo

In light of (2.13), this will follow from

sup EIP0k)112 < co; 0

(9.1)

T < co,

a fact which we prove by induction. For k = 0, (9.1) is obvious. Assume (9.1) for some value of k. Proceeding similarly to the proof of Theorem 2.5, we obtain the

bound for 0 5 t < T: EI1V+"112 < 9E11 112 + 9(T + 1)K2

(9.2)

(1 + E11X?)112)ds,

which gives us (9.1) for k + 1. From (9.2) we also have

f

EMk+i)112 < C(1 + E11112) + C

E11X?)112ds;

0 < t < T,

o

where C depends only on K and T Iteration of this inequality gives

(CO'

r +1)112 < C(1 + Eg112)[1 + Ct + ("2) 2

(k + 1)!

and (2.17) follows. 2.11. We will obtain (2.4) by letting k -+ co in (2.16), once we show that the two integrals

on the right-hand side of (2.16) converge to the proper quantities. With T > 0, (2.21) gives maxo,,,T1,1X,(co) - )0k)(o))11 < 2', V k > N(co). Consequently, 2

b(s, X?))ds - ft b(s, X5) ds

< K2T

f

X s12 ds 0

converges to zero a.s. for 0 < t < T, as k In order to deal with the stochastic integral, we observe from (2.19) that for fixed 0 < t < T, the sequence of random variables {.)0},T_I is Cauchy in L2(52, P), and since V) -+ X, a.s., we must have

Ell r) - X,I12 -0 as k- co. Moreover, (2.17) shows that Ell r)II 2 is uniformly bounded for 0 < t < T and k > 0, and from Fatou's lemma we conclude that E11X,112 k, if II

115 k,


389

and set 6, = 1{,4 (t - s)-' and every A e Ws_ft,/), we can find B e A+(0,) such that P(A A B) = 0 (Problem 2.7.3). The martingale property for {Mt,f, gk; 0 < u < co} implies that (9.5)

E [{ f(y(t))

- f(y(s +

I

) ) -J n

t

(s1 J) ±11,)

= E[ff ( y(t)) -f(y (s + -)) -..1 n

1A]

(sit,: f )( y) du}

,,d =0.

s+(tin)

If follows that the expectation in (9.5) is equal to zero for every A e.f.s = Ws+ . We can then let n -oo and obtain the martingale property E[(Mir - MS )1A] = 0 from the bounded convergence theorem. 4.25. Suppose jud (P(x)Pi(dx) = 1Rd 9(x)/12(dx) for every ys, Cr (Rd), and take ik e Co(111°). Let p E Co (Rd) satisfy p > 0, cRd p(x) dx = 1, and set (p(x) A juld tk(x + (y /n))p(y) dy = n IRdt1/(z)p(nz - nx) dz. Then cp e q°(Rd) and 9(x) --1//(x) for every x e Old. It follows from the bounded convergence theorem that .fRa ii/(x)111 (dx) = jRa ti/(x)/12(dx), for every 0 a Co(Rd). Now suppose G c Rd is open and bounded. Let 0(x) = 1 A infy# G nlly - xII. Then On E Co( ) for all n,

and ifr 11,. It follows from the monotone convergence theorem that it,(G) = i12(G) for every bounded open set G. The collection of sets Cif = {BE At(Rd); PI(B) = 142(B)} form a Dynkin system, and since every bounded, open set is in ', the Dynkin System Theorem 2.1.3 shows that elf = g(ge).

5.3. For t

0, let Et = {Po' a2(Xs) ds = co}. Using the method of Solution 3.4.11, we can show that lim

AS. = XO

a(Xs)dRis = o o,

lim

a.s. on E

n-.c0

AS

ct(Xs)dW = - co, as. on E,. lim XrAS = X0 + lim ft n-,co .-.0 0 But X, is continuous in the topology of the extended real numbers, so P(E,) = 0. Consequently, X, ,,s = X0 roA 0.(xs) a'His is real-valued a.s., for every t 0, so S = co a.s.

5.27. For s > 0 and c+s<x 1 has a unique strong solution for any bounded, Borel-measurable drift b(t, x).

In another important development, Nakao (1972) showed that pathwise uniqueness holds for the equation (5.1), provided that the coefficients b, a are bounded and Borel-measurable, and a is bounded below by a positive constant and is of bounded variation on any compact interval. For further extensions of this result (to time-dependent coefficients), see Veretennikov (1979), Nakao (1983), and Le Gall (1983). The material of Subsection C is fairly standard; we relied on sources such

5.10. Notes

397

as McKean (1969), Kallianpur (1980), and Ikeda & Watanabe (1981), particularly the latter. A generalization of the Feller test to the multi-dimensional case is due to Khas'minskii (1960) and can be found in Chapter X of Stroock & Varadhan (1979), together with more information about explosions. A complete characterization of strong Markov processes with continuous

sample paths, including the classification of their boundary behavior, is possible in one dimension; it was carried out by Feller (1952-1957) and appears in Ito & Mc Kean (1974) and Dynkin (1965), Chapters XV-XVII. See also Meleard (1986) for an approach based on stochastic calculus. The recurrence and ergodic properties of such processes were investigated by Maruyama

& Tanaka (1957); see also §18 in Gihman & Skorohod (1972), as well as Khas'minskii (1960) and Bhattacharya (1978) for the multi-dimensional case.

Section 5.6: Langevin (1908) pioneered an approach to the Brownian movement that centered around the "dynamical" equation (6.22), instead of relying on the parabolic (Fokker-Planck-Kolmogorov) equation for the transition probability density. In (6.22), X, represents the velocity of a free particle with mass m in a field consisting of a frictional and a fluctuating force, a is the coefficient of friction, and 62 = 2ockT/m, where T denotes (absolute) temperature and k the Boltzmann constant. Langevin's ideas culminated in the Ornstein-Uhlenbeck theory for Brownian motion; long considered a purely heuristic tool, unsuitable for rigorous work, this theory was placed on firm mathematical ground by Doob (1942). Chapters IX and X of Nelson (1967) contain a nice exposition of these matters, including the Smoluchowski equation for Brownian movement in a force field. Section 5.7: The monograph by Freidlin (1985) offers excellent follow-up reading on the subject matter of this section, as well as on degenerate and quasi-linear partial differential equations and their probabilistic treatment. In the setting of Theorem 7.6 with k

0, g E: 0 it is possible to verify directly,

under appropriate conditions, that the function (10.1)

u(t, x) = Ef(Xx))

on the right-hand side of (7.15) possesses the requisite smoothness and solves the Cauchy problem (7.12), (7.13). We followed such an approach in Chapter

4 for the one-dimensional heat equation. Here, the key is to establish "smoothness" of the solution V) to (7.1) in the initial conditions (t, x) so as to allow taking first and second partial derivatives in (10.1) under the expectation sign; see Friedman (1975), p. 124, for details. Questions of dependence on the initial conditions have been investigated extensively. The most celebrated of such results is the diffeomorphism theorem (Kunita (1981), Stroock (1982)), which we now outline in the context of the stochastic integral equation (4.20). Under Lipschitz and linear growth conditions as in Theorem 2.9, this equation has, for every initial position x e Rd,

a unique strong solution IX,(x); 0 "i: t < col. Consider now the (d + 1)dimensional random field d = {X t(x , w); (t, x) e [0, co) x Rd, co e ill. It can be

shown, using the Kolmogorov-entsov theorem (Problem 2.2.9) in conjunc-

398


tion with Problems 3.3.29 and 3.15, that there exists a modification such that: (i) (t, x) i- gt(x, w) is continuous, for a.e. w e CI; (ii) For fixed t > 0, xi-, IC,(x, w) is a homeomorphism of D

of I

into itself for a.e.

We

Furthermore, if the coefficients b, a have bounded and continuous derivatives of all orders up to k > 1, then for every t > 0, (iii) x F- fet(x, w) is a C" -diffeomorphism for a.e. w e e.

For an application of these ideas to the modeling issue of Subsection 5.2.D, see Kunita (1986).

Malliavin (1976, 1978) pioneered a probabilistic approach to the questions of existence and smoothness for the probability densities of Brownian func-

tionals, such as strong solutions of stochastic differential equations. The resulting "functional" stochastic calculus has become known as the stochastic calculus of variations, or the Malliavin calculus; it has found several exegeses and applications beyond its original conception. See, for instance, Watanabe (1984), Chapter V in Ikeda & Watanabe (1981), and the review articles of Ikeda & Watanabe (1983), Zakai (1985) and Nualart & Zakai (1986). For applications of the Malliavin calculus to partial differential equations, see Stroock (1981, 1983) and Kusuoka & Stroock (1983, 1985).

Section 5.8: The methodology of Subsection A is new, as is the resulting treatment of the option pricing and consumption/investment problems in Subsections B and C, respectively. Similar results have been obtained independently by Cox & Huang in a series of papers (e.g., (1986, 1987)). For Subsection B, the inspiration comes in part from Harrison & Pliska (1981) and Bensoussan (1984); this latter paper, as well as Karatzas (1988), should be consulted for the pricing of American options. Material for Subsection C was drawn from more general results in Karatzas, Lehoczky & Shreve (1987). The problem of Subsection D was introduced by Samuelson (1969) and Merton (1971); it has been discussed in Karatzas et al. (1986) on an infinite horizon and with very general utility functions. An application of these ideas to equilibrium analysis is presented in Lehoczky & Shreve (1986). See also Duffie (1986) and Huang (1987).

CHAPTER 6

P. Levy's Theory of Brownian Local Time

6.1. Introduction This chapter is an in-depth study of the Brownian local time first encountered

in Section 3.6. Our approach to this subject is motivated by the desire to perform computations. This is manifested by the inclusion of the conditional Laplace transform formulas of D. Williams (Subsections 6.3.B, 6.4.C), the derivation of the joint density of Brownian motion, its local time at the origin and its occupation time of the positive half-line (Subsection 6.3.C), and the computation of the transition density for Brownian motion with two-valued drift (Section 6.5). This last computation arises in the problem of controlling the drift of a Brownian motion, within prescribed bounds, so as to keep the controlled process near the origin. Underlying these computations is a beautiful theory whose origins can be traced back to Paul Levy. Levy's idea was to use Theorem 3.6.17 to replace the study of Brownian local time by the study of the running maximum (2.8.1) of a Brownian motion, whose inverse coincides with the process of first passage times (Proposition 2.8.5). This latter process is strictly increasing, but increases by jumps only, and these jumps have a Poisson distribution. A precise statement of this result requires the introduction of the concept of Poisson random measure, a notion which has wide application in the study of jump processes. Here we use it to provide characterizations of Brownian local time in terms of excursions and downcrossings (Theorems 2.21, 2.23). In Section 6.3 we take up the study of the independent, reflected Brownian motions obtained by looking separately at the positive (negative) parts of a standard Brownian motion. These independent Brownian motions are tied together by their local times at the origin, a fact which does not violate their independence. Exactly this situation was encountered in the Discussion of

6. P. Levy's Theory of Brownian Local Time

400

F. Knight's Theorem 3.4.13, where we observed that intricately connected processes could become independent if we time-change them separately and then forget the time changes. The first formula of D. Williams (Theorem 3.6) is a precise statement of what can be inferred about the time change from observing one of these reflected Brownian motions. Section 6.4 is highly computational, first developing Feynman-Kac formulas involving Brownian local time at several points, and then using these formulas to perform computations. In particular, the distribution of local time at several spatial points, when the temporal parameter is equal to a passage time, is computed and found to agree with the finite-dimensional distribution of one-half the square of a two-dimensional Bessel process. This is the RayKnight description of local time; it allows us finally to prove the DvoretzkyErdos-Kak utani Theorem 2.9.13.

6.2. Alternate Representations of Brownian Local Time In Section 3.6 we developed the concept of Brownian local time as the density

of occupation time. This is but one of several equivalent representations of Brownian local time, and in this section we present two others. We begin with a Brownian motion W = {147,, ,F,; 0 < t < co} where P[Wo = 0] = 1 and {Ft} satisfies the usual conditions, and we recall from Theorem 3.6.17 (see, in particular, (3.6.34), (3.6.35)) that (2.1)

where B, = (2.2)

= M, - B,, 2L,(0) = Mt; V 0

t < co] = 1,

sgn(Ws) AK is itself a Brownian motion,

M,= max 135; 0 < t < co, o<s

and L,(0) is the local time of W at the origin: (3.6.2)

L,(0) = lim

1 meas {0 < s < t; I147,1 < e }.

Thus, a study of the loca;ti0 time 4 of W at the origin can be reduced to a study of the more easily conceived process M. The idea of this reduction and much of what follows originated with P. Levy (1939, 1948).

A. The Process of Passage Times The process M of (2.2) is continuous, nondecreasing, and "flat" (constant) during excursions of the reflected Brownian motion I WI away from the origin. Because the Lebesgue measure of the set {0 < t < co; I W11 = 0} is zero, P-a.s.

6.2. Alternate Representations of Brownian Local Time

401

(Theorem 2.9.6), the process M has time derivative almost everywhere equal to zero; it is very much like the Cantor function. We may regard M as a new clock, which stops when I WI is away from the origin and runs at an accelerated rate when I WI is at the origin. In order to pass from this new clock to the original one, we introduce the right-continuous inverse of M, defined as in Problem 3.4.5: (2.3)

Sb

inf t

0; M, > b1 = inf {t

0; B, > b};

0

b < oo.

Each Sb is an optional (and hence also a stopping) time of the right-continuous

filtration f 0, the time in se [0, for which B, = M, is almost surely unique. Thus, for each w in an event SI* of probability one, this assertion holds for every rational t. Fix w e Se, a positive number t (not necessarily rational), and define (2.5)

y,(w)

sup {s e [0, t]; Ws(co) = 0} = sup {s e [0, a Bs(co) = M,(w)1,

(2.6)

13,(w)

inf {s e [t, co); W,(w) = 0} = inf {s e [t, co); Bs(co) = M,(01.

If 14/,(co) = 0, then /3,(w) = t = y,(w). We are interested in the case W,(w) # 0, which implies (2.7)

< t < A(0). In this case, the maximum of Bs(w) over 0 < s < t is attained uniquely at s = y,(w), hence 7,,,,,(w)(w) = y,(w). Similarly Tww),(w) = Sw,)(w) = f3,(w), for otherwise there would be a rational q > 13,((o) such that Bfit(w)(w) = Bm,)(w) =

WO, a contradiction to the choice of w e Sr. We see then that for w e and t chosen so that W(w) 0 0, the size of the jump in Tb(w) at b = M,(w) is the length of the excursion interval (y,(w), Ma))) straddling t: (2.8)

TAftoo+ (0) - Tut( (0(0 =

- Y*0)

It is clear from (2.4) that Tb is strictly increasing in b, and To = 0. It is less clear that T grows only by jumps. To see this, consider the zero set of W(w), namely 22'1, A {0 < t < co; Wt(w) = 0},

which is almost surely closed, unbounded, and of Lebesgue measure zero (Theorem 2.9.6). As with any open set, £ n (0, co) can be written as a countable


402

union of disjoint open intervals

27. n (0, co ) = U 4(0, ae A

and each of these intervals contains a number ta(w). In the notation of (2.5), (2.6), we have a G A.

la = (ye., A.);

Because meas(Aeb,) = 0 for P-a.e. w e S/, we have

=E

me A

yi.) =aeAE (Tm. -

°

°

E (Tx, - Tx); 0 < t < co,

x<M,

almost surely. Now set t = Ta to obtain

E (Tx, - Tx),

a < oo;

x 0,

P[-i- > b] = e-'i6;

0 < b < co.

After T,, we may begin the wait for the next jump of T whose size exceeds e, i.e., the wait for I W+7-,1 = +7. to have another excursion of WT I

duration exceeding e. It is not difficult to show, using the strong Markov property, that the additional wait t2 is independent of Tl. Indeed, the "interarrival times" T1, t2, r3, ... are independent random variables with the same (exponential) distribution. Recalling the construction in Problem 1.3.2 of the Poisson process, we now see that for fixed b > 0, the number of jumps of the process T on [0, b], whose size exceeds e, is a Poisson random variable. To

formalize this argument and obtain the exact distributions of the random variables involved, we introduce the concept of a Poisson random measure.

B. Poisson Random Measures A Poisson random variable takes values in No A {0, 1, 2, ... and can be thought of as the number of occurrences of a particular incident of interest. Such a concept is inadequate, however, if we are interested in recording the occurrences of several different types of incidents. It is meaningless, for example, to keep track of the number of jumps in (0, b] for the process S of (2.3), because

there are infinitely many of those. It is meaningful, though, to record the number of jumps whose size exceeds a positive threshold e, but we would like


404

to do this for all positive simultaneously, and this requires that we not only count the jumps but somehow also classify them. We can do this by letting v((0, b] x A) be the number of jumps of S in (0, b] whose size is in A e.4((0, co)),

and then extending this counting measure from sets of the form (0, b] x A to the collection .4((0, co)2). The resulting measure v on ((0, co)2, .4((0, co)2)) will

of course be random, because the number of jumps of {S.; 0 < a < b} with sizes in A is a random variable. This random measure v will be shown to be a special case of the following definition.

2.3 Definition. Let (S1,F, P) be a probability space, (H, Yr) a measurable space, and v a mapping from SI to the set of nonnegative counting measures on (H, Jr), i.e., v,,,(C)e No u {co} for every co ES./ and C e Jr. We assume that the mapping coi-+ v,,,(C) is .F/M(No u {co})-measurable; i.e., v(C) is an No u {co}-valued random variable, for each fixed Ce Ye. We say that v is a Poisson random measure if:

(i) For every C E A', either P[v(C) = co] = 1, or else A(C) -A Ev(C) < co

and v(C) is a Poisson random variable: P[v(C)

=

e--A(c)(A(C))" n!

;

neNo.

(ii) For any pairwise disjoint sets C1, ..., Cm in Jr, the random variables v(C, ),

v(Cm) are independent.

The measure A(C) = Ev(C); Cele, is called the intensity measure of v. 2.4 Theorem (Kingman (1967)). Given a o--finite measure A on (H, Jr), there exists a Poisson random measure v whose intensity measure is A. PROOF. The case A(H) < co deserves to be singled out for its simplicity. When it prevails, we can construct a sequence of independent random variables = A(C)/01(11); C E dr, as well as an . with common distribution PN1 E

independent Poisson random variable N with P[N = n] = e-411)(A(H))7n!; n e No. We can then define the counting measure v(C) A hi i=i

1c(4 C E

It remains to show that v is a Poisson random measure with intensity A. Given a collection C1, ..., Cm of pairwise disjoint sets in Y9, set Co = H Uir=i C so E,7=0 v(C,,) = N. Let no, n1, nm be nonnegative integers with n = no +

ni +

+ n,. We have P[v(C0) = no, v(C1) = n1, ..., v(Cm) = nm]

= P[N = n] P[v(C0) = no, v(Ci)

ni,

v(Cm) = n,,,IN


=e

n!

(A(C0) yo

n!

(.1,(H))"

?T (1.(H) no! ni!... ?T.!

)

405

0.(Ci ) r 0.(Cm)r" A(H) )

A(H)

nn e_A(co(A(Ck)r" nk!

k =0

and the claim follows upon summation over no E No.

2.5 Problem. Modify the preceding argument in order to handle the case of o- finite A.

C. Subordinators 2.6 Definition. A real-valued process N = {Ng; 0 < t < co} on a probability space (1/, 0, a > 0, (o,c0)

for a constant m

0 and a cr-finite measure /..i on (0, co) for which the integral in (2.14) is finite. Furthermore, if i7 is a Poisson random measure on (0, c0)2 (on a possibly different space (0, g", P)) with intensity measure

A(dt x de) = dt it(de),

(2.15)

then the process (2.16)

N,

ev((0,t] x de); 0 < t < co

A mt + J (0, co)

is a subordinator with the same finite-dimensional distributions as N.

2.8 Remark. The measure it in Theorem 2.7 is called the Levy measure of the

subordinator N. It tells us the kind and frequency of the jumps of N. The simplest case of a Poisson process with intensity A > 0 corresponds to m = 0

406


andµ which assigns mass A. to the singleton { 1 }. If y does not have support on a singleton but is finite with A. = 14(0, co)), then N is a compound Poisson process; the jump times are distributed just as they would be for the usual Poisson process with intensity ;4., but the jump sizes constitute a sequence of independent, identically distributed random variables (with common distribution it(de')/),), independent of the sequence of jump times. The importance of Theorem 2.7, however, lies in the fact that it allows /.4 to be a-finite; this is exactly the device we need to handle the subordinator S = {Sb; 0 < b < which has infinitely many jumps in any finite interval (0, b]. PROOF OF THEOREM 2.7. We first establish the representation (2.14). The stationarity and independence of the increments of N imply that for a > 0, the nonincreasing function p °(t) A Ee-xl'4; 0 < t < co, satisfies the functional equation pc,(t + s) = pc,(t)p(s). It follows from Problem 2.2 that Ee'N' = e-"I1(2);

(2.17)

t > 0, a > 0,

holds for some continuous, nondecreasing function 0: [0, co) -* [0, co) with 0(0) = 0. Because the function g(x) A xe-";x > 0, is bounded for every a > 0, we may differentiate in (2.17) with respect to a under the expectation sign to obtain the existence of ti/ and the formula (a)e-"1/(2) =

1

t

1

-t

E[N,C°N1

8e-21 P[N,Ed1];

a > 0, t > 0.

Consequently, we can write

Oa) = lim kck J

(2.18)

k-'c c

(1 + [0,00)

where

(

E[1 +1N 1/, 1, p,(&) A clk

ck

1 -I-

( P[N,/, e de].

0 a.s., the theorem is trivially true, so we may assume the contrary and choose a > 0 so that c E(IVI 1{bri

E[

A

1

+1N i m

1

1{N

)] >

1

1+a

E[N11,1{N P[Nfik - Nu_of < e; j = 1,

k]

= 1P[Nlik

and thus, using (2.19), we may write

(2.20) MK cc))

1+a

k(1 + a)

kP[N,,,, >

(P[N, < tykl.

{1

= -log 0 < < 1, we can make the right-hand

Because limk k(1 -

side of (2.20) as small as we like (uniformly in k), by taking ( large. Prohorov's Theorem 2.4.7 implies that there is a subsequence {p,5}1°=, which converges

weakly to a probability measure p on ([0, co), d([0, co))). In particular, be+ e)e-e is bounded for every positive a, we must cause the function e have (1 + e)e-2? pk.,(d()=

lim

(1 + e)e-cie p(do. [0.c.)

10.c.)

Combined with (2.18), this equality shows that kick, converges to a constant c > 0, so that (2.21)

1//(a) = c

(1 + e)e-at p(cl()

J

= cp( {O})

c

(1 + e)e'ep(de); 0 < a < co.

J

op..)

Note, in particular, that i//' is continuous and decreasing on (0, co). From the fundamental theorem of calculus, the Fubini theorem, (2.21), and /i(0) = 0, we obtain now (2.22)

Oa) = acp({0}) + c f (0..)1 + e (1

e-2e)p(de);

0 < a < co.

The representation (2.14) follows by taking m = cp({0}) and (2.23)

p(de) =

c(1 +

p(de);

> 0,

the latter being a a-finite measure on (0, co). In particular, we have from (2.22): (2.24)

tp(c) = ma +

(1 - C'e')12(de) < co;

0 < a < cc.

(0.00)

We can now use Theorem 2.4 with H = (0, co)2 and lt = ge(H), to construct P) a random measure 17 with intensity given by on a probability space (n, P) via (2.16). (2.15); a nondecreasing process 13 can then be defined on (n, It is clear that Ro = 0, and because of Definition 2.3 (ii), 13 has independent increments (provided that (2.25), which follows, holds, so the increments are


408

defined). We show that IV is a subordinator with the same finite-dimensional distributions as N. Concerning the stationarity of increments, note that for a

nonnegative simple function cp(f)=E7=iail,(l) on (0, x), where A1, ..., are pairwise disjoint Borel sets, the distribution of (o..)

t + h] x (10=

tp(t)

+ h] x Ai) i =1

is a linear combination of the independent, Poisson (or else almost surely infinite) random variables Iii((t, t + h] x Ai)}7=1 with respective expectations {h1t(Ai)}7=1. Thus, for any nonnegative, measurable co, the distribution of .1.0,00(p(I)v((t,t + h] x dl) is independent of t. Taking (Xi) = t, we have the stationarity of the increment A +h - A. In order to show that IV is a subordinator, it remains to prove that

A < co; 0

(2.25)

t < oo

and that lS is right-continuous, almost surely. Right-continuity will follow from (2.25) and the dominated convergence theorem applied in (2.16), and (2.25) will follow from the relation

Re'" = e-0(x); a > 0, t > 0,

(2.26)

= (6("1, /1")]; 1 < j < 4", we have from the monotone and bounded convergence theorems: where (I, is as in (2.24). With n > 1, and (1.) A j2-", (2.27)

E exp -

f

4^

( 17( 0, t] x de)} = lim

(0.oD)

exp - E 4!),

t] x

/.;"))} .

j= 2

n-.co

But the random variables ii((0, t] x II")); 2 < j < 4" are independent, Poisson, with expectations ty(11")); 2 < j < 4", and these quantities are finite because the integral in (2.14) is finite. The expectation on the right-hand side of (2.27) becomes 4"

Eexp{ -aer, i7((0,

x /l"))} =exp -t

! =2

4"

(1 -

)µ(/1"))

,

.1 =2

which converges to exp{ -t f(o)(1 - e'f)ki(d1)} as n co. Relation (2.26) follows and shows that for each fixed t > 0, A has the same distribution as N. The equality of finite-dimensional distributions is a consequence of the independence and stationarity of the increments of both processes. Theorem 2.7 raises two important questions:

(I) Are the constant m > 0 and the Levy measure it unique?

(II) Does the original subordinator N admit a representation of the form (2.16)?

One is eager to believe that the answer to both questions is affirmative; for

409


the proofs of these assertions we have to introduce the space of RCLL functions, where the paths of N belong. 2.9 Definition. The Skorohod space D[0, co) is the set of all RCLL functions from [0, oo) into R. We denote by AD [0, co)) the smallest a-field containing all finite-dimensional cylinder sets of the form (2.2.1).

2.10 Remark. The space D[0, co) is metrizable by the Skorohod metric in such a way that M(D [0, co)) is the smallest a-field containing all open sets (Parthasarathy (1967), Chapter VII, Theorem 7.1). This fact will not be needed here.

2.11 Problem. Suppose that P and P are probability measures on (D[0, co), .4(D [0, co))) which agree on all finite-dimensional cylinder sets of the form

Y(WernI, < co, and Fie M(R); i = 1,

1Y GDP, GO; Ati)Gri,

where n > 1, 0 < t, < t2
fro u {co} be defined by (2.28)

n(y;

A # {(t,e)E C; IY(t)

-

-)I = el,

where # denotes cardinality. In particular, n(y; (t, t + h] x ((, co)) is the number of jumps of y during (t,t + h] whose sizes exceed e. Show that n(; C) is .11(D[0, co))-measurable, for every Ce.4((0, co)2). (Hint: First show that n(; (0, t] x (i' , co)) is finite and measurable, for every t > 0, ( > 0.) Returning to the context of Theorem 2.7, we observe that the subordinator

N on (SI,,F,P) may be regarded as a measurable mapping from (12,") to (D [0, co), .4(D [0, co))). The fact that Ar defined on (5 , g, P) by (2.16) has the same finite-dimensional distributions as N implies (Problem 2.11) that N and induce the same measure on D[0, co): PEN e A] = P[KI e A]; V A e .4(D [0, oo)).

We say that N under P and under P have the same law. Consequently, for C1, C2, C, in gf((0, co)2), the distribution under P of the random vector . , n(N; Cm)) coincides with the distribution under 15 of (n(N; C1), (n(N; , n(N; Cm)). But (2.29)

n(g ; C) = ("(C);

C e *(0, 00) x

00))

is the Poisson random measure (under P) of Theorem 2.7, so (2.30)

v(C) A n(N; C) = # {(t, eC;

N,

- Nt_ = e}; C e .4((0, al')

is a Poisson random measure (under P) with intensity given by (2.15).


410

We observe further that for t > 0, the mapping qv D[0, cc) -* [0, cc] defined by (My) A

f

in(y; (0, t] x do)

(o,o0)

is Af(D [0, oo))/.4([0, co])-measurable, and co,(N) =

f

(v((0, t] x do,

(MR) =

f

mo,t] x de).

co,c0)

co,c0)

It follows that the differences {N, - pi(N); 0 < t < col and {A - cp,(R); 0 < t < col have the same law. But A - co,(N). mt is deterministic, and thus N, - q),(N) = mt as well. We are led to the representation (2.31)

N, = mt +

i

0v((0, t] x dO);

0 < t < 00.

We summarize these remarks as two corollaries to Theorem 2.7. 2.13 Corollary. Let N = {Nt; 0 < t < co} be a subordinator with moment generating function (2.14). Then N admits the representation (2.31), where v given by (2.30) is a Poisson random measure on (0, cc)2 with intensity (2.15).

2.14 Corollary. Let N = INt; 0 < t < col be a subordinator. Then the constant m .. 0 and the a-finite Levy measure it which appear in (2.14) are uniquely determined. PROOF. According to Corollary 2.13, a(A) = Ev((0, 1] x A); A E MO, 00), where

v is given by (2.30); this shows that it is uniquely determined. We may solve (2.31) for m to see that this constant is also unique.

2.15 Definition. A subordinator N = {Nt; 0 < t < co} is called a one-sided stable process if it is not almost surely identically zero and, to each a > 0, there corresponds a constant 11(a) 0 such that {aNt; 0 < t < co } and {Now; 0 < t < col have the same law. 2.16 Problem. Show that the function Ma) of the preceding definition is continuous for 0 < a < cc and satisfies (2.32)

Nay) = Q(a)NY);

cc > 0, Y > 0

as well as (2.33)

i I t (a) = r i 3 (a);

a > 0,

where r = it/(1) is positive and 4./ is given by (2.17), or equivalently, (2.24). The unique solution to equation (2.32) is 13(a) = a', and from (2.33) we see that for a one-sided stable process N,


411

0 < a < oo.

Ee'N. = exp{ - till(a)} = exp{ - traE};

(2.34)

The constants r, c are called the rate and the exponent, respectively, of the process. Because b is increasing and concave (cf. (2.21)), we have necessarily 0 < e < 1. The choice E = 1 leads to m = r, t = 0 in (2.14). For 0 < E < 1, we have

m = 0, m(dt) =

(2.35)

tit r(1rE E) e' +g;

e

D. The Process of Passage Times Revisited The subordinator S of Theorem 2.1 is one-sided stable with exponent E = (1/2).

Indeed, for fixed a > 0, (1/ot)Sb,A; is the first time the Brownian motion (1/.,/a) Wcu; 0 < t < oo } reaches level b; i.e.,

(Lemma 2.9.4 (i)) W* = {14/,* 1

- Sb,A,=

inf It

0; 147,* > bl.

az

Consequently, {otSb; 0 < b < co} has the same law as laSt, = Sb,";; 0 < b < col, from which we conclude that Noz) appearing in Definition 2.15 is NAi Comparison of (2.11) and (2.34) shows that the rate of S is r = .\/2, and (2.35) gives us the Levy measure

de it(de) = /2n83;; 2e3

e > 0.

Corollary 2.13 asserts then that eq(0,13] x de);

Sb

0 < b < oo,

to. co)

where, in the notation of (2.1)-(2.4), (2.30) (2.36)

v(C) = # {(b,e) e C; Sb

Sb- = el = # {(b,e) e C; WI has an excursion of duration e starting at time Tb }; CeI((0, co )2 ),

is a Poisson random measure with intensity measure (dt de'1,/27re'3). In particular, for any I eM((0, co)) and 0 < 5 < E < oo, we have (2.37)

Ev(I x [5,0) = meas(/)

dA

27c,13

= meas(/)

2(

For 0 c, completed by W at or before time t." This expression differs from v((0, Mt] x [e, co)) by at most one excursion, and such a discrepancy in counting would be of no consequence in formulas (2.48), (2.49): in the former, it would be eliminated by the factor ,,/i as c 10; in the latter, the effect on the

integral would be at most e, and even after being divided by 4, the effect would be eliminated as E 10. Recalling the identifications (2.1), we obtain the following theorem.

2.21 Theorem (P. Levy (1948)). The local time at the origin of the Brownian motion W satisfies L,(0) = lim \i-c,l.o

=

40

8

# excursion intervals away from the origin, of duration -_c, completed by {Ws; 0

s

t}

Total duration of all excursion intervals away from the origin of individual duration 0, let us define recursively the stopping times To an

4 inf {t > tn-i; I II = e },

to

0 and

itif{t > an; III = 0}

for n > 1. With nn = an - rn -1 = rn an, we have from the strong Markov property as expressed by Theorem 2.6.16 that the pairs (n 1> are independent and identically distributed. Moreover, Problem 2.8.14 asserts that

E1 =

(2.52)

E2'

and we also have

{ j < NO} = {-5 < t} is independent of {(rh, ,)},T=J+,.

(2.53)

Suppose that t E [an, t,,) for some n > 1; then n-i -1wQ,11+ 147,I IlWtAt,} -1WtAn,11 = j=1

J =1

= -EDi(E)

e

1Wil

aDt(e). In either case,

If t U,T=1 [an, r), then ET=1{1141tAr,} -1WtAn,1} = (2.54)

W,

T

I - iWt,n,11 = -0,(a) +

WI

- a)

J =1

l[ni,)(t) J=1

On the other hand, the local time L.(0) is flat on U;,'°_1 [a, c) (cf. Problem 3.6.13 (ii)), and thus from the Tanaka formula (3.6.13) we obtain, a.s.: (2.55) J =1

{1WtA

- I WtAt7,11 = 1Wt1

From (2.54), (2.55) we conclude that

2L,(0) - E

J=1

i" tAr.,

sgn(147,)dWs.

417


el),(e) - 2L,(0) = -I WI

(2.56)

,,

-

+E

11,,r5)(t) J=1

J=1

sgn(14/s)dWs,

a.s.

A 5_,

.i=1

2.24 Problem. Conclude from (2.56) that, for some positive constant C(t) depending only on t, we have EleD,(E) - 2L,(0)12 < C(t)s.

(2.57)

Ceby§ev's inequality and (2.57) give

P[In-2A(n-2) - 2L,(0)I > n-114] < C(t)n- 312 ,

and this, coupled with the Borel-Cantelli lemma, implies

lim n-2D,(n-2) = 2L,(0), a.s. n-oco

But for every 0 < E < 1, one can find an integer n = n(e)

1 such that

(n + 1)-2 < E < n-2, and obviously (n + 1)-2D,(n-2) < eD,(c) < n-2 D,((n + 1)-2) holds. Thus, (2.51) holds for every fixed t E [0, o0); the general statement follows

from the monotoncity in t of both sides of (2.51) and the continuity in t of 2.25 Remark. From (2.51) and (2.1), (2.2) we obtain the identity (2.58)

lim ED[o ti(0; E; M(w)

B(w)) = 114,(w);

0 < t < oo

E

for P-a.e. w E a The gist of (2.58) is the "miraculous fact," as Williams (1979)

puts it, that the maximum-to-date process M of (2.2) can be reconstructed from the paths of the reflected Brownian motion M - B, in a nonanticipative way. As Williams goes on to note, "this reconstruction will not be possible for any picture you may draw, because it depends on the violent oscillation of the Brownian path." You should also observe that (2.58) offers just one way of carrying out this reconstruction; other possibilities exist as well. For instance, we have from Theorem 2.21 that lim o

-# 2

lim

40

excursion intervals away from the origin, of duration > c, completed by {M, - 135; 0

s

total duration of all excursion intervals, away from the origin, of individual duration 0, 14/1-,1(.0 0 hold a.s. P°. It follows then that (3.15)

W +(t) = Wrt,i(r) = L +(t) -B +(t) = max B ±(u) -B ±(t) O 0 simultaneously. From Theorem 3.6.17, each of the processes W+ is a reflected Brownian motion starting at the origin; W+ are independent because B are.

3.2 Remark. Theorem 3.6.17 also yields that the pairs {(WAT), L +(T)); 0 < t < CO} have the same law as {(1W,I, 24(0)); 0 < t < col, under P°. In particular, we obtain from (3.6.36) and (3.14), (3.15): 1

(3.16)

L + (T) = lim

40 LE

meas {0 5_ o- < t; W+ (0)
b} .

inf {z > 0; L +(r) > 6} = inf

0 0, t > 0 we have (3.22)

E°[e'+'(')I W,(u); 0< u< co] =

.12aL,(v);

a.s. P°.

PROOF. The argument hinges on the important identity (3.23)

a.s. P°

f+-1 (r) = t + L:1 (L,(T)) = z + TL-Ao;

which expresses the inverse occupation time TV (r) as t, plus the passage time of the Brownian motion B_ to the level L,(r). But L +(t) is a random variable measurable with respect to the completion of a(W +(u); 0 < u < co), and hence independent of the Brownian motion B_ . It follows from Problem 2.7.19 (ii) that (3.24)

Lii (L+(r)) =

a.s. P°,

and this takes care of the second identity in (3.23). The first follows from the string of identities (see Problem 3.7) (3.25)

L 11 (L,(r)) = inf t

0; L_(t) > L,(T)}

= inf It > 0; L(Fil(t)) > L(r VW)} = inf It

0; 1".=1(t), FV(r)}

=1--(1"-Ti(T))=F;1(t)- r+(rV(T)) ET1(7)- t, a.s. P°. Now the independence of {B_(u); 0 < u < a)} and { W,(u); 0 < u < cc}, along with the formula (2.8.6) for the moment generating function for Brownian passage times, express the left-hand side of (3.22) as E° CiTb b= +(t)

e-cts-.1.2iLAr)

a.s. P°.

111

3.7 Problem. Establish the third and fourth identities in (3.25). Following McKean (1975), we shall refer to (3.22), or alternatively (3.23), as

the first formula of D. Williams. This formula can be cast in the equivalent

422


forms (3.26)

PITZ1(r) < t I W+(u); 0 < < ao] =

[Tb < t -

lb-L+(t)

= 2 [1 -(I) ( Li+ (t) )1 t

rr

=

L+(t)

e -Li (r)/2(0-0

2140 - r)3

dO

a.s. P°, which follow easily from (3.23) and (2.8.4), (2.8.5). We use the notation (120(z) = e'2/2 du.

We offer the following interpretation of Williams's first formula. The reflected Brownian motion { W+(u); 0 < u < co} has been observed, and then a time r has been chosen. We wish to compute the distribution of TV (r) based on our observations. Now 147_, consists of the positive part of the original Brownian motion W, but W+ is run under a new clock which stops whenever W becomes negative. When r units of time have accumulated on this clock corresponding to W+, r +' (r) units of time have accumulated on the original clock. Obviously, F+-1(r) is the sum of T and the occupation time 1-1(1-+-1(r)). Because W_ is independent of the observed process W+, one might surmise that nothing can be inferred about r_ (t) from W. However, the independence between W+ and W_ holds only when they are run according to their respective clocks. When run in the original clock, these processes are intimately con-

nected. In particular, they accumulate local time at the origin at the same rate, a fact which is perhaps most clearly seen from the appearance of the same process L in both the plus and minus versions of (3.2). After the time changes (3.12) which transform W ± into W+, this equal rate of local time accumulation finds expression in (3.13). (From (3.16) we see that L+ is the local time of W+.) In particular, when we have observed W+ and computed its local time L+(r), and wish to know the amount of time W has spent on the negative half-line

before it accumulated r units of time on the positive half-line, we have a relevant piece of information: the time spent on the negative half-line was enough to accumulate L+(t) units of local time. Suppose L+(c) = b. How long should it take the reflected Brownian motion W- to accumulate b units of local time? Recalling from Theorem 3.6.17 that the local time process for a reflected Brownian motion has the same distribution as the maximum-to-date process of a standard Brownian motion, we see

that our question is equivalent to: How long should it take a standard Brownian motion starting at the origin to reach the level b? The time required is the passage time Tb appearing in (3.23), (3.26). Once L +(T) = b is known,

nothing else about W. is relevant to the computation of the distribution of 1--(r+-1(-0).

3.8 Exercise. Provide a new derivation of P. Levy's arc-sine law for F+ (t) (Proposition 4.4.11), using Theorem 3.6.

6.3. Two Independent Reflected Brownian Motions

423

C. The Joint Density of (W(t), L(t), r+(t)) Here is a more interesting application of the first formula of D. Williams. With T > 0 fixed, we obtain from (3.26): P°[1-_-,-i CO E dt1W+(t) = a; L+(t) = b] -

(3.27)

be-b2/2(`-') ,

- \ / 2n(t - -03

dt;

t 0,

and in conjunction with (3.27): P° [ W, (r) e da; L_F(T)e db;IT1(t)e dt] = f(a, b; t, 'Oda db dt;

(3.28)

a > 0,

b > 0, t > r

where

f(a, b; t, t) _°

(3.29)

( + b) ba

nr3/2 (t - -0312 exP 1

b2

2(t - r)

(a + 6)21 2t j .

We shall employ (3.28) in order to derive, at a given time t e (0, co), the trivariate density for the location W, of the Brownian motion; its local time L(t) = L1(0) at the origin; and its occupation time r+(t) of (0, co) as in (3.3), up to t. 3.9 Proposition. For every finite t > 0, we have (3.30)

P°[ W, e da; L(t) e db; r+(t)e dt] f(a, b; t, -r)da db dr; a > 0, b > 0, 0 0, 0 0 need be established; the one for

a < 0 follows from the former and from the observation that the triples (W L(t), r,(0) and ( -W L(t), t - F+(t)) are equivalent in law. Now in order to establish (3.30) for a > 0, one could write formally 1

dtr[147+(t)e 1

da; L+(r)e db; rvcoe dt]

= -PqW e da; L(t)e db;1-.,(t)e dt]; a > 0, b > 0, 0 < r < t, dt '


424

and then appeal to (3.28). On the left-hand side of this identity, r is fixed and we have a density in (a, b, t); on the right-hand side, t is fixed and we have a density in (a, b, r). Because the two sides are uniquely determined only up to sets of Lebesgue measure zero in their respective domains, it is not clear how this identity should be interpreted. We offer now a rigorous argument along these lines; we shall need to recall the random variable fl, of (2.6), as well as the following auxiliary result. 3.11 Problem. For a e gl, t > 0, c > 0 we have P°

(3.31)

[

max Ws > a + c; min Ws < a - ci = o(h) t<s 0: P°[W+(r) > a; t

(3.32)

TV (r) < t + h; fl, < t + h] = o(h)

P°[14/, > a; fl, < t + h] = o(h)

(3.33) as h 1 O.

PROOF OF PROPOSITION 3.9. For arbitrary but fixed a > 0, b > 0, t > 0, r e (0, t)

we define the function co

F(a, b; t, r) A

(3.34)

f(a, 13; t, r) da d13 6

a

which admits, by virtue of (3.28), (3.32), the interpretation (3.35)

F(a, b; t, r) 1

= lim -P°[W+(r) > a; L+(t) > b; t _ a; L+(t) > b; t ._ t + hl,

we obtain (3.37)

P°[W.,(r)> a+ e; L +(r) >b; t 51T1(r) a; A]

< P° [ max Ws > a + c; 241-P°[ min t<s a;

Al = o(h),

6.4. Elastic Brownian Motion

425

by virtue of (3.31). Dividing by h in (3.37) and then letting h 10, we obtain in conjunction with (3.35), (3.32): (3.38)

1

F(a + e, b; t, t) < lim - PqW, > a; L(t) > b; t -h < r+(t) < a T-1.7i h

Similarly, (3.38)

P°[Wt> a; A] - PG1W+(z)> a- e; L+(z)>b; tl-Z1(T) t+h]

< P° [ max Ws > a; /1]- P°[ min Ws > a -E; Al = o(h), t<s a; L(t) > b; T - h < F+(t)

t].

h.Lo h

Letting E ,I, 0 in both (3.38), (3.39) we conclude that 1

F(a, b; t, T) = lim -P°[W, > a; L(t) > b; i - h < F,(1) _. 0 follows in a straightforward manner.

0

3.12 Remark. From (3.30) one can derive easily the bivariate density (3.40)

P° [24(0) e db; r+ (0 e dr] =

bte-'21"`-') 4irr 3/2 (t - -r)3/2

db di;

b > 0, 0 < r < t

as well as the arc-sine law of Proposition 4.4.11.

The reader should not fail to notice that for a < 0, the trivariate density of (3.30) is the same as that for (W M 0,) in Proposition 2.8.15, for M, = maxo.(s , Ws, et = sup {s .5 t; Ws = AI. This "coincidence" can be explained by an appropriate decomposition of the Brownian path {Ws; 0 < s < t }; cf. Karatzas & Shreve (1987).

6.4. Elastic Brownian Motion This section develops the concept of elastic Brownian motion as a tool for computing distributions involving Brownian local time at one or several points. This device allows us to study local time parametrized by the spatial variable, and it is shown that with this parametrization, local time is related to a Bessel process (Theorem 4.7). We use this fact to prove the DvoretzkyErdi5s-Kakutani Theorem 2.9.13.


426

We employ throughout this section the notation of Section 3.6. In particular, W = {W, F; 0 5_ t < co}, (C/, F), {Px}x RI will be a one-dimensional Brownian family with local time (4.1)

1

L,(a) = lim - meas {0 40 4a

t; 1W- al 5 a}; 05t 0; L,(a,) > r}; (4.2)

r > 0, and

0; L,(a,) > R, for some 1 _ 0; Lt(0) > 4;

(4.41)

For fixed b 0 0, the process N, A Lp.(b); P°, and Eoe-aNs

(4.42)

= exp

0 ._ s < co is a subordinator under as

a, s > 0.

1 + alb'

PROOF. Let Lt = L,(0), and recall from Problem 3.6.18 that lim, L, = a), P° a.s. The process N is obviously nondecreasing and right-continuous with N, = 0, a.s. P°. (Recall from Problem 3.6.13 (iii) that po = 0 a.s. P°.) We have

the composition property p, = p3 + p, o O; 0 < s < t, which, coupled with the additive functional property of local time, gives P ° -a.s.:

0 < s < t.

isi, - Als = Lpt_.08..(b) 0 Op.;

Note that W, = 0 a.s. According to the strong Markov property as expressed in Theorem 2.6.16, the random variable Lp,_.00..(b) o Op. is independent of Fi,. and has the same distribution as L ts (b) = N,,. This completes the proof that

N is a subordinator. As for (4.42), we have from Problem 3.4.5 (iv) for a > 0, /3 > 0: q --'A E°

ro° exp{ -As - ocLp.(b)} ds = E°

f

exp{ - 134 - aL,(b)} dL,

o

E0

T1,

f

e-pc., dLt

o

+ E° [e-PLTb .i'm exp { -13L, 0 OTb - aL,(b) o 0Tb} d(L, 0 OT,). o

The first expression is equal to [1 - E° e-PLTb]/ 11, and by the strong Markov property, the second is equal to ee-flurb times

436


Eb J

exp {

134 - aL,(b)} dL,

03

= Eb f

exp{

/IL, - aL,(b)} dL,

= Eb[e-21-70(b)

f exp{ - fL, o °To

To

aLt(b) OTo} d(Lt o 0To)

03

= Eqe-"-To(b)]- E° f

oo

exp { -13L, - aL,(b)} dL,

= q Eb[e'LT0(b)] = q- E°[e'LTb]. Therefore, 1

q = -[1 - E°e-PLTb] + q E° [e-I3LTb] E°[e-2LTb], and (4.40) allows us to solve for i

CO

e-fisE°(e-"i's)ds =

(13

+

q

1 + 001

Inversion of this transform leads to (4.42).

4.13 Remark. Recall the two independent, reflected Brownian motions WI_ of Theorem 3.1, along with the notation of that section. If b < 0 in Lemma 4.12, then the subordinator N is a function of W_, and hence is independent of W.+. To see this, recall the local time at 0 for W_: L_(t) A Li(t)(0), and let Lb_(T)

L,w(b)

be the local time of W_ at b. Both these processes can be constructed from W_ (see Remark 3.2 for L_), and so both are independent of The same is true for

FAO = inf {T > 0: LAO > and hence also for Lb-(F-(P.0) = Lrit(rApo)(b).

But Wp. = 0 a.s., and so fi(ps + a) > 11(0 for every e > 0 (Problem 2.7.18 applied to the Brownian motion W o Op.). It follows from Problem 3.4.5 (iii) that TT' (1"_(ps)) = p and so Lb_.(f_(ps))=-- N,. A comparison of (4.42) with (2.14) shows that for the subordinator N, we have m = 0 and Levy measure u(d() =

4.14 Proposition (D. Williams (1969)). In the notation of Section 3, and for fixed numbers a > 0, t > 0, b < 0, we have a.s. P°:

437

6.5. Transition Probabilities of Brownian Motion with Two-Valued Drift

(4.43)

E° [exp { - al,,,I(t)(b)} I W+(u); 0 

'(i); Ls(0) = L

= inf {s _.- IT'(x); Ws = 0},

i (0(0)}

a.s. P°.

Because W. > 0 for r_ +' (0 _< u _< p,r.v.(0) = pw,), the local time 4(6) must be constant on this interval. Therefore, with N the subordinator of Lemma 4.12, Nwo = L +(,)(b) = 1-;,(0(b), a.s. P°. Remark 4.13, together with relation (4.42), gives

E° [exp { -akso} I W+ (u); 0< u< co] = E° [exp{ - aNt} ]It=1.-,.(r) (xL +(t)

= exp { 1

}

+ albl '

a.s. P°.

4.15 Exercise.

(i) Show that for # > 0, a > 0, b < 0, we have E0 [exp { - fil.. T(b)} I W+(u); 0

u < co }]

fiL T.(0) 1 1

1 + flibi Y

a.s. P°.

(ii) Use (i) to prove that for a > 0, /3 > 0, a > 0, b < 0, Eb[expl -13LT.(b) - aLT.(0)}]

1

f2(a, a; a + Ibl, 11)'

where f2 is given by (4.22), (4.23). (This argument can be extended to provide an alternate proof of Lemma 4.9; see McKean (1975)).

6.5. An Application: Transition Probabilities of Brownian Motion with Two-Valued Drift Let us consider two real constants Coo < Os, and denote by V the collection of Borel-measurable functions b(t, x): [0, co) x 01 [On, Or]. For every b E W and

x ell, we know from Corollary 5.3.11 and Remark 5.3.7 that the stochastic integral equation (5.1)

X, = x +

1 t b(s, Xs) ds + 14(t;

0 s t < co

o

has a weak solution (X, W), (CI,

,Px), {,i} which is unique in the sense of


438

probability law, with finite-dimensional distributions given by (5.3.11) for 0 ti < t2 < < t = t < co, I- e ROI"): (5.2)

Px[(X,,,.

.

. , xtn)

= Ex [1{(}v, ,

e F] }f,

)E ,-) exp

If

t

b(s, Ws)dW, -

1

1 b2(s, Ws) ds}1. 0

o

Here, {147 .97,}, (0,39, {P},,E R is a one-dimensional Brownian family. We shall take the point of view that the drift b(t, x) is an element of control,

available to the decision maker for influencing the path of the Brownian particle by "pushing" it. The reader may wish to bear in mind the special case 00 < 0 < 0,, in which case this "pushing" can be in either the positive or the negative direction (up to the prescribed limit rates 0, and 0,, respectively). The goal is to keep the particle as close to the origin as possible, and the decision maker's efficacy in doing so is measured by the expected discounted quadratic deviation from the origin

J(x; b) = Ex J

e- xi X, dt o

for the resulting diffusion process X. Here, a is a positive constant. The control problem is to choose the drift b. e 0/i for which J(x; b) is minimized over all b eqi:

J(x; b.) = min J(x; b), V x e R.

(5.3)

bell

This simple stochastic control problem was studied by Bend, Shepp & Witsenhausen (1980), who showed that the optimal drift is given by b.(t,x) = u(x); 0 < t < co, x e R and (5.4)

u(x) 4- 101; Bo;

x < 6} x>6 '

1

6

'-4-

1

N/ q + 2a + 0, q + 2aL - Bo

This is a sensible rule, which says that one should "push as hard as possible

to the right, whenever the process Z, solution of the stochastic integral equation (5.5)

Z, = x + f t u(Zs) ds + W,; 0 < t < oo, o

finds itself to the left of the critical point 6, and vice versa." Because there is no explicit cost on the controlling effort, it is reasonable to push with full force up to the allowed limits. If 0, = -0, = 0, the situation is symmetric and 6 = 0. Our intent in the present section is to use the trivariate density (3.30) in order to compute, as explicitly as possible, the transition probabilities (5.6)

fit(x, z) dz = Ex [Z, E dz]


439

of the process in (5.5), which is a Brownian motion with two-valued, statedependent drift. In this computation, the switching point 6 need not be related to 00 and 0,. We shall only deal with the value 6 = 0; the transition probabilities for other values of 6 can then be obtained easily by translation. The starting point is provided by (5.2), which puts the expression (5.6) in the form (5.7)

z)dz = Ex[l- {w, caz} exP I

f

u(Ws) dllis -

,1

J u2(Ws) ds}1.

L

o

,1

Further progress requires the elimination of the stochastic integral in (5.7). But if we set

u(y)dy =

f(z)

0 z; z < 0 100z; z > 0

we obtain a piecewise linear function, for which the generalized Ito rule of Theorem 3.6.22 gives

f(W) = J(Wo) +

u(His)dW, + (00 - 0,)L(t)

J 0

where L(t) is the local time of W at the origin. On the other hand, with the notation (3.3) we have (Ws) ds = tfi t + (01, - EDF.,(t), 0

and (5.7) becomes (5.8)

/3,(x; z)dz = exp[f(z)

- f(x) - Od

itiO /10 exp

{NO,

- 00) + (eq -

I:" [W, e dz; L(t)e db; 1-+(t)e dr].

It develops then that we have to compute the joint density of (W L(t), 1-+(t)) under F.', for every x e R, and not only for x = 0 as in (3.30). This is accomplished with the help of the strong Markov property and Problem 3.5.8; in the notation of the latter, we recast (3.30) for b > 0, 0 < t < t as

P°[W,eda; L(t) e db;1-+(t)e dr]

J2h(t; b, 0)h(t - r; b - a, 0) da db d-r; 12h(t b, 0)h(r; b + a, 0) da db d-r;

-

and then write, for x > 0 and a < 0:

a 0,

440


Px[W,e da; L(t) e db; 1"÷(t)e d-r]

(5.9)

= Px[W, e da; L(t) e db;F+(t)e di; T0

= jo Px[Wie da; L(t)e db; 1-+(t)e d-r1T0 = s] 1' [T0 e ds]

= J i P° [W,_, e da; L(t - s) e db; F +(t - s)e dt - s]- h(s; x, 0) ds 0

= 2h(t; b + x, 0)h(t - t; b - a, 0) da db ch. For x > 0, a > 0 a similar computation gives (5.10)

Px[W,e da; L(t)e db; FAO e dr] = 2h(t - t; b, 0)h(t; b + a + x, 0)dadb dr, and in this case we have also the singular part

Px[Wieda; L(t) = 0, FAO = t] = Px[W,e da; T0 > t] = p_(t; x,a)da

(5.11)

[p(t; x, a) - p(t; x, - a)] da (cf. (2.8.9)). The equations (5.9)-(5.11) characterize the distribution of the triple L(t),1",(t)) under Px. Back in (5.8), they yield after some algebra: f),(x, z) =

(5.12) 2

x > 0, z < 0

c° ft e2be, h(t - r; b - z, -0,)h(t; x + b, -00)d-c db; J0

200

J.te200,+zoo)h(t_ t; b, -01)h(t; x + b + z, -00)dr db

2

0

1:0

1

2nt

[exp

(X -Z

0002} 2t

exp

(x

z2

t

"2

4 ];

x > 0, z > 0.

Now the dependence on 00, 0, has to be invoked explicitly, by writing z; 00,0,) instead of 15,(x, z). The symmetry of Brownian motion gives (5.13)

/5,(x, z; 00, 0, ) = fit( -x, -z; -0,, -00),

and so for x < 0 the transition density is obtained from (5.12) and (5.13). We conclude with a summary of these results.

5.1 Proposition. Let u: R [00, 0,] be given by (5.4) for arbitrary real 5, and let Z be the solution of the stochastic integral equation (5.5). In the notation of (5.6), + 5, z + 5) is given for every z e R, 0 < t < oo by (i) the right-hand side of (5.12) if x Z 0, and


441

(ii) the right-hand side of (5.12) with (x, z, 00, 01) replaced by (- x, - z, -01, - 00), if x 0.

5.2 Remark. In the special case 01 = -00 = 0 > 0 = (5, the integral term in the second part of (5.12) becomes 2

f

t

so

e20(b-z)h(t - T; b, - 0)h(t; x + b + z, 0)dt db

0

0

ft e_20zh(t - T; b, 0)h(t; x + b + z, 0) dt d0

=2

0

0

\/2nt

=2

e'h(t; x + 2b + z, 0)db

1

(x + z + 002

[exp

2t

0021

(v

96,-26.1 c° exp

20x}

dvi,

x+.

where we have used Prob em 3.5.8 again. A similar computation simplifies also the first integral in (5.12); the result is (5.14)

"Mx, z) = 1

,/2nt

[exp

(x -z - 002}

exp

9e-20z

2t

(v - 0021 dvi 2t

x+ z

x 1

.12nt

(x -z

[exp 120x

002/

exp

2t

z>a

(V -2t9t)2}dvi;

x-z

x > 0, z < O. When 0 = 1 and x = 0, we recover the expression (3.19).

5.3 Exercise. When 01 = -00 = 1, S = 0, show that the function v(t, x) Ex(Z,2); t > 0, x ER is given by (5.15)

1

v(t, x) = +

.N/ t

{03(1

ezixi

- t - 1)exp

-

t

(I xi 2-t t)2

.1}(121(lxi

_)[i 2

(D(Ixl

t)

442


with 41)(z) = (1/.\/2n)Sz e-u212 du, and satisfies the equation

v, = iv - (sgn x) v,

(5.16)

as well as the conditions lim v(t, x) = x2,

lim v(t, x) =

(4.0

t-oco

1

2

5.4 Exercise (Shreve (1981)). With 01 = -0o = 1, 6 = 0, show that the function v of (5.15) satisfies the Hamilton-Jacobi-Bellman equation v, =

(5.17)

-2 vx + min (uvx) 1

on (0, co) x R.

1u15.1

As a consequence, if X is a solution to (5.1) for an arbitrary, Borel-measurable [ - 1, 1] and Z solves (5.5) (under Ex in both cases), then b: [0, co) x (5.18)

0 < t < CO.

Ex Z,2 < Ex X i2;

In particular, Z is the optimally controlled process for the control problem (5.3).

6.6. Solutions to Selected Problems 2.5. If A is a-finite but not finite, then there exists a partition {1-11}711 g_ if of H with 0 < 2(H1) < co for every i > 1. On a probability space (SI, P), we set up independent sequences 10); j e IN,}711 of random variables, such that for every i 1:

(1) a), are independent, (ii) N, A a) is Poisson with EN; = 2(H1), and (iii) P[c ji)E C] = 2(C n H;)12(H;); = 1, 2, ....

E7±1 1,(0)); CE if, is a Poisson random measure for every i > 1, and v1, v2, ... are independent. We show that v v1 is a Poisson random measure with intensity A. It is clear that Ev(C) = E Er_1 v;(C n =

As before, MC)

Er=, 2(C n H;) = A(C) for all C E if, and whenever 2(C) < o o , v(C) has the proper distribution (the sum of independent Poisson random variables being Poisson). Suppose 2(C) = co. We set 2 = 2(C n H.), so v(C) is the sum of the independent, Poisson random variables Ivi(C)IT_I, where Evi(C) = A,. There is a number a > 0

such that 1 -e > 2/2; 0 < 2 < a, and so with b A 2(1 - ca): CO

CO

E P[1,;(C) > 1] = E (1 1=1

1=1

-

1

co

- E (A, Ab) =oo 2 i=1

because Er=, 2 = 2(C) = co. By the second half of the Borel-Cantelli lemma (Chung (1974), p. 76), P[v(C) = co] = P[vi(C) 1 for infinitely many i] = 1. This completes the verification that v satisfies condition (i) of Definition 2.3; the verification of (ii) is straightforward.


443

2.12. If for some ye D[0, co), t > 0, and e > 0 we have n(y; (0, t] x ((, co)) = co, then we can find a sequence of distinct points {tk }T=1 g (0, t] such that I y(th ) y(tk - )1 > (; k > 1. By selecting a subsequence if necessary, we may assume without loss of generality that {tk}ic_, is either strictly increasing or else strictly decreasing to a limit Be [0, t]. But then y(t k)} ,T=, and { y(t, -) }k 1 converge to the same limit (which is y(0 -) in the former case, and y(0) in the latter). In either case we obtain a contradiction. For any interval (t, t + h], the set Ar.,+h(e)

h] such that I y(s) - y(s -)I > e}

{ y E D [0, co); 3 s E (I, .0

.0

=un

k=1 m=1

1

D[O, oo); y(q) - y(r)I > e

U

i a; t

I (r) < t + h; Mt) < t + h]

< P° [ max Ws > a; min W < 0], t<s0 4 universal

Sample path (see also Brownian sample path) I Sample space

1

Samuelson, P.A. 398 Scale function 339, 343 Schauder functions 57 Schmidt, W. 329, 332, 335, 396 Schwarz, G. 174 Sekiguchi, T. 237 Semigroup 395 Semimartingale (see Continuous semimartingale) Sharpe, M.J. 237 Shepp, L.A. 438, 445, 446 Shift operators 77, 83

73

Simon, B.

280 Simple process 132

Skorohod, A.V.

236, 323, 395, 397 Skorohod equation 210 Skorohod metric 409 Skorohod space of RCLL functions 409 Smoluchowski equation 361, 397 Sojourn time (see Occupation time) Speed measure 343, 352 as an invariant measure 353, 362 Square-integrable martingale cross-variation of 31, 35 ItO integral representation of Brownian 182 metric on the space of 37

orthogonality of 31 quadratic variation of Stability theory 395 State space

S

23, 46, 200, 236, 395

31

1

Stationary increments 48 Stationary process 103 Statistical communication theory 395 Statistical tests of power one 279 Stochastic calculus 128, 148, 150 Stochastic calculus of variations 398 Stochastic control (see Stochastic optimal control) Stochastic differential 145, 150, 154 Stochastic differential equation 281ff, 394

approximation of 295ff, 395 comparison of solutions of 293, 395, 446 explosion time of 329 functional 305

468

Index

Stochastic differential equation (cont.) Gaussian process as a solution of 355, 357 linear 354ff one-dimensional 329ff Feller's test for explosions 348350, 396 invariant distribution for 353 nonexplosion when drift is zero 332 pathwise uniqueness 337, 338, 341, 353, 396 strong existence 338, 341, 396 strong uniqueness 338, 341, 396 uniqueness in law 335, 341 weak existence 334, 341 weak solution in an interval 343 pathwise uniqueness 301, 302, 309310 strong existence 289, 310-311, 396 strong solution 283, 285 strong uniqueness 286, 287, 291, 396 uniqueness in law 301, 302, 304305, 309, 317 weak existence 303, 310, 323, 332 weak solution 129, 300 weak solution related to the martingale problem 317-318 well-posedness 319-320 with respect to a semimartingale 396 Stochastic integral 129ff characterization of 141-142 definition of 139 with respect to a martingale having absolutely continuous quadratic variation 141 Stochastic integral equation (see Stochastic differential equation) of Volterra type 396 Stochastic maximum principle 446 Stochastic optimal control 284, 379, 395, 438, 446 Stochastic partial differential equations 395 Stochastic process adapted to a filtration 4 finite-dimensional distributions of 2 Gaussian 103 1

355ff

Gauss-Markov

LCRL 4 measurable 3 modification of 2

of class D 24 of class DL 24 progressively measurable

4

RCLL 4 sample path of simple

I

132

state space of stationary 103 stopped at a stopping time 9 zero-mean 103 Stochastic systems 395 Stone, C. 253, 278 Stopping time 6 accessible 46 predictable 46 totally inaccessible 46 Stratonovich integral (see Fisk-Stratonovich integral) Stricker, C. 168, 189 Strong existence (see Stochastic differential equation) Strong Markov family 81-82 universal filtration for 93 Strong Markov process 81-82 augmented filtration for 90-92 classification of boundary behavior of 397 Strong Markov property 79, 127 equivalent formulations of 81-84 extended 87 for Brownian motion 86, 127 for Poisson processes 89 for solutions of the martingale problem 322 Strong solution (see Stochastic differential equation) Strong uniqueness (see Stochastic differential equation) Stroock, D.W. 127, 236, 238, 283, 311, 322, 323, 327, 395, 396, 397, 398, 416 Submartingale (see also Martingale) 11 backward 15 convergence of 17-18 1

469

Index

Doob-Meyer decomposition of 24ff inequalities for 13-14 last element of 12-13, 18 maximal inequality for 14 optional sampling of 19-20 path continuity of 14, 16 regular 28 uniform integrability of 18 Subordinator 405 Supermartingale (see also Martingale) 11 exponential 147, 153, 198ff not a martingale 201 last element of 13 Sussmann, H. 395 Synonimity of processes 46

T Taksar, M. 446 Tanaka, H. 237, 301, 397 Tanaka formulas for Brownian motion 205, 215 Tanaka-Meyer formulas for continuous semimartingales 220 Taylor, H. 395, 445 Taylor, S.J. 445 Teicher, H. 46 Tied-down Brownian motion (seeBrownian bridge) Tightness of a family of probability measures 62 Time-change for martingales 174 for one-dimensional stochastic differential equations 330ff for stochastic integrals 176-178 Time-homogeneous martingale problem (see Martingale problem) Toronjadze, T.A. 395 Trajectory Transformation of drift (see Girsanov's theorem) Transition density for absorbed Brownian motion 98 for Brownian motion 52 for diffusion process 369 for reflected Brownian motion 97 1

Trotter, H.F. 207, 237 Trudinger, N.S. 366 Tychonoff uniqueness theorem

255

U

Uhlenbeck, G.E. 358 Uniformly elliptic operator 364 Uniformly positive definite matrix 327 Uniqueness in law (see Stochastic differential equation) Universal filtration 93 Universal cr-field 73 Universally measurable function 73 Uperossings 13-14 Usual conditions (for a filtration) 10 379 Utility function

V

Value function 379 Van Schuppen, J. 237 Varadhan, S.R.S. 127, 236, 238, 279, 283, 311, 322, 323, 327, 396, 397

Variation of a process 32 Ventsel, A.D. 237 Veretennikov, A. Yu. 396 Ville, J. 46

Wald identities 141, 168, 197 Walsh, J.B. 395 Wang, A.T. 212, 237 Wang, M.C. 358 Watanabe, S. 46, 130, 149, 156, 232, 236, 237, 238, 291, 292, 293, 299, 306, 308, 309, 395, 396, 397, 398, 445, 446 Weak convergence of probability measures 60-61 Weak solution (see Stochastic differential equation) Weak uniqueness (see Stochastic differential equation, uniqueness in law) Wentzell, A.D. 39, 82, 117, 127 White noise 395

470 Widder, D.V. 256. 277, 279 Widder's uniqueness theorem 261 Wiener. N. 48. 110. 126, 236. 278 Wiener functional (see Brownian functional) Wiener martingale (see Brownian martingale) Wiener measure d-dimensional 72 one-dimensional 71 Wiener process (see Brownian motion) Williams. D. 127. 445

first formula of 421-422. 423 second formula of 436-437 Williams, R. 236 Wintner, A. 279 Witsenhausen, H.S. 438. 446 Wong. E. 169, 237. 395

Index

Y

Yamada. T. Yan. J . A .

291. 308, 309. 395. 396 189

Yor, M. 44, 46. 126. 127. 168. 237, 238. 445 Yushkevich, A.A. 98. 127, 242

Zakai, M. 169. 395. 398 Zaremba, S. 278 Zaremba's cone condition 250 Zero-one laws Blumenthal 94 Engelbert-Schmidt 216, 396 Levy 46 Zvonkin, A.K. 396 Zygmund, A. 110. 236

Brownian Motion and Stochastic Calculus (Graduate Texts in Mathematics)

Brownian Motion and Stochastic Calculus (Graduate Texts in Mathematics)

Brownian Motion and Stochastic Calculus