Stochastic Mechanics Random Media Signal Processing and Image Synthesis Mathematical Econ omics and Finance Stochastic Optimization
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21
Stochastic Control Stochastic Models in Life Sciences
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Applications of Mathematics 1 Fleming/Rishel, Deterministic and Stochastic Optimal Control (1975)
2 Marchuk, Methods of Numerical Mathematics 1975, 2nd. ed. 1982) 3 Balakrishnan, Applied Functional Analysis (1976, 2nd. ed. 1981)
4 Borovkov, Stochastic Processes in Queueing Theory (1976)
5 Liptser/Shiryaev, Statistics of Random Processes I: General Theory (1977, znd. ed. 2001) 6 Liptser/Shiryaev, Statistics of Random Processes II: Applications (1978, 2nd. ed. 2001) 7 Vorob'ev, Game Theory: Lectures for Economists and Systems Scientists (1977) 8 Shiryaev, Optimal Stopping Rules (1978) 9 Ibragimov/Rozanov, Gaussian Random Processes (1978) 10 Wonham, Linear Multivariable Control: A Geometric Approach (1979, 2nd. ed. 1985) u
Hida, Brownian Motion (1980)
12 Hestenes, Conjugate Direction Methods in Optimization (1980) 13 Kallianpur, Stochastic Filtering Theory (1980) 14 Krylov, Controlled Diffusion Processes (1980) 15 Prabhu, Stochastic Storage Processes: Queues, Insurance Risk, and Dams (1980) 16 17 18 19 20 21
Ibragimov/Has'minskii, Statistical Estimation: Asymptotic Theory (1981) Cesari, Optimization: Theory and Applications (1982) Elliott, Stochastic Calculus and Applications (1982) Marchuk/Shaidourov, Difference Methods and Their Extrapolations (1983) Hijab, Stabilization of Control Systems (1986) Protter, Stochastic Integration and Differential Equations (1990, znd. ed. 2003)
22 Benveniste/Metivier/Priouret, Adaptive Algorithms and Stochastic Approximations (1990) 23 Kloeden/Platen, Numerical Solution of Stochastic Differential Equations (1992, corr. 3rd printing 1999) 24 Kushner/Dupuis, Numerical Methods for Stochastic Control Problems in Continuous Time (1992) 25 Fleming/Soner, Controlled Markov Processes and Viscosity Solutions (1993) 26 Baccelli/Bremaud, Elements of Queueing Theory (1994, 2nd ed. 2003) 27 Winkler, Image Analysis, Random Fields and Dynamic Monte Carlo Methods (1995, 2nd. ed. 2003) 28 Kalpazidou, Cycle Representations of Markov Processes (1995) 29 Elliott/Aggoun/Moore, Hidden Markov Models: Estimation and Control (1995) 30 Hermlndez-Lerma/Lasserre, Discrete-Time Markov Control Processes (1995) 31 Devroye/Gyorfi/Lugosi, A Probabilistic Theory of Pattern Recognition (1996) 32 Maitra/Sudderth, Discrete Gambling and Stochastic Games (1996) 33 Embrechts/Kliippelberg/Mikosch, Modelling Extremal Events for Insurance and Finance (1997, corr. 4th printing 2003) 34 Duflo, Random Iterative Models (1997) 35 Kushner/Yin, Stochastic Approximation Algorithms and Applications (1997) 36 Musiela/Rutkowski, Mar tingale Methods in Financial Modelling (1997) 37 Y in, Continuous-Time Markov Chains and Applications (1998) 38 Dembo/Zeitouni, Large Deviations Techniques and Applications (1998) 39 Karatzas, Methods of Mathematical Finance (1998) 40 Fayolle/Iasnogorodski/Malyshev, Random Walks in the Quarter-Plane (1999) 41 Aven/Jensen, Stochastic Models in Reliability (1999) 42 Hernandez-Lerma/Lasserre, Further Topics on Discrete-Time Markov Control Processes (1999) 43 Yong/Zhou, Stochastic Controls. Hamiltonian Systems and HJB Equations (1999) 44 Serfozo, Introduction to Stochastic Networks (1999) 45 Steele, Stochastic Calculus and Financial Applications (2001) 46 Chen/Yao, Fundamentals of Queuing Networks: Performance, Asymptotics, and Optimization (2001) 47 Kushner, Heavy Traffic Analysis of Controlled Queueing and Communications Networks (2001) 48 Fernholz, Stochastic Portfolio Theory (2002) 49 Kabanov/Pergamenshchikov, Two-Scale Stochastic Systems (2003) so Han, Information-Spectrum Methods in Information Theory (2003)
(continued after index)
Philip E. Protter
Stochastic Integration and Differential Equations Second Edition
Springer
Author Philip E. Protter Cornell University School of Operations Res. and Industrial Engineering Rhodes Hall 14853 Ithaca, NY U SA e-mail:
[email protected] Managing Editors B. Rozovskii
M. Yor
Center for Applied Mathematical Sciences University of Southern California 1042 West 36 th Place, Denney Research Building 308 Los Angeles, CA 90089, USA
Universite de Paris VI Laboratoire de Probabilites et Modeles Aleatoires 175, rue du Chevaleret 75013 Paris, France
Mathematics Subject Classification (2ooo): PRIMARY: 60H05, 60H10, 60H20 SECONDARY: 60G07, 60G17, 60G44, 60G51 Cover pattern by courtesy of Rick Durrett (Cornell University, Ithaca) Cataloging-in-Publication Data applied for A catalog record for this book is available from the Library of Congress.
Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at http://dnb.ddb.de
ISSN 0172-4568 ISBN 3-540-00313-4 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer- Verlag. Violations are liable for prosecution under the German Copyright Law. Springer-Verlag Berlin Heidelberg New York a member of BertelsmannSpringer Science + Business Media GmbH http://www.springer.de C Springer-Verlag Berlin Heidelberg 2004
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The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: Erich Kirchner, Heidelberg
Typesetting by the author using a Springer TEX macro package Printed on acid-free paper 41/3142DB-543210
To Diane and Rachel
Preface to the Second Edition
It has been thirteen years since the first edition was published, with its subtitle "a new approach." While the book has had some success, there are still almost no other books that use the same approach. (See however the recent book by K. Bichteler [15] .) There are nevertheless of course other extant books, many of them quite good, although the majority still are devoted primarily to the case of continuous sample paths, and others treat stochastic integration as one of many topics. Examples of alternative texts which have appeared since the first edition of this book are: [32] , [44], [87] , [110] , [186] , [180] , [208] , [216] , and [226] . While the subject has not changed much, there have been new developments, and subjects we thought unimportant in 1990 and did not include, we now think important enough either to include or to expand in this book. The most obvious changes in this edition are that we have added exercises at the end of each chapter, and we have also added Chap. VI which intro duces the expansion of filtrations. However we have also completely rewritten Chap. III. In the first edition we followed an elementary approach which was P. A. Meyer's original approach before the methods of Doleans-Dade. In or der to remain friends with Freddy Delbaen, and also because we now agree with him, we have instead used the modern approach of predictability rather than naturality. However we benefited from the new proof of the Doob-Meyer Theorem due to R. Bass, which ultimately uses only Doob's quadratic martin gale inequality, and in passing reveals the role played by totally inaccessible stopping times. The treatment of Girsanov's theorem now includes the case where the two probability measures are not necessarily equivalent, and we include the Kazamaki-Novikov theorems. We have also added a section on compensators, with examples. In Chap. IV we have expanded our treatment of martingale representation to include the Jacod-Yor Theorem, and this has allowed us to use the Emery-Azema martingales as a class of examples of mar tingales with the martingale representation property. Also, largely because of the Delbaen-Schachermayer theory of the fundamental theorems of mathe matical finance, we have included the topic of sigma martingales. In Chap. V
VIII
Preface to the Second Edition
we added a section which includes some useful results about the solutions of stochastic differential equations, inspired by the review of the first edition by E. Pardoux [191]. We have also made small changes throughout the book; for instance we have included specific examples of Levy processes and their corresponding Levy measures, in Sect. 4 of Chap. I. The exercises are gathered at the end of the chapters, in no particular order. Some of the (presumed) harder problems we have designated with a star ( * ) , and occasionally we have used two stars ( ** ) . While of course many of the problems are of our own creation, a significant number are theorems or lemmas taken from research papers, or taken from other books. We do not attempt to ascribe credit, other than listing the sources in the bibliography, primarily because they have been gathered over the past decade and often we don't remember from where they came. We have tried systematically to refrain from relegating a needed lemma as an exercise; thus in that sense the exercises are independent from the text, and (we hope) serve primarily to illustrate the concepts and possible applications of the theorems. Last, we have the pleasant task of thanking the numerous people who helped with this book, either by suggesting improvements, finding typos and mistakes, alerting me to references, or by reading chapters and making com ments. We wish to thank patient students both at Purdue University and Cornell University who have been subjected to preliminary versions over the years, and the following individuals: C. Benes, R. Cont, F. Diener, M. Di ener, R. Durrett, T. Fujiwara, K. Giesecke, L. Goldberg, R. Haboush, J . Ja cod, H. Kraft, K. Lee, J . Ma, J. Mitro, J. Rodriguez, K. Schiirger, D. Sezer, J. A. Trujillo Ferreras, R. Williams, M. Yor, and Yong Zeng. Th. Jeulin, K. Shimbo, and Yan Zeng gave extraordinary help, and my editor C. Byrne gives advice and has patience that is impressive. Over the last decade I have learned much from many discussions with Darrell Duffie, Jean Jacod, Tom Kurtz, and Denis Talay, and this no doubt is reflected in this new edition. Finally, I wish to give a special thanks to M. Kozdron who hastened the ap pearance of this book through his superb help with M'-TEX, as well as his own advice on all aspects of the book. Ithaca, NY August 2003
Philip Protter
Preface to the First Edition
The idea of this book began with an invitation to give a course at the Third Chilean Winter School in Probability and Statistics, at Santiago de Chile, in July, 1984. Faced with the problem of teaching stochastic integration in only a few weeks, I realized that the work of C. Dellacherie [42] provided an outline for just such a pedagogic approach. I developed this into a series of lectures (Protter [201]), using the work of K. Bichteler [14], E. Lenglart [145] and P. Protter [202], as well as that of Dellacherie. I then taught from these lecture notes, expanding and improving them, in courses at Purdue University, the University of Wisconsin at Madison, and the University of Rouen in France. I take this opportunity to thank these institutions and Professor Rolando Rebolledo for my initial invitation to Chile. This book assumes the reader has some knowledge of the theory of stochas tic processes, including elementary martingale theory. While we have recalled the few necessary martingale theorems in Chap. I, we have not provided proofs, as there are already many excellent treatments of martingale the ory readily available (e.g. , Breiman [23], Dellacherie-Meyer [45, 46], or Ethier Kurtz [71]). There are several other texts on stochastic integration, all of which adopt to some extent the usual approach and thus require the general theory. The books of Elliott [63], Kopp [130], Metivier [158], Rogers-Williams [210] and to a much lesser extent Letta [148] are examples. The books of McK ean [153], Chung-Williams [32], and Karatzas-Shreve [121] avoid the general theory by limiting their scope to Brownian motion (McKean) and to contin uous semimartingales. Our hope is that this book will allow a rapid introduction to some of the deepest theorems of the subject, without first having to be burdened with the beautiful but highly technical "general theory of processes." Many people have aided in the writing of this book, either through dis cussions or by reading one of the versions of the manuscript. I would like to thank J. Azema, M. Barlow, A. Bose, M. Brown, C . Constantini, C. Dellache rie, D. Duffie, M. Emery, N. Falkner, E. Goggin, D. Gottlieb, A. Gut, S. He, J . Jacod, T. Kurtz, J. de Sam Lazaro, R. Leandre, E. Lenglart, G. Letta,
X
Preface to the First Edition
S . Levantal, P. A. Meyer, E. Pardoux, H. Rubin, T. Sellke, R. Stockbridge, C. Stricker, P. Sundar, and M. Yor. I would especially like to thank J. San Mar tin for his careful reading of the manuscript in several of its versions. Svante Janson read the entire manuscript in several versions, giving me support, encouragement, and wonderful suggestions, all of which improved the book. He also found, and helped to correct, several errors. I am extremely grateful to him, especially for his enthusiasm and generosity. The National Science Foundation provided partial support throughout the writing of this book. I wish to thank Judy Snider for her cheerful and excellent typing of several versions of this book.
Philip Protter
Contents
.................................................. 1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1 Basic Definitions and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3 The Poisson Process and Brownian Motion . . . . . . . . . . . . . . . . 1 2 4 Levy Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 5 Why the Usual Hypotheses? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 6 Local Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 7 7 Stieltjes Integration and Change of Variables . . . . . . . . . . . . . . . 39 8 Naive Stochastic Integration Is Impossible . . . . . . . . . . . . . . . . . . 43 Bibliographic Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 Exercises for Chapter I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 Semimartingales and Stochastic Integrals . . . . . . . . . . . . . . . . . 51 1 Introduction to Semimartingales . . . . . . . . . . . . . . . . . . . . . . . . . . 51 2 Stability Properties of Semimartingales . . . . . . . . . . . . . . . . . . . . 52 3 Elementary Examples of Semimartingales . . . . . . . . . . . . . . . . . . . 54 4 Stochastic Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 5 Properties of Stochastic Integrals . . . . . . . . . . . . . . . . . . . . . . . . . 60 6 The Quadratic Variation of a Semimartingale . . . . . . . . . . . . . . 66 7 Ito's Formula (Change of Variables) . . . . . . . . . . . . . . . . . . . . . . . 78 8 Applications of Ito's Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 Bibliographic Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 Exercises for Chapter II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 Semimartingales and Decomposable Processes . . . . . . . . . . . . . 101 101 1 Introduction 2 The Classification of Stopping Times 103 3 The Doob-Meyer Decompositions . 105 4 Quasimartingales . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . 1 16
Introduction I
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Contents
XII
Compensators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 The Fundamental Theorem of Local Martingales . . . . . . . . . . . . 12 4 Classical Semimartingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 Girsanov's Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 The Bichteler-Dellacherie Theorem . . . . . . . . . . . . . . . . . . . . 143 Bibliographic Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 Exercises for Chapter III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 7
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IV General Stochastic Integration and Local Times . . . . . . . . . . . 153 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 2 Stochastic Integration for Predictable Integrands . . . . . . . . . . . . 153 3 Martingale Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 4 Martingale Duality and the Jacod-Yor Theorem on Martingale Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 5 Examples of Martingale Representation . . . . . . . . . . . . . . . . . . . 2 00 6 Stochastic Integration Depending on a Parameter . . . . . . . . . . . 2 05 7 Local Times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 10 8 Azema' s Martingale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 7 9 Sigma Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 33 Bibliographic Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 35 Exercises for Chapter IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 36 .
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V Stochastic Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 243 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 2 The HP Norms for Semimartingales . . . . . . . . . . . . . . . . . . . . . . . . 244 3 Existence and Uniqueness of Solutions . . . . . . . . . . . . . . . . . . . . . 249 4 Stability of Stochastic Differential Equations . . . . . . . . . . . . . . . 25 7 5 Fisk-Stratonovich Integrals and Differential Equations . . . . . . . . 270 6 The Markov Nature of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 7 Flows of Stochastic Differential Equations: Continuity and Differentiability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 8 Flows as Diffeomorphisms: The Continuous Case . . . . . . . . . . . . 310 9 General Stochastic Exponentials and Linear Equations . . . . . . . 32 1 10 Flows as Diffeomorphisms: The General Case . . . . . . . . . . . . . . . 328 1 1 Eclectic Useful Results on Stochastic Differential Equations . . . 338 Bibliographic Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 Exercises for Chapter V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349 .
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VI Expansion of Filtrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 2 Initial Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356 3 Progressive Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369 4 Time Reversal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 77 Bibliographic Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 Exercises for Chapter VI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384 .
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XIII
Contents
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389 403 407
Introduction
In this book we present a new approach to the theory of modern stochastic integration. The novelty is that we define a semimartingale as a stochastic pro cess which is a "good integrator" on an elementary class of processes, rather than as a process that can be written as the sum of a local martingale and an adapted process with paths of finite variation on compacts: This approach has the advantage over the customary approach of not requiring a close analysis of the structure of martingales as a prerequisite. This is a significant advantage because such an analysis of martingales itself requires a highly technical body of knowledge known as "the general theory of processes." Our approach has a further advantage of giving traditionally difficult and non-intuitive theorems (such as Stricker's Theorem) transparently simple proofs. We have tried to capitalize on the natural advantage of our approach by systematically choos ing the simplest, least technical proofs and presentations. As an example we have used K . M. Rao's proofs of the Doob-Meyer decomposition theorems in Chap. III, rather than the more abstract but less intuitive Doleans-Dade measure approach. In Chap. I we present preliminaries, including the Poisson process, Brown ian motion, and Levy processes. Naturally our treatment presents those prop erties of these processes that are germane to stochastic integration. In Chap. II we define a semimartingale as a good integrator and establish many of its properties and give examples. By restricting the class of integrands to adapted processes having left continuous paths with right limits, we are able to give an intuitive Riemann-type definition of the stochastic integral as the limit of sums. This is sufficient to prove many theorems (and treat many applications) including a change of variables formula ( "Ito's formula" ) . Chapter III is devoted to developing a minimal amount of "general the ory" in order to prove the Bichteler-Dellacherie Theorem, which shows that our "good integrator" definition of a semimartingale is equivalent to the usual one as a process X having a decomposition X = M + A, into the sum of a local martingale M and an adapted process A having paths of finite variation on compacts. Nevertheless most of the theorems covered en route (Doob-
2
Introduction
Meyer, Meyer-Girsanov) are themselves key results in the theory. The core of the whole treatment is the Doob-Meyer decomposition theorem. We have followed the relatively recent proof due to R. Bass, which is especially simple for the case where the martingale jumps only at totally inaccessible stopping times, and in all cases uses no mathematical tool deeper than Doob's quadratic martingale inequality. This allows us to avoid the detailed treatment of nat ural processes which was ubiquitous in the first edition, although we still use natural processes from time to time, as they do simplify some proofs. Using the results of Chap. III we extend the stochastic integral by continu ity to predictable integrands in Chap. IV, thus making the stochastic integral a Lebesgue-type integral. We use predictable integrands to develop a theory of martingale representation. The theory we develop is an £2 theory, but we also prove that the dual of the martingale space 1i1 is BMO and then prove the Jacod-Yor Theorem on martingale representation, which in turn allows us to present a class of examples having both jumps and martingale representation. We also use predictable integrands to give a presentation of semimartingale local times. Chapter V serves as an introduction to the enormous subject of stochastic differential equations. We present theorems on the existence and uniqueness of solutions as well as stability results. Fisk-Stratonovich equations are pre sented, as well as the Markov nature of the solutions when the differentials have Markov-type properties. The last part of the chapter is an introduction to the theory of flows, followed by moment estimates on the solutions, and other minor but useful results. Throughout Chap. V we have tried to achieve a balance between maximum generality and the simplicity of the proofs. Chapter VI provides an introduction to the theory of the expansion of fil trations (known as "grossissements de filtrations" in the French literature) . We present first a theory of initial expansions, which includes Jacod's Theorem. Jacod's Theorem gives a sufficient condition for semimartingales to remain semimartingales in the expanded filtration. We next present the more diffi cult theory of progressive expansion, which involves expanding filtrations to turn a random time into a stopping time, and then analyzing what happens to the semimartingales of the first filtration when considered in the expanded filtration. Last, we give an application of these ideas to time reversal.
I Preliminaries
1 Basic Definitions and Notation
We assume as given a complete probability space (0, :F, P). In addition we are given a filtration (:Ft)or} 1 is a cadlag process, and Xr = lA. Hence A 1
1A
is the indicator function of A
:
1A (w)
=
{
1
'
0,
wE A, w �A.
E
Q, and :Fr
c
Q.
6
I Preliminaries
Next let X be an adapted dtdlag process. We need to show Xr is Fr measurable. Consider X(s, w) as a function from [0, oo) x 0 into R Construct cp : { T < t} � [0, oo) x 0 by cp(w) = (T(w), w) . Then since X is adapted and cadlag, we have Xr = X ocp is a measurable mapping from ( {T < t}, Ftn{T < t}) into (IR, !3) , where l3 are the Borel sets of R Therefore {w: X(T(w),w) E
is in Ft, and this implies Xr
E
B} n {T < t}
Fr . Therefore Q C Fr.
D
We leave it to the reader to check that if S < T a.s., then Fs C Fr, and the less obvious (and less important) fact that Fs nFr = Fs11r· If X andY are cadlag, then Xt = yt a.s. each t implies that X andY are indistinguishable, as we have already noted. Since fixed times are stopping times, obviously if Xr = Yr a.s. for each finite stopping time T, then X and Y are indistinguishable. If X is cadlag, let .6.X denote the process .6.Xt = Xt - Xt - . Then .6.X is not cadlag, though it is adapted and for a.a. w , t � .6.Xt = 0 except for at most countably many t. We record here a useful result. 7.
Let X be adapted and cadlag. If .6.Xr1 { r 0} is countable a.s. since X is cadlag. Moreover Theorem
:
1 {t : I .6.Xtl > 0} = u {t : 1 .6.Xt l > -} n n= 1 and the set { t I.6.Xt l > 1 / } must be finite for each n, since to < oo. Using Theorem 4 we define stopping times for each n inductively as follows: Tn ' 1 = . f{ t > 0: I.6.Xtl > -n1 } 1 k n , k > n r , inf{ t > T 1 I.6.Xt l n }. Then Tn , k > Tn k 1 a.s. on { Tn , k 1 < oo }. Moreover, 00
n
:
m
=
,
-
-
:
-
{f.6.Xt l > 0} = U {f .6.Xrn , d{rn,k 0}, n ,k where the right side of the equality is a countable union. The result follows. Corollary. Let T, .6.Xr 1 { r E{Xt} is constant, and hence it has a right continuous modification. Corollary. If X = (Xt)o < t< oo is a martingale then there exists a unique modification Y of X which is dtdlag. Since all martingales have right continuous modifications, we will always assume that we are taking the right continuous version, without any special mention. Note that it follows from this corollary and Theorem 2 that a right continuous martingale is cadlag. Theorem 10 (Martingale Convergence Theorem) . Let X be a right continuous supermartingale, supo< tn} a
Let (Ua )aEA be a subset of L1. The following are equivalent: {i) (Ua)aEA is uniformly integrable. {ii) supaEA E{J Ua l } < oo, and for every c > 0 there exists 8 > 0 such that A E F, P(A) < 8, imply E{IUalA I } < c. 2 Vo:o;t E{Xr lFs} a.s. (resp. = ) . If Tis a stopping time, then so is t 1\ T min (t, T) , for each t > 0. Definition. Let X be a stochastic process and let Tbe a random time. xr is said to be the process stopped at T if xr Xti\ T · Note that if X is adapted and dtdlag and if T is a stopping time, then Theorem
=
=
is also adapted. A martingale stopped at a stopping time is still a martingale, as the next theorem shows. Theorem 18. Let X be a uniformly integrable right continuous martingale, and let T be a stopping time. Then xr = (XtAT )o n (that is, the sequence is a refining sequence). Suppose more over that limn ->oo mesh( 7rn ) = 0. Let 7rn B = "':.t E JBt - Bt; )2• Then limn->oo 7rn B = t a.s., for a standard Brownian motion B . ,
7r
i+
I
18
I Preliminaries
Proof.
We first show convergence in mean square. We have 7rnB - t =
2::= {(Bt'+, - Bt; ) 2 - (t i+ 1 - ti ) } t iE 'Trn i
where Yi are independent random variables with zero means. Therefore
E{(7rn B - t) 2 } = E{ ( l::= Yi ) 2 } = 2::= E{Y? }. i i
Next observe that (Bti + l - BtJZ/ (ti+1 - ti) has the distribution of Z2 , where Z is Gaussian with mean 0 and variance 1. Therefore
which tends to 0 as n tends to oo. This establishes L 2 convergence (and hence convergence in probability as well). To obtain the a.s. convergence we use the Backwards Martingale Conver gence Theorem (Theorem 14). Define
Nn (w) = Ln B = L (Bt; +1 (w) - Bt; (w)) 2 , tiE 'IT-n
for n = -1, -2, -3, . . . . Then it is straightforward (though notationally messy) to show that
E{Nn l Nn
-
1
, Nn - 2 , . . . } = Nn - 1 ·
Therefore Nn is a martingale relative to On = { Nk k < n}, n = -1, -2, . . . By Theorem 14 we deduce limn---+ - oo Nn = limn---+ oo 7rn B exists a.s., and since 1r B converges to t in £2 , we must have limn ---+ oo 1rn B t a.s. as well. 0 n u
,
.
=
Comments. As noted in the proofs, the proof is simple (and half as long) if we conclude only £2 convergence (and hence convergence in probability) , instead of a.s. convergence. Also, we can avoid the use of the Backwards Martingale Convergence Theorem (Theorem 14) in the second half of the proof if we add the hypothesis that Ln mesh(7rn ) < oo The result then follows, after having proved the L 2 convergence, by using the Borel-Cantelli Lemma and Chebsyshev's inequality. Furthermore to conclude only £2 convergence we do not need the hypothesis that the sequence of partitions be refining. .
Theorem 28 can be used to prove that the paths of Brownian motion are of unbounded variation on compacts. It is this fact that is central to the difficulties in defining an integral with respect to Brownian motion (and martingales in general) .
4 Levy Processes
19
For almost all the sample paths t Bt ( ) of a standard Brownian motion B are of unbounded variation on any interval. Proof. Let A = [a, b] be an interval. The variation of paths of B is defined to w,
Theorem 29.
�
w
be
where P are all finite partitions of [a, b]. Suppose P(VA < oo) > 0. Let 1r n be a sequence of refining partitions of [a, b] with lim n mesh(7rn ) = 0. Then by Theorem 28 on {VA < oo } , lim b - a = n�oo
=0
""" L.-t
ti E'Tr n
(Bt;+1 - Bt; ) 2
'
since SUPt; E 7rn IBt 0, for all finite ( u 1 , . . . , un; a 1 , . . . , an), then there exists a Levy process corresponding to f.) In particular it follows that if X is a Levy process then for each t > 0, Xt =
'
,
=
=
=
=
=
has an infinitely divisible distribution. Inversely it can be shown that for each infinitely divisible distribution p, there exists a Levy process X such that p, is the distribution of xl . Theorem 30. Let X be a Levy process. There exists a unique modification
Y of X which is cadlag and which is also a Levy process. Proof. Let M;: J, ( u) . For each fixed u in Q, the rationals in IR, the process (Mtu )o dtdlag functions. Let
t
f--->
A = {(w ' u) E n
X
e iuX, (w) ,
with
tE
21
IQl+ , are the restrictions to IQl+ of
IR .. e iuX, (w) ' t E /(1\ "'!+ , is not the restriction of a dtdlag function } .
One can check that A is a measurable set. Furthermore, we have seen that J l A (w, u)P(dw) = 0, each u E R By Fubini's Theorem
J 1: l A (w, u)duP(dw) 1: J A (w, u)P(dw)du 1
=
=
0,
hence we conclude that for a.a. w the function t f---> e iu X, (w) , t E Q+ is the restriction of a dtdlag function for almost all u E R We can now conclude that the function t f---> Xt ( w) ' t E IQl+ ' is the restriction of a cadlag function for every such w , with the help of the lemma that follows the proof of this theorem. Next set yt(w) = lim s E IQI s t Xs(w) for all w in the projection onto n of {D x IR} \ A and yt = 0 on A, all t. Since :Ft contains all the P-null sets of :F and (:Ft)Ooo X n k - Xm k = 0, as claimed.
We will henceforth always assume that we are using the ( unique) dtdlag version of any given Levy process. Levy processes provide us with examples of filtrations that satisfy the "usual hypotheses," as the next theorem shows.
22
I Preliminaries
Let X be a Levy process and let Ot :f1 vN, where (:f1)o t and ( u 1 , . . . , Un ), we give the proof for n = 2 for nota tional convenience. Therefore let z > v > t, and suppose given U 1 and U z . We have E{ ei(u rXv +u2Xz ) I Ot+ }
lim E{ ei (u rXv +u 2Xz) I Ow } wlt eiu 2Xz i r = lim E{ e u Xv ( uz ) l 9w } z wlt ( fz U z eiu2 Xv r = lim E{ ei u Xv fv ( u z z (U z ) IOw }, wl t =
/ /
· g t hat MU2 usm v
=
• le. Combming eiu2 Xv is a martmga " terms the above becomes fv (u 2 )
and the same martingale argument yields = =
=
+ uz )fz - v (uz ) wl e i (u r+ u 2 )Xt fv - t( U I + U2 )fz - v (U 2 ) limt ei (u r +u )Xw fv - w (U I 2
E { ei(u r Xv +u2X z ) I Ot }.
It follows that E{ ei EuJ X•J I Ot+ } = E{ ei EuJ X• J IOt} for all (s 1 , . . . , sn ) and all (u 1 , . . . , un ), whence E{ ZIOt+ } = E{ ZIOt} for every bounded Z E Vo <s 0. D The next theorem shows that a Levy process "renews itself" at stopping times.
4 Levy Processes
23
Let X be a Levy process and let T be a stopping time. On the process Y (Yt)o 0 : �Xt
r;; + 1
=
inf{t > T;i : �Xt E A } .
26
I Preliminaries
Since X has dtdlag paths and 0 ¢:. A, the reader can readily check that {T;t > t} E :Ft+ = :Ft and therefore each T;t is a stopping time. Moreover 0 ¢:. A and cadlag paths further imply Tj > 0 a.s. and that limn _,oo TA = oo a.s. We define 00
n=l
and observe that N A is a counting process without an explosion. It is straight forward to check that for 0 < s < t < oo,
NtA - N:- E u{Xu - Xv ; s < v < u < t}, and therefore NtA - Nf is independent of :Fs ; that is, NA has independent increments. Note further that N/ - Nf" is the number of jumps that Zu Xs +u - Xs has in A, 0 < u < t - s. By the stationarity of the distributions of X, we conclude N/ - Nf" has the same distribution as Nl- s· Therefore N A is a counting process with stationary and independent increments. We conclude that NA is a Poisson process. Let v (A) E{Nf} be the parameter of the Poisson process N A ( v ( A ) < oo by the proof of Theorem 34). Theorem 35. The set function A NtA (w) defines a a--finite measure on lR \ {0} for each fixed (t, w) . The set function v (A ) E{Nf} also defines a a--finite measure on lR \ { 0}. Proof The set function A NtA (w) is simply a counting measure: p, (A ) { number of s < t .!::l. Xs (w) E A } . It is then clear that v is also a measure. 0 =
=
f---+
=
f---+
:
Definition. The measure v defined by
E{Nf} E{ L I A (f:::l.Xs) } O<s< l is called the Levy measure of the Levy process X. v ( A)
=
=
We wish to investigate further the role the Levy measure plays in governing the jumps of X. To this end we establish a preliminary result. We let Nt (w, dx) denote the random measure of Theorem 35. Since Nt (w, dx) is a counting measure, the next result is obvious. Theorem 36. Let A be a Borel set of IR, 0 ¢:. A, f Borel and finite on A.
Then
1A f(x)Nt (w, dx)
=
L f( i:::l.X8)1 A (f:::l.X8). O<s< t
Just as we showed that NtA has independent and stationary increments, we have the following consequence.
4 Levy
Corollary. Let A be a Borel set of IR with 0 ¢: A, and let finite on A. Then
i f(x)Nt( · , dx)
is a Levy process.
Processes
f
27
be Borel and
For a given set A (as always, 0 ¢: A), we defined the associated jump process to be Jf = 2:::: �Xsl A ( �X. ).
O< s 0. Theorem 37. Given A, 0 ¢: A, the process Xt - JtA is a Levy process.
Proof
It is clear that we need only check the independence and stationarity of the increments. But
Xt - Jf - (Xs - J: ) = Xt - Xs - 2:::: �Xu l A (�Xu ) s< u a } > O<s 0. The advantage of doing this is that y a then has jumps bounded by a, and hence has finite moments of all orders (Theorem 34). We can choose any a > 0; we arbitrarily choose a = 1. Note that
It _- Xt Jt(-oo,-l]U[l,oo) = Xt - Jr > XNt (·, dx) . lxl l
·v·l
_
The next theorem gives an interpretation of the Levy measure as the expected rate at which the jumps of the Levy process fall in a given set. Theorem 38. Let A and let fl A E L2(dv) .
be Borel with 0 ¢: A. Let v be the Levy measure of X, Then
i
E{ f(x)Nt(·, dx)} = t
i f(x)v(dx)
28
I Preliminaries
and also Proof.
First let f =
Lj aj 1 Aj , a simple function. Then A E{2:>1Nt 1 } = :�:::::>j E{ N/1 } j = t L a1 v ( A1 ) , j
since N/i is a Poisson process with parameter v ( A1 ). The first equality follows easily. For the second equality, let Mf = Nf' -tv(Ai ) · The Mf are LP martingales, all p > 1 , by the proof of Theorem 34. Moreover, E{ Mf } = 0. Suppose A i , AJ are disjoint. We have
E{MfMl } = E {l: )ML+ l - ML ) 2:)M1t+ I - M/t )} k
i
0 = to < t 1 < · · · < tn = t. Using the martingale property E{MjM/ } = E{ L(ML+I - ML )(M/k+I - M/J }.
for any partition we have
k
Using the inequality
lab I < a2 + b2 , we have
2)ML+r - MfJ (ML+r - M/J < i:: ) ML+r - MjJ 2 + l: ) ML+r - M/J 2 . k
k
k
However Lk( Mfk+ r - MfJ 2 < ( N/!' ) 2 + v(Ai )2t2; therefore the sums are dominated by an integrable random variable. Since Mj and M/ have paths of finite variation on [0, t] it is easy to deduce that if we take a sequence ( 7rn ) n > l of partitions where the mesh tends to 0 we have
L (ML+I - Mfk )( ML+ r - M/J = L D.M;D. M{ nl!_.� t tk+ O<s 1, E{I Wtl n } exists (Theorem 34), and the stationary increments of W implies E{Wt} = at, for a = E{Wl }. (Recall E{Wo} = 0.) We set yt = Wt - E{Wt } · Then Y has independent increments and mean 0; it is a martingale. Setting Zt = Jt + at completes the proof. 0 While Theorem 40 is the most important result about Levy processes Theorem 40.
=
-
from the standpoint of stochastic integration, the next two theorems provide a better understanding of Levy processes themselves. Theorem 41. Let X be a Levy process with jumps bounded by a. That is, sup8 j � Xs l < a a.s. Let Zt = Xt - E{Xt} · Then Z is a martingale and Zt = Zf + Zf where zc is a martingale with continuous paths, zd is a martingale,
Zf = f x(Nt(-, dx) - tv(dx)), }{lxi < a } and zc and zd are independent Levy processes. Proof. Z has mean zero and independent increments so it is a martingale, as well as a Levy process. For a given set A we define
l xNt (· , dx) - t i xv(dx) = L �Xs 1 A ( �Xs ) - t J xv(dx). A O<s l} (1 - eiux )v(dx) + }{lr xl < l} (1 - eiux + iux )v(dx). o:, the corresponding Levy process is unique in distri Moreover given v, 2, IJ bution.
The Levy-Khintchine formula has another, more common expression. ( For a complete discussion of the various formulae of Levy and Khintchine for the case of infinitely divisible distributions, see Feller [72, pages 558-565] .) Indeed, in the n-dimensional case we can express the Levy-Khintchine formulas as follows ( letting ) denote the standard inner product for
]Rn ). (, Theorem 44. Let X be a Levy process in JRn , with X0 0 . Then there exists n JR such a convolution semigroup of probability -measures on that C(X J. l t) t· t The characteristic function of J.lt is e 'I/J (u) , where 'lj; is given nin terms of a v on ]R such that positive definite n n matrix E and a JRn and bounded away from 0; on if A is Borel ((iii) v(A) 1..k Pk , where Ak > 0, each k, and "£": Ak = 1 . Then X is a semimartingale under R as well. Proof Suppose Hn E S converges uniformly to H E S. Since X is a Pk semimartingale for all Pk , Ix(Hn ) converges to Ix(H) in probability for every 0 Pk. This then implies Ix(Hn ) converges to Ix (H) under R. Theorem 2.
m
1
1
Let X be a semimartingale for the filtmtion lF. Let G be a subfiltmtion of lF, such that X is adapted to the G filtmtion. Then X is a G semimartingale. Proof For a filtration JBI, let S (JBI) denote the simple predictable processes for the filtration lBI = (1tt)t>o · In this case we have S ( G) is contained in S(JF) . Theorem 4 ( Stricker's Theorem) .
The theorem is then an immediate consequence of the definition.
0
Theorem 4 shows that we can always shrink a filtration and preserve the property of being a semimartingale (as long as the process X is still adapted) , since we are shrinking as well the possible integrands; this, in effect , makes it "easier" for the process X to be a semimartingale. Expanding the filtra tion, therefore, should be--and is-a much more delicate issue. Expansion of filtrations is considered in much greater detail in Chap. VI. We present here an elementary but useful result. Recall that we are given a filtered space (n, :F, lF, P) satisfying the usual hypotheses. Theorem 5 (Jacod's Countable Expansion) . Let A be a collection of events in :F such that if Aa" A,a E A then Aa n A,a = 0, (a =/= f3) . Let 1tt be
the filtmtion genemted by :Ft and A. Then every (lF, P) semimartingale is an (JBI, P) semimartingale also. Proof Let An E A. If P(A n ) = 0, then An and A� are in :Fo by hypothesis. We assume, therefore, that P(A n ) > 0. Note that there can be at most a countable number of An E A such that P(A n ) > 0. If A = Un >1 An is the union of all An E A with P(An ) > 0, we can also add Ac to A without loss of generality. Thus we can assume that A is a countable partition of n with P(An ) > 0 for every An E A. Define a new probability Qn by Qn ( · ) = P( · IAn ) , for An fixed. Then Qn « P, and X is a (lF, Qn ) semimartingale by Theorem 2 . If we enlarge the filtration lF by all the :F measurable events that have Q n probability 0 or 1 , we get a larger filtration Jfn = (:Jt)t>o, and X is a (Jfn , Qn ) semimartingale. Since Qn (A m ) = 0 or 1 for =/= n, we have :Ft C 1tt C :Jt, for t > 0, and for all n. By Stricker's Theorem (Theorem 4) we conclude that X is an (JBI, Qn ) semimartingale. Finally, we have dP = L n > 1 P(An ) dQn , m
54
II Semimartingales and Stochastic Integrals
where X is an (IHI, Qn ) semimartingale for each we conclude X is an (IHI, P) semimartingale.
n. Therefore by Theorem 3 0
Corollary. Let A be a finite collection of events in :F, and let IHI = (1tt)t>o be the filtration generated by :Ft and A. Then every (IF, P) semimartingale is an (IHI, P) semimartingale also.
Proof. Since A is finite, one can always find a ( finite ) partition II of n such that 7tt = :Ft V II. The corollary then follows by Theorem 5. 0 Note that if B = (Bt)t> O is a Brownian motion for a filtration (:Ft)t>o, by Theorem 5 we are able to add, in a certain manner, an infinite number of "future" events to the filtration and B will no longer be a martingale, but it
will stay a semimartingale. This has interesting implications in finance theory (the theory of continuous trading) . See for example Duffie-Huang [60] . The corollary of the next theorem states that being a semimartingale is a "local" property; that is, a local semimartingale is a semimartingale. We get a stronger result by stopping at Tn - rather than at Tn in the next theorem. A process X is stopped at T- if X[ - = Xtl{o 0. Define Rn = Tn l{ Tn t}· Then
Theorem 6.
But P(Rn < oo) = P(Tn < t) , and since Tn increases to oo a.s., P(Tn < t) --+ 0 as n --+ oo. Thus if Hk tends to 0 in Su , given E > 0, we choose n so that P(Rn < oo) < c/2, and then choose k so large that P{I Icx n ) < (H k ) l > c} < Ej2. Thus, for k large enough, P{ l lx< (Hk )l > c} < E. o
Corollary. Let X be a process. If there exists a sequence (Tn ) of stopping times increasing to 00 a.s., such that X Tn (or XTn l{Tn >O}) is a semimartin gale, each n, then X is also a semimartingale. 3 Elementary Examples of Semimartingales
The elementary properties of semimartingales established in Sect. 2 will allow us to see that many common processes are semimartingales. For example , the Poisson process, Brownian motion, and more generally all Levy processes are semimartingales.
3 Elementary Examples of Semimartingales
55
Each adapted process with cadlag paths of finite variation on compacts (of finite total variation} is a semimartingale (a total semimartin gale}. Proof. It suffices to observe that llx (H)I < IIHIIu J;' ldXsl, where f000 l dXs l
Theorem 7.
denotes the Lebesgue-Stieltjes total variation and
IIH II u
=
SUP (t , w) IH(t, w) j .
0
Each L2 martingale with cadlag paths is a semimartingale. Proof. Let X be an L2 martingale with X0 = 0, and let H E S. Using Doob's Optional Sampling Theorem and the L2 orthogonality of the increments of L 2 martingales, it suffices to observe that Theorem 8.
n
n
E{(Ix ( H) )2} = E{(I: Hi (Xri+, - XrJ)2} = E{I: Hl(Xr,+, - XrJ2} i=O
I converges to a process H uni formly on compacts in probability (abbreviated ucp) if, for each t > 0, SUPo<s t implies (Hk · X); = (Hk · X);, we have ----+
c
----+
4 Stochastic Integrals
P((H k · X ); > J ) < P((Hk · X); > J) + P(Rk < < + P((Hk ); > rJ)
�
t)
< c:
if k is large enough, since lim k -H'o P((Hk ); > ry )
=
59
D
0.
We have seen that when X is a semimartingale, the integration operator Jx is continuous on Su cp, and also that Su cp is dense in Lu cp · Hence we are able to extend the linear integration operator Jx from S to L by continuity, since ID>u cp is a complete metric space.
Definition. Let X be a semimartingale. The continuous linear mapping Jx : Lu cp ---> ID>u cp obtained as the extension of Jx S ---> ID> is called the stochastic integral. The preceding definition is rich enough for us to give an immediate example of a surprising stochastic integral. First recall that if a process (At ) t >o has continuous paths of finite variation with A0 = 0, then the Riemann-Stieltjes integral of J; A8dA8 yields the formula (see Theorem 54 of Chap. I) :
1t A8dA8 0
=
1
- A2t . 2
Let us now consider a standard Brownian motion B = (Bt)t>o with B0 = 0. The process B does not have paths of finite variation on compacts, but it is a semimartingale. Let (1r ) be a refining sequence of partitions of [0, oo) with limn -+oo mesh(7rn ) = 0. Let Bf = L t .E 11"n Bt . l (t . , t • +tl · Then B E L for each n. Moreover, B n converges to B m ucp. Fix t > 0 and assume that t is a partition point of each 7rn · Then n
JB ( B
n
n
) t = L Bt . (Bt k+ t - B t • ) ttkEk 0 (with the convention that X0_ 0), we can decompose [X, X] path-by-path into its continuous part and its pure jump part. =
=
Definition. For a semimartingale X, the process by-path continuous part of [X, X].
[X, X] c denotes the path
We can then write
[X, X]t
=
=
[X, X] f + xJ + L (�Xs) 2 O<soo = lim E{[M, M]t ATn } n -->oo = E{[M, M]t},
oo.
74
II Semimartingales and Stochastic Integrals
where the last result is by the Monotone Convergence Theorem. In particular we have that E{[M, M]t} < oo. For the converse, we now assume E{[M, M]t} < oo, all t > 0. Define stopping times by yn = inf {t > 0 : IMtl > n} 1\ n. Then yn increase to oo a.s. Furthermore (MT n )* < n + I�Mrn l < n + [M, M];/ 2 , which is in £ 2 . By Theorem 51 of Chap. I, MTn is a uniformly integrable martingale for each n. Also we have that E{(Mr) 2 } < E{((MTn ) * ) 2 } < oo, for all t > 0. Therefore MTn satisfies the hypotheses of the first half of this n theorem, and E{(Mr) 2 } = E{[MT , Mr n]t } · Using Doob's inequality we have E{(Mt*A Tn ) 2 } < 4E{(MtTn ) 2 } = 4E{[MTn , MTn ]t}
4E{[M, M] rnA t} < 4E{[M, M]t}.
=
The Monotone Convergence Theorem next gives < 4 E{[M, M]t}
E{[M, M]o} = 1 since [M, M]t is increasing. There fore we cannot have E{Ml} = E{[M, M]t} for all t. Indeed, by Corollary 3 and the preceding we see that we must have E{[M, M]t} = oo for all t > 0. In conclusion, M = IIB II - 1 is a continuous local martingale with E{Ml} < oo for all t which is both not a true martingale and for which E{Ml} < oo while E{[M, M]t} = oo for all t > 0. (Also refer to Exercise 20 at the end of this chapter.)
6 The Quadratic Variation of a Semimartingale Corollary 5. Let X be a continuous local martingale. Then have the same intervals of constancy a.s.
Proof
X
and
75
[X, X]
Let r be a positive rational, and define
Then M = XTr - x r is a local martingale which is constant. Hence [M, M] = [X, X] Tr - [X, Xt is also constant. Since this is true for any rational r a.s. , any interval of constancy of X is also one of [X, X]. Since X is continuous, by stopping we can assume without loss of generality that X is a bounded martingale (and hence square integrable). For every positive, rational r we define
Br = inf{ t > r [X, X]t > [X, X] r }. :
Then
E{(Xsr - Xr) 2 } = E{X�r } - E{X; }
by Doob's Optional Sampling Theorem. Moreover
E{X�J E{X; } = E{[X, X]sr - [X, X] r } = 0, by Corollary 3. Therefore E{(Xsr - Xr) 2 } = 0, and Xsr = Xr a.s. Moreover this implies Xq = Xs a.s. on {Sq = Br} for each pair of rationals (r, q), and therefore we deduce that any interval of constancy of [X, X] is also one of X. o Note that the continuity of the local martingale X is essential in Corol lary 5. Indeed, let Nt be a Poisson process, and let Mt = Nt t. Then M is a martingale and [M, M]t = Nt; clearly M has no intervals of constancy while N is constant except for jumps. Theorem 28. Let X be a quadratic pure jump semimartingale. Then for any semimartingale Y we have [X, Y]t = XoYo + L �Xs�Ys . O<s H ( ucp), letting n ----> oo we have
zn ----> Z
in
ucp,
and
J j Y Z - J Y_dZ - J Z_dY
n , Y] = YZ - Y- HdX - z_dY [z n---> oo lim
= =
[Z, Y],
again by integration by parts. Since limn __, 00 J H�d[X, Y]s = J H8d[X, Y]8, we have [Z, Y] = [H · X, Y] = J H8d[X, Y]s, and the proof is complete. 0
Example. Let Bt be a standard Wiener process with Eo = 0, (i.e., Brownian motion). Bl - t is a continuous martingale by Theorem 27 of Chap. I. Let H E IL be such that E{f H'fds } < oo, each t > 0. By Theorem 28 of Chap. I
�
6 The Quadratic Variation of a Semimartingale
[B , B ] t
77
[H · B , H B] t I� I H8dB8 t E{(l H8dB8) 2 } E{[H B , H · B] t } E{ fot H?ds}.
we have t, hence H?ds. By the martingale · preservation property, is also a continuous local martingale, with 0. By Corollary 3 to Theorem 27 ·
(H B )0
=
=
=
=
·
=
It was this last equality,
that was crucial in K. Ito's original treatment of a stochastic integral. Theorem 30. Let be a cadlag, adapted process, and let be
H semimartingales. Let
X, Y two be a sequence of random partitions tending to the identity. Then n t , r yr 1 TI+ l ) x )(Y H (X Tt+ ,n : :: l: r (0 < T0 < T]' < converges in ucp to I H8_d [X, Y] s (Ho - 0) . Here · · · < Tt < · · · < Tf:J . Proof. By the definition of quadratic variation, [X, Y] XY -X_· Y-Y_ ·X, where X_ · Y denotes the process (J� X8_dYs ) t >o· By the associativity of the stochastic integral ( Theorem 19) H_ . [X, YJ H_ . (XY) - H_ . (X_ . Y) - H_ . (Y_ . X) H_ . (XY) - (H_X_) . Y - (H_Y_) . X H_ . (XY) - (HX)_ . Y - (HY)_ . X. an
_
_
=
an
=
=
=
=
=
By Theorem 2 1 the above is the limit of
i
=
- yTt (XTI+ l - XT,n )} 2::::: HTt ( xT,''p - xr,n ) (YTDt-1 - yTt ) i
0
0
78
II Semimartingales and Stochastic Integrals
7 Ito's Formula ( Change of Variables)
A H
Let be a process with continuous paths of finite variation on compacts. If E lL we know by Theorem 17 that the stochastic integral agrees a.s. with the path-by-path Lebesgue-Stieltjes integral J In Sect. 7 of Chap. I (Theorem 54) we proved the change of variables formula for f E C 1 , namely
H8dA8•
We also saw at the end of Sect. process with 0,
B0 =
B
Taking f(x)
HA ·
4 of this chapter that for a standard Wiener
1B 1 . t 8 BsdB = 2 'f - 2 t Jo
x2 /2 , the above formula is equivalent to t t 1 f ( Bt ) - f ( Bo ) = f ( Bs )dBs + -2 f (Bs)ds ,
=
1 0
1 0
I
II
which does not agree with the Lebesgue-Stieltjes change of variables formula ( *). In this section we will state and prove a change of variables formula valid for all semimartingales. We first mention, however, that the change of variables formula for con tinuous Stieltjes integrals given in Theorem 54 of Chap. I has an extension to right continuous processes of finite variation on compacts. We state this result as a theorem but we do not prove it here because it is merely a special thanks to Theorem case of Theorem
32,
26.
Let V be an F V process with right continuous paths, and let f be such that f1 exists and is continuous. Then ( f( Vt ) ) t>o is an FV process and
Theorem 31 (Change of Variables) .
f( Vt ) - f(Vo) =
t !'Ws- )dVs +
Jo
+
J�+
2:.::
( { ) ) ) } s Vs s Vs 'W fW � f · ! st
O<
R. + is an increasing function with limulO r(u) = 0, which is valid for f E C 2 defined on a compact I
II
set. We now prove separately the continuous case and the general case. Proof for the continuous case. We first restrict our attention to a continuous semimartingale X , since the proof is less complicated but nevertheless gives the basic idea. Without loss of generality we can take = 0. Define stopping times Rm = inf{t : J Xtl > m } . Then the stopped process X R = is bounded by m , and if Ito's formula is valid for X R= for each m , it is valid for X as well. Therefore we assume that X takes its values in a compact set. We fix a t > 0, and let an be a refining sequence of random partitions5 of [0, t] tending to the identity [an = (0 = Tf) < Tf < . . . < Tf:n = t)]. Then
Xo
f(Xt) - f(Xo ) = L { f (Xr,+ , ) - f (Xr,n ) } kn
i=O
(Xr;:.-1 - Xr,n) + � l:: f" (Xr,n ) (xr,+ 1 - Xr,n f + l:: R (Xr,n Xr,+ 1 )
= I: !' ( Xr,n ) t
i
5
i
,
Note that it would suffice for this proof to restrict attention to deterministic partitions.
80
II Semimartingales and Stochastic Integrals
The first sum converges in probability to the stochastic integral I� f' (X8 _ ) dX8 by Theorem 2 1 ; the second sum converges to � I� f"(X8 ) d[X , X] s in probabil ity by Theorem 30. It remains to consider the third sum Ei Xr,n , Xr,f-, . But this sum is majorized, in absolute value, by
R(
)
and since L i (Xr,n+ , - Xr,n ) 2 converges in probability to [X, X]t (Theo rem 22), the last term will tend to 0 if limn ---> oo supi r ( IXT,f- 1 - Xr,n I ) = 0. However s f-* X8 (w ) is a continuous function on [0, t], each fixed w, and hence uniformly continuous. Since limn ---> 00 supi ITI.f- 1 = 0 by hypoth esis, we have the result. Thus, in the continuous case, f(Xt ) - f(X0 ) = I� f'(Xs-) dX8 + � I� f"(X8_ ) d[X, X8], for each t, a.s. The continuity of the paths then permits us to remove the dependence of the null set on t, giving the complete result in the continuous case. Proof for the general case. X is now given as a right continuous semimartin gale. Once again we have a representation as in ( ) but we need a closer analysis. For any t > 0 we have Eo<s 0, let A = A(c:, t) be a set of jump times of X that has a.s. a finite number of times s, and let B = B(c:, t) be such that Ls EB (�Xs) 2 < c2 , where A and B are disjoint and A U B exhaust the jump times of X on (0, t]. We write
rr l
* * *
,
f(Xt ) - f(Xo ) = I:i { f(Xr,f- 1 ) - f(Xr,n ) } =
L { f (XT,f- 1 ) - f(Xr,n ) } + L { f(Xr,f- , ) - f(Xr,n ) }
""' {f(Xs) - f(Xs_)} , f(Xr1.T>+ 1 ) - f(XrT>1. )} � { sEA i,A
lim ""' � n
i,B
i,A
and by Taylor ' s formula
=
{ f(Xr,f-, ) - f(Xr,n ) } L i,B Li f'(Xr,n )(Xr,f-, - Xr,n ) + � Li f" (Xr,n )(Xr,f-, - Xr,n )2 - I: { !' (Xr,n ) (Xr,f- , - Xr,n ) + � !" (Xr,n ) (Xr;f-, - Xr,n ) 2 } i,A =
+ L R(Xr,n , Xr,+ J·
i,B
)
7 Ito's Formula (Change of Variables
81
As in the continuous case, the first two sums on the right side above converge respectively to ) dX8 and � f" (Xs-) d[X , X]8• The third sum converges to
I�+
I�+ f' (Xs-
{ s EA
- I: f'(X.-)�X.
+ � f" (X8_ )(�X.) 2 } .
Assume temporarily that J Xs l < k, some constant k, all s < t. Then uniformly continuous, and using the right continuity of X we have
()
c
where r(c+) is lim supo!e: r J . Next let tend to 0. Then to 0, and
f"
is
r(c+ ) [X, X] t tends
{ f(X.) - f(X._) - f'(X.- )�X. - � J"(X._)(�X.)2 } L s EA(e:, t)
tends to the series in ( * * * ), provided this series is absolutely convergent. Let vk inf t > 0 : IXt l > k}, with Xo 0. By first establishing * * * for X l[o,Vk)' which is a semimartingale since it is the product of two semimartin gales Corollary 2 of Theorem 22), it suffices to consider semimartingales taking their values in intervals of the form [-k, k]. For f restricted to [- k , k] we have lf(y) - f(x) - (y - x)f'(x)l < C(y - x )2 . Then =
(
{
( )
=
I: lf (Xs) - f(Xs-) - f'(X.-)�X. I < C I: (�X.) 2 < C[X, X] t < oo, and Lo<s o need not be right continuous, but one can take Hs = 9s + = to obtain one. Note further that { [M, M]t < s = {Ts > t , hence ([M, M]t)t>o are stopping times for the filtration g = (9s)s> O · = s < oo, By Corollary 3 of Theorem 27 we have E{Mf.J = E{[M, M] r. = s identically because is continuous. Thus the time since
:Fr, ([M, M] Mr,.
M
:Fr
M, M] :Fr. +
8
}
}
[ M , M] [ M , changed process is square integrable. Moreover
-
M] rJ
by the Optional Sampling Theorem. Also
E{B� - B;l9s} = E{(Bu - Bs ) 2 l 9s } = E{ ( = E{ = u - s.
Mr -MrJ2 I :FrJ [M, M]ru - [M, M]r.I :Fr,} u
8 Applications of Ito's Formula
89
Therefore BI - s is a martingale, whence [B, B] s = s provided B has contin uous paths, by Corollary 2 to Theorem 27. We want to show that B8 = Mrs has continuous paths. However by Corol lary 5 of Theorem 27 almost surely all intervals of constancy of [M, MJ are also intervals of constancy of M. It follows easily that B is continuous. It remains to show that Mt = B[M,M]t . Since Bs = Mrs , we have that B[M, M] t = Mr[ M , MJ t , a.s. Since (Ts )s>O is the right continuous inverse of [M, M], we have that T[M, M] t > t, with equality holding if and only if t is a point of right in crease of [M, M]. (If (Ts )s>O were continuous, then we would always have that T[M,M]t = t.) However T[M,M] t > t implies that t [M, M]t is constant on the interval (t, T[M,M] t ); thus by Corollary 5 of Theorem 27 we conclude M is constant on (t , T[M,MJ , ) . Therefore B[M,M], = Mr[M , Ml t = Mt a.s., and we are done. 0 f-+
Another application of the change of variables formula is the determination of the distribution of Levy 's stochastic area process. Let Bt = (Xt, yt) be an IR2-valued Brownian motion with (X0 , Yo) = (0, 0). Then during the times s to s + ds the chord from the origin to B sweeps out a triangular region of area �R8dN8, where
R8 = VX2s + Y28
and
Ys dXs + Xs dY8• dNs = - Rs Rs
Therefore the integral At = J� R8dN8 = J�(-Y8dXs+XsdYs) is equal to twice the area swept out from time 0 until time t. Paul Levy found the characteristic function of At and therefore determined its distribution. Theorem 43 is known as Levy's stochastic area formula. Theorem 43. Let Bt = (Xt, yt) be an IR2-valued Brownian motion, B o (0, 0), u E JR. Let At = f� XsdYs - f� YsdXs . Then 1 A . - oo < u < oo. ' ew O < t < oo, } E{ ' cosh (ut ) -
Proof Then
Let
a(t), f3(t) be C 1 functions, and set . a(t) vt = zuAt - 2 (Xt2 + �2 ) + jJ (t). . At - a ' (t) (xt2 + �2 )dt dvt = zud 2 - a(t){XtdXt + ytdyt + dt } + f3' (t)dt = ( -iuyt - a(t)Xt)dXt + (iuXt - a(t)yt)dyt 1 - 2 dt{a'(t)Xf + a'(t) �2 + 2a(t) - 2/]' (t)}.
90
II Semimartingales and Stochastic Integrals
Next observe that, from the above calculation,
d[V, V]t = (-iuyt - a(t)Xt) 2 dt + (iuXt - a(t) Yt) 2 dt = (a(t) 2 - u2 ) (X1 + �2 )dt. Using the change of variables formula and the preceding calculations
dev' = e v, (dvt + 21 d[V, V]t) = ev' (-iuyt - a(t)Xt)dXt + ev' (iuXt - a(t)yt)dyt 1 + 2 ev'dt{ (a(t ) 2 - u2 - a'(t )) (X; + �2 ) + 2f3'(t ) - 2a(t)} . Therefore e v, is a local martingale provided
a' (t) = a(t) 2 - u2 (J'(t) = a(t). Next we fix t 0 > 0 and solve the above ordinary differential equations with a(t0) = (J(to) = 0. The solution is
a(t) = u tanh(u(to - t)) (J(t) = - log cosh (u(to - t)), where tanh and cosh are hyperbolic tangent and hyperbolic cosine, respec tively. Note that for 0 < t < t0 ,
Thus ev, , 0 < t < t0 is bounded and is therefore a true martingale, not just a local martingale, by Theorem 51 of Chap. I. Therefore, However, yt0 = iuAt 0 since a(to) = (J(to) = 0. As follows that V0 = - log cosh( ut0 ). We conclude that
Ao
=
Xo
=
Yo
= 0, it
E{ eiuA 'o } = exp{- log cosh (uto) } 1 - cosh( uto) '
and the proof is complete.
0
There are of course other proofs of Levy's stochastic area formula (e.g. , Yor [246] , or Levy [1 50] ). As a corollary to Theorem 43 we obtain the density for the distribution of At ·
8 Applications of Ito's Formula
91
Corollary. Let Bt = (Xt , yt) be an IR2 -valued Brownian motion, B0 = (0, 0), and set At = XsdYs - Y8dX8 • Then the density function for the distri bution of At is
J�
J�
1 !A , (x) = 2t cosh(nxj2t) '
- 00
<X
x > -r,
e - iux
cr2 c3
1f
r cr4
-> y > 0, t 1f
-
as shown in Fig. 1 below. y
x= Fig. 1 .
= jt 1f
x=r
-r
The closed curve of integration
Cr = C¢ + c;_ + c;_ + C�.
Therefore fer f(z)dz = 2ni Res (!, zo) . Along c: the integral is
- r e - i( u+ 1 -
";i ) x
cosh( ut)
r j du = e -, �X
e - iUX
du. cosh( ut)
-r r The integrands on C'; and c; are dominated by 2e - r, and therefore ----
92
II Semimartingales and Stochastic Integrals
lim r
r--+oo
}
Cr
f ( z ) dz = (1 + e x n
ft )
-u
e i x du 1 _ 00 COSh ( ut )
= 2ni Res(!, zo )
x/ t e = 2n t
-n
Finally we can conclude
00
2
1 1 00 e -iux 1 du 2n _ 00 cosh(ut) = 2n
(
2ne¥t-
t(1 + e7 ) 1
t( e ;: + e 2�x ) 1 2t cosh( ;� ) ·
) 0
The stochastic area process A shares some of the properties of Brownian motion, as is seen by recalling that At = R8dN8 , where N is a Brownian motion by Levy's Theorem (Theorem 39) , and N and R are independent (this must be proven, of course) . For example A satisfies a reflection principle. If one changes the sign of the increments of A after a stopping time, the process obtained thereby has the same distribution as that of A. One can use this fact to show, for example, that if Bt = SUPo<s 0.
J�
-
-
Bibliographic Notes The definition of semimartingale and the treatment of stochastic integration as a Riemann-type limit of sums is in essence new. It has its origins in the fundamental theorem of Bichteler [13, 1 4] , and Dellacherie [42] . The peda gogic approach used here was first suggested by Meyer [1 76] , and it was then outlined by Dellacherie [42] . Dellacherie 's outline was further expanded by Lenglart [145] and Protter [201, 202] . These ideas were also present in the work of M. Metivier and J. Pellaumail, and to a lesser extent Kussmaul, al though this was not appreciated at the time by us. See for example [160] , [162] , and [ 1 38] . A similar idea was developed by Letta [ 1 48] . A recent treatment is in Bichteler [15] . We will not attempt to give a comprehensive history of stochastic inte gration here, but rather just a sketch. The important early work was that of Wiener [228, 229] , and then, of course, Ito [93, 94, 95, 97, 98] . However, it has recently been discovered, by B. Bru and M. Yor, that the young French mathematician W. Doeblin6 had essentially established Ito's formula in early 6
Wolfgang Doeblin (Doblin ) was born in 1915 in Berlin, and after acquiring French citizenship in 1936 was known as Vincent Dobbin. He was killed in June 1940.
Bibliographic Notes
93
1940, but was killed later that year in action in the Second World War, and thus his results were lost until recently [26, 27] . Doob stressed the martingale nature of the Ito integral in his book [55] and proposed a general martingale integral. Doob's proposed development depended on a decomposition theo rem (the Doob-Meyer decomposition, Theorem 8 of Chap. III) which did not yet exist . Meyer proved this decomposition theorem in [163, 164] , and com mented that a theory of stochastic integration was now possible. This was begun by Courrege [36] , and extended by Kunita and Watanabe [134] , who revealed an elegant structure of square integrable martingales and established a general change of variables formula. Meyer [166, 167, 168, 169] extended Kunita and Watanabe's work, realizing that the restriction of integrands to predictable processes is essential. He also extended the integrals to local mar tingales, which had been introduced earlier by Ito and Watanabe [102] . Up to this point, stochastic integration was tied indirectly to Markov processes, by the assumption that the underlying filtration of O"-algebras be "quasi-left continuous." This hypothesis was removed by Doleans-Dade and Meyer [53], thereby making stochastic integration a purely martingale theory. It was also in this article that semimartingales were first proposed in the form we refer to as classical semimartingales in Chap. III. A different theory of stochastic integration was developed independently by McShane [154, 155] , which was close in spirit to the approach given here. However it was technically complicated and not very general. It was shown in Protter [200] (building on the work of Pop-Stojanovic [194] ) that the theory of McShane could for practical purposes be viewed as a special case of the semimartingale theory. The subject of stochastic integration essentially lay dormant for six years until Meyer [171] published a seminal "course" on stochastic integration. It was here that the importance of semimartingales was made clear, but it was not until the late 1970's that the theorem of Bichteler [13, 14] , and Del lacherie [42] gave an a posteriori justification of semimartingales. The seem ingly ad hoc definition of a semimartingale as a process having a decomposi tion into the sum of a local martingale and an FV process was shown to be the most general reasonable stochastic differential possible. (See also Kuss maul [1 38] in this regard, and the bibliographic notes in Chap. III.) Most of the results of this chapter can be found in Meyer [171 ] , though they are proven for classical semimartingales and hence of necessity the proofs are much more complicated. Theorem 4 (Stricker's Theorem) is (of course) due to Stricker [21 8] ; see also Meyer [1 73] . Theorem 5 is due to Meyer [175] . There are many other methods of expanding a filtration and still preserving the semimartingale property. For further details, see Chap. VI on expansion of filtrations. Theorem 1 4 is originally due to Lenglart [142] . Theorem 16 is known to be true only in the case of integrands in lL. The local behavior of the integral (Theorems 1 7 and 18) is due to Meyer [171 J (see also McShane [154]) . The a.s.
Exercises for Chapter II
94
Kunita-Watanabe inequality, Theorem 25, is due to Meyer [171] , while the ex pected version (the corollary to Theorem 25) is due to Kunita-Watanabe [134) . That continuous martingales have paths of infinite variation or are con stant a.s. was first published by Fisk [73] (Corollary 1 of Theorem 27) . The proof given here of Corollary 4 of Theorem 27 (that a continuous local martin gale X and its quadratic variation [X, X] have the same intervals of constancy) is due to Maisonneuve [151] . The proof of Ito's formula (Theorem 32) is by now classic; however we benefited from Follmer's presentation of it [75] . A popular alternative proof which basically bootstraps up form the formula for integration by parts, can be found (for example) on pages 57-58 of J. Jacod and A. N. Shiryaev [110] . The Fisk-Stratonovich integral was developed independently by Fisk [73] and Stratonovich [21 7] , and it was extended to general semimartingales by Meyer [171 J . Theorem 35 is inspired by the work of Getoor and Sharpe [215] . The stochastic exponential of Theorem 37 is due to Doleans-Dade [49] . It has become extraordinarily important. See, for example, Jacod-Shiryaev [1 10] . The pretty formula of Theorem 38 is due to Yor [237] . Exponentials have of course a long history in analysis. For an insightful discussion of exponentials see Gill-Johansen [82] . That every continuous local martingale is the time change of a Brownian motion is originally due to Dubins-Schwarz [59] and Dambis [37] . The proof of Levy's stochastic area formula (Theorem 43) is new and is due to Janson. See Janson-Wichura [1 13] for related results. The original result is in Levy [150] , and another proof can be found in Yor [246]. Exercises for Chapter II
Exercise 1. Let B be standard Brownian motion and let f be a function
mapping IR -+ IR which is continuous except for one point where there is a jump discontinuity. Show that Xt = f(Bt) cannot be a semimartingale.
Exercise 2. Let Q
! } · n
Show that xn is a semimartingale, each not be a semimartingale.
Exercise 18. (Continuation of Exercise motion and let
n,
but that
Y0 = 0 and with
limn _,oo xn
=
Y
need
17.) Let B be a standard Brownian
xn = n JL ! Bsds1 {t> ! }· Solve
dZ;' = Z�dX�, Z'(; = 1 , and show that limn ->oo X[' = Bt a.s., each t, but that limn _,oo zn =/=- Z, where Z solves the equation dZt = ZtdBt. Exercise 19. (Related to Exercises 17 and 18.) Let Af = ;. sin( nt) , 0 < t < 1rj2. Show that A n is a semimartingale for each n, and that J02 ldA� I = 1, each n (this means the total variation process, path-by-path, of A), but that sup IAfl = 0. ->oo O IR be given by u(y) = IIYII- 1 . Assume Bo = x a.s., where x =/=- 0, show the following. (a) Mt = u(Bt) is a local martingale. (b) limt oo E x (u(Bt)) = 0. (c) u(Bt) E L2, each t > 0. (d) SUPt>o Ex (u(Bt) 2) < 00. ->
u(B) is not a martingale. Exercise 21. Let A, C be two non-decreasing cadlag processes with Ao Co = 0. Assume that both A00 = limt_, 00 At and Coo = limt->oo Ct are finite. (a) Show that AooCoo = j000 (A00 - As)dCs + j000 (Coo - Cs - )dA8• Conclude that
(b) Deduce from part (a) the general formula
Note that the above formulae are not symmetric. See also Theorem 3 Chap. VI.
m
Exercises for Chapter II
Exercise 22. Let A be non-decreasing and continuous, and assume A00 limt->oo At is finite, Ao = 0 . Show that for integer p > 0,
97 =
*Exercise 23 (expansion of filtrations) . Let B be a standard Brownian motion on a filtered probability space (0, :F, IF, P), and define a new filtration
Q2 = :Ft V a(BI) and gt = nu >t g� . Show that E{Bt - Bs iQs} = � � (B 1 - B8) for 0 < s < t < 1 .
G by
(a) (b) Show that
Bt =
where
t/\1 1 B l - Bs ds f3t +
1-s
0
f3 is a G-Brownian motion. (Note in particular (Bt - J� 8i=�· ds)oO be a standard Brownian motion. Let a, b > 0 and T = inf{ t > 0 : Bt E (- a, b)c}. (a) Show that
is a local martingale. (b) Show that
98
Exercises for Chapter II
(c) For 0
<e< � a
b,
show that X T is a positive supermartingale, and deduce
cos(
a + b e)E{ exp{ 1
l 2T}}
2
(d) Use (c) to show that martingale. (e) Conclude that
< cos( a -2 b e) .
XT = sups O be standard Brownian motion, and let T = inf{t > 0 : Bt E (-d, d) c }. Let M = B T . Show that (a) if d
< ; , then E{ exp{ � [M, M] T } } < oo; but (b) if d = ; , then E{expg [M, M] T }} = oo. (Hint: Use Exercise 25.) Exercise 27. Let (Bt, .Ft)t> O be standard Brownian motion, and let Xt = e- t (Xo+a J; e "8 dB8 ) . Show that X is a solution to the stochastic differential
equation
dXt = - Xtdt + adBt. o:
Exercise 28. Let B be a standard Brownian motion and let £(B) denote the stochastic exponential of B. Show that limt-> £(B)t = 0 a.s. Exercise 29. Let X be a semimartingale. Show that oo
1
d( £(X) ) Exercise 30. Let
=
-dX + d[X , X] £ (X)
B be a standard Brownian motion.
(a) Show that M is a local martingale, where
(b) Calculate [M, M]t , and show that M is a martingale. (c) Calculate E{eB• }.
The next eight problems involve a topic known as changes of time. For these problems, let ( 0, .F, IF, P) satisfy the usual hypotheses. A change of time R = (Rt)t> O is a family of stopping times such that for every w E n, the function R. (w) is non-decreasing, right continuous, Rt < oo a.s., and Ro = 0. Let gt = .FRt . Change of time is discussed further in Sect. 3 of Chap. IV.
Exercise 31. Show that
G = (Qt)t> O satisfies the usual hypotheses.
Exercise 32. Show - that if M is an IF uniformly integrable martingale and Mt := MR. , then M is a G martingale.
99
Exercises for Chapter II
Exercise 33. If M is an IF (right continuous) local martingale, show that
is a
G semimartingale.
M
*Exercise 34. Construct an example where M is an IF local martingale, but
M is not a G local martingale. (Hint: Let (Xn )n EN be an adapted process. It is a local martingale if and only if IXn l dP is a a-finite measure on Fn - 1 , and E{Xn iFn - d = Xn - 1 , each n > 1 . Find Xn where IXn l dP is not a-finite on Fn - 2 , any and let Rn = 2n.) Rs continuous, strictly *Exercise 35. Let R be a time change, with 0, and Rt < oo, each t > 0. Show that for a continuous increasing, R0 n,
s f---+
=
semimartingale X, for bounded
HEL
*Exercise 36. Let
R and X be as in Exercise 35. No longer assume that Rt < oo a.s. , each t > 0, but instead assume that X is a finite variation process. Let At = inf{ > 0 : Rs > t}. (a) Show that R strictly increasing implies that A is continuous. (b) Show that R continuous implies that A is strictly increasing. (c) Show that for general R, RA, > t, and if R is strictly increasing and continuous then RA, t. (d) Show that for bounded H E IL we have s
=
(e) Show that for bounded H E
IL
we have
1Rt H8dA8 1ti\Aoo HR.ds. =
Ro See in this regard Lebesgue 's change of time formula, given in Theorem 45 of 0
Chap. IV.
R be a change of time and let G be the filtration given by gt = :FR, · Let At = inf { > 0 : R8 > t}. Show that A = (At)t>O is a change of time for the filtration G. Show also that if t --+ Rt is continuous a.s., Ro 0, and Roo oo, then RA, t a.s. , t > 0. *Exercise 38. Let A, G, be as in Exercise 37 and suppose that RA, t a.s., t > 0. Show that QA, C :Ft each t > 0. *Exercise 39. A function is Holder continuous of order o: if lf(x) - f(y)i < Klx -yic.. Show that the paths of a standard Brownian motion are a.s. nowhere locally Holder continuous of order o: for any o: > 1/2. (Hint: Use the fact that lim n oo L 7rn[o ,t] (Bti+ 1 - Bt ; )2 t.) *Exercise 37. Let
s
=
=
=
=
,
->
=
III Semimartingales and Decomposable Processes
1 Introduction
In Chap. II we defined a semimartingale as a good integrator and we developed a theory of stochastic integration for integrands in IL, the space of adapted processes with left continuous, right-limited paths. Such a space of integrands suffices to establish a change of variables formula (or "Ito's formula" ) , and it also suffices for many applications, such as the study of stochastic differential equations. Nevertheless the space IL is not general enough for the consider ation of such important topics as local times and martingale representation theorems. We need a space of integrands analogous to measurable functions in the theory of Lebesgue integration. Thus defining an integral as a limit of sums�which requires a degree of smoothness on the sample paths�is inade quate. In this chapter we lay the groundwork necessary for an extension of our space of integrands, and the stochastic integral is then extended in Chap. IV. Historically the stochastic integral was first proposed for Brownian motion, then for continuous martingales, then for square integrable martingales, and finally for processes which can be written as the sum of a locally square integrable local martingale and an adapted, cadlag processes with paths of finite variation on compacts; that is, a decomposable process. Later Doleans Dade and Meyer [53] showed that the local square integrability hypothesis could be removed, which led to the traditional definition of a semimartingale (what we call a classical semimartingale) . More formally, let us recall two definitions from Chaps. I and II and then define classical semimartingales.
Definition. An adapted, cadlag process A is a finite variation process (FV) if almost surely the paths of A are of finite variation on each compact
interval of [0, oo ) . We write J000 ldA8 I or is the total variation of the paths of A.
I Aioo for the random variable which
Definition. An adapted, cadlag process X is decomposable if there exist processes
N, A such that
102 with
III Semimartingales and Decomposable Processes
No = Ao = 0, N a locally square integrable local martingale, and A an
FV process.
Y is a classical semimartingale if there exist processes N, B with No = Bo = 0 such that Definition. An adapted, cadlag process
N is a local martingale and B is an FV process. Clearly an FV process is decomposable, and both FV processes and de composable processes are semimartingales (Theorems 7 and 9 of Chap. II). The goal of this chapter is to show that a process X is a classical semimartin gale if and only if it is a semimartingale. To do this we have to develop a small
where
amount of "the general theory of processes." The key result is Theorem 25 which states that any local martingale M can be written
where N is a local martingale with bounded jumps (and hence locally square integrable), and A is an FV process. An immediate consequence is that a classical semimartingale is decomposable and hence a semimartingale by The orem 9 of Chap. II. The theorem of Bichteler and Dellacherie (Theorem 43) gives the converse: a semimartingale is decomposable. We summarize the results of this chapter, that are important to our treat ment, in Theorems 1 and 2 which follow.
Theorem 1.
Let X be an adapted, cadlag process. The following are equiva
lent: (i) X is a semimartingale; {ii) X is decomposable; (iii) given j3 > 0, there exist M, A with M0 = A0 = 0, M a local martingale with jumps bounded by /3 , A an FV process, such that Xt = X0 + Mt + At ; (iv) X is a classical semimartingale. p on JR.+
n is the smallest a algebra making all processes in IL measurable. We also let P (resp. bP ) denote Definition. The predictable a-algebra
X
the processes (resp. bounded processes) that are predictably measurable.
The next definition is not used in this chapter, except in the Exercises, but it is natural to include it with the definition of the predictable a-algebra.
Definition. The optional a-algebra 0 on JR.+ X n is the smallest a-algebra making all cadlag, adapted processes measurable. We also let 0 (resp. bO )
denote the processes (resp. bounded processes) that are optional.
2 The C lassification of Stopping Times
103
Let X be a semimartingale. If X has a decomposition Xt Xo + Mt + At with M a local martingale and A a predictably measurable FV process, M0 = A0 = 0 , then such a decomposition is unique. Theorem 2.
=
In Theorem 1 , clearly (ii) or (iii) each imply (iv), and (iii) implies (ii), and (ii) implies (i) . That (iv) implies (iii) is an immediate consequence of the Fundamental Theorem of Local Martingales (Theorem 25). While Theo rem 25 (and Theorems 3 and 22) is quite deep, nevertheless the heart of Theo rem 1 is the implication (i) implies (ii), essentially the theorem of K. Bichteler and C. Dellacherie, which itself uses the Doob-Meyer decomposition theorem, Rao's Theorem on quasimartingales, and the Girsanov-Meyer Theorem on changes of probability laws. Theorem 2 is essentially Theorem 30. We have tried to present this succession of deep theorems in the most direct and elementary manner possible. In the first edition we were of the opinion that Meyer's original use of natural processes was simpler than the now universally accepted use of predictability. However, since the first edition, R. Bass has published an elementary proof of the key Doob-Meyer decompo sition theorem which makes such an approach truly obsolete. We are pleased to use Bass' approach here; see [11]. 2 The Classification of Stopping Times
We begin by defining three types of stopping times. The important ones are predictable times and totally inaccessible times.
Definition. A stopping time T is predictable if there exists a sequence of
stopping times (Tn ) n > l such that Tn is increasing, Tn < T on {T > 0}, all and limn _,00 Tn = T a.s. Such a sequence (Tn ) is said to announce T.
n,
If X is a continuous, adapted process with Xo = 0, and T = inf{t : IXtl > c}, for some c > 0, then T is predictable. Indeed, the sequence Tn = inf { t : I Xt I > c - � } /\ n is an announcing sequence. Fixed times are also predictable.
Definition. A stopping time T is accessible if there exists a sequence (Tk)k>l of predictable times such that 00
P( U {w : n (w) = T(w) < oo } ) = P(T < oo). k= l
Such a sequence (Tk)k>l is said to envelop T.
Any stopping time that takes on a countable number of values is clearly ac cessible. The first jump time of a Poisson process is not an accessible stopping time (indeed, any jump time of a Levy process is not accessible) .
Definition. A stopping time T is totally inaccessible if for every pre dictable stopping time
S,
104
III Semimartingales and Decomposable Processes
P{w : T(w) = S(w)
< oo} = 0.
Let T be a stopping time and A E FT. We define TA (w) =
{
T(w), oo ,
if w E A, if w rf. A.
It is simple to check that since A E FT , TA is a stopping time. Note further that T = min(TA, TAc) = TA 1\ TAc . A simple but useful concept is that of the graph of a stopping time.
Definition. Let T be a stopping time. The graph of the stopping time T is the subset of R+ X n given by { (t , w) : 0 < t = T(w) < oo}; the graph
of T is denoted by [T] .
Theorem 3. Let T be a stopping time. There exist disjoint events A, B such that A U B = {T < oo} a.s., TA is accessible and Ts is totally inaccessible,
and T = TA 1\ T8 a.s. Such a decomposition is a.s. unique.
Proof If T is totally inaccessible there is nothing to show. So without loss of generality we assume it is not. We proceed with an inductive construction: Let R1 = T and take o: 1 = sup{P(S = R1
< oo)
: S is predictable}.
O:i = sup{ P( S = Ri
< oo)
: S is predictable}
Choose S1 predictable such that P(S1 = Rt < oo) > T and set V1 = S1 . Define R2 = R1 {v1 # R d · For the inductive step let If o:i = o we stop the induction. Otherwise, choose Si predictable such that P(Si = Ri < oo) > T and set Vi = Si {s, does not equal any of vj , l <j < i- 1} " Let Ri+l = Ri { V, #R; } . The graphs of the Vi are disjoint by construction (the finite valued parts, of course) . Moreover the sets {Ri+1 =1- Ri , Ri < oo} are disjoint. Thus I: �' < I: P(Ri + 1 =1- Ri, Ri < oo) < 1, which implies that O:i __, 0. The Ri form a non-decreasing sequence, so set U = limi_, oo Ri. If U is not totally inaccessible, there exists a predictable time W such that P(W = U) > 0 and hence P(W = U) > o: i for some i. This contradicts how we chose Si at step i of the induction. Therefore U is totally inaccessible, and B = {U = T < oo}. D
A beautiful application of Theorem 3 is Meyer's Theorem on the jumps
of Markov processes, a special case of which we give here without proof. (We use the notation established in Chap. I, Sect. 5 and write IF�' = (:Ff)o 0
E{Coo - Ck iFk } < N a.s. for all k. Then E{C!} < 2N2. Proof of Lemma 1. First observe E{Coo} = E{E{Coo - GolFo}} < N. Let ting Ck = Ck+ l - Ck > 0, we obtain by rearranging terms: such that
Thus it follows that
107
3 The Doob-Meyer Decompositions
Choose and fix a constant
Dn = { k2 -n 0 < k2 -n < } Lemma 2. Let T be a totally inaccessible stopping time. For 6 > 0, let R(6) = supt< v P(t < T < t + 61Ft) · Then R(6) ----> 0 in probability as 6 ----> 0. Proof of Lemma 2. Let a > 0 and Sn (6) = inft{t E Dn : P(t < T < t + 6lFt) > a} !\ . First we assume Sn (6) is less than T. Since Sn (6) is countably valued, it is accessible, and since T is totally inaccessible, P(Sn (6) = T) = 0. Suppose r c {T < t}, and also T E Ft. Then E{E{l{t< T 0. Let r = { T < t, Sn (6) = t}. Then from the definition of Sn (6) we v
E Z+ and let
:
v
.
v
obtain
a contradiction. Thus we conclude Sn (6 ) < T a.s. Next we define a stopping time S as follows. Let S(6) = infn Sn (6) and S = supn S(�). Fix n , so that S(�) = infk Sk ( � ) . Thus on { S = T}, S(�) < S. Hence since S = supn S(�), we have S is accessible on { S = T}, which implies that T is accessible on { S = T }, which in turn implies P(S = T) = 0. Consequently E{l {S=T } IFs - } = 0 a.s. Suppose now that the result is not true. That is suppose there exists an c > 0 such that P(R(6) > a) > c, for all 6 tending to 0. Let (3 > 0 and 6 < (3. For n sufficiently large
P(E{l{ Sn (8 ) < T<Sn(8 ) +8 } 1Fsn(8) } > a) > c. We also have P(T = S(6)+6) = 0, since if not S(6)+6-,!. could announce part of T which would contradict that T is totally inaccessible. Thus P( { Sn ( 6) < oo where the symbol � T < Sn (6) + 6} � {S(6) < T < S(6) + 6}) ----> 0 as denotes the symmetric difference set operation. Since the symmetric difference tends to 0, if P(Sn (6) < T < Sn (6) + 6IFsn (8) > a) > c for any 6 > 0, then n ---->
we must also have
since otherwise were it to tend to 0 along with the symmetric difference tend ing to 0, then we would have ( * ) ----> 0, a contradiction. Thus we have P( S( 6) < T < S(6) + f31Fs ( .s) ) > a on a set A, with P(A) > c. From the definition of S and working on the set A, this implies P(E{l{ s < T < S+.B} IFs - } > a) > c. Let 0 (3 ----> 0 to obtain P(T = S I Fs- ) > a, and we have a contradiction.
108
III
Semimartingales and Decomposable Processes
Lemma 3. Let the hypotheses of Theorem 7 hold and also assume
IZI < N
where N > 0 is a constant, and further that the paths of Z are constant after a stopping time v. Let W(6) = supt 0 since Z is a supermartingale, and also ak E FJ:_ 1 . Let Af: 2::;= 1 aj . Then L k = Z 2"n + A'k is an :FJ: discrete time martingale. Define
=
1 k k n n Bt Ak 1"f 2 n < t n. Since Bm and B n are constant on intervals of the form (k2- m , (k + 1)2 - m ], the sup of the difference occurs at some k2 - m . Fix t and let u = inf{s : s E Dn and > t}. Observe that E{A� Ak' I :Fk } = E{Zt - Zooi:Ft } is bounded by 2N. Also E{A � - B�I:Ft} E{E{A� - B� I:Fu }i:Ft} = E{E{Zu - Zooi:Fu }I:Ft} = E{Zu - ZooiFt} which is bounded by 2N. This implies E{B:;:; - B� - ( Bf - Bf) I:Ft } = E{Zt - Zui:Ft} , with the right side being nonnegative and bounded by W(2 -n ), and of course also by E{W( 2 -n ) I:Ft}. Using Lemmas 1 and 2 we get that E{supt IB� Bfl 2 } --. 0 as n, m --. oo. Therefore the sequence Bf is a Cauchy sequence and we denote its limits by At . Next we show that the process A has continuous paths. We do this by analyzing the jumps of the approximating processes B n . Note that for t k2 -n llB� = E{ Z k-1 - Z.2JLn !Rk- d which are bounded by W(T n ). But then supt l ll Bf l --. 0 in L2 . Thus there exists a subsequence (nj ) such that supt I�B:j I --. 0 a.s. , and hence the limit A is continuous. It remains to show that Zt + A t is a uniformly integrable martingale. Both =
s
=
=
'
�
are square integrable, so we only need concern ourselves with the martingale property. Let s, t E Dn , s < t, and A E :F8 • Then E{(Zt + At) 1A} = E{(Zs + A8)1A} · The result follows by taking limits of Zt + Bf. At the beginning of the proof we made the simplifying assumption that general case. Choose N and let Z had bounded jumps. We now treat the T = inf{t > 0 : IZt l > N} 1\ N. Then zr (that is, z stopped at the time T) has at most one jump bigger than 2 N . By localization and the uniqueness of the decomposition, it suffices to prove the result for this case. Thus without
1 10
III Semimartingales and Decomposable Processes
loss of generality we assume Z has at most one jump greater than or equal to 2N in absolute value, that it occurs at the time T, and that Z is constant after the time T. We let
z: = I: �Zs l{ t�Z. > 2N }
zt- =
s T}
=
�Zrl{�ZT < - 2N.} l{ t>T}
Since -z+ and z - both have decreasing paths, they are both supermartin gales. Suppose we can show the theorem is true for -z+ and z - . Let -z+ = M + - A+ and z- = M - - A - be the two decompositions, with A + and A both continuous. Then Z = Z + ( -z+ + A+ ) - ( Z - -J:: A - t = Z,+ M+ - M- is a supermartingale with jumps bounded by 2N. Let Z = M - A be the unique decomposition of Z which we now know exists. Then Z = Z + M+ - M - , and therefore Z = CM M + - M - ) - A is the desired (unique) Doob-Meyer decomposition. Thus it remains to show that - z+ and z- both have Doob-Meyer de compositions. First observe that I�Zr l < IZr - 1 + IZrl < N + IZrl E L 1 , and hence E{ I� Zrl} < oo. Choose 0. Next choose R N so large that E{ I�Zrl l {I .6. ZT I> R} } < The cases for when the jump is positive and when the jump is negative being exactly the same except for a minus sign, we only treat the case where the jump is negative. Let Zf = � Zr l{t > T }l{ - .6-ZT > R } , and Zf = zt- - Zf. We define Bn- , BnR , Bnd analogously to the way we defined Bn in equation ( * ) above. We first show Bf - converges uniformly in t in probability. We have:
+
t:a.
P(sup i B� -
a, t: >
>
- B;" - I > a )
(* * )
t < P(sup t I B�d - B;"d l >
a ) + P(sup IB�R I > a ) + P(sup IB;"R I > a ) t t 3 3 3 The second term on the right side of ( * * ) above is less than 3t::
3t:
The third term on the right side of ( ** ) is shown to be less than similarly. Since I ZR I is bounded by R, the first term on the right side of ( * * ) can be made arbitrarily small by taking m and n large enough, analogous to what we did at the beginning of the proof for Bn . Thus as we did before in this proof, we can conclude that B�+ converges uniformly in t in probability as n ----> oo, and we denote its limit process by Ai.
3 The Doob-Meyer Decompositions
111
We prove continuity of A exactly as before. Finally by taking a subsequence n1 such that Bnj + converges almost surely, we get
+ } < lim inf E{B� + } = -E{Z�,} B� E{A;t,} = E{ njlim ---+ oo n1 ---+ oo by Fatou's Lemma. From this it easily follows that z + + A+ is a uniformly
integrable martingale, and the proof is complete.
0
While Theorem 7 is sufficient for most applications, the restriction to su permartingales having jumps only at totally inaccessible stopping times can be insufficient for some needs. When we move to the general case we no longer have that A is continuous in general (however see Exercises 24 and 26 for supplemental hypotheses that assure that the increasing process A of the Doob-Meyer decomposition is in fact continuous) . Without the continuity we lose the uniqueness of the decomposition, since there exist many martingales of finite variation (for example, the compensated Poisson process) that we can add to the martingale term and subtract from the finite variation term of a given decomposition, to obtain a second, new decomposition. Instead of the continuity of A we add the condition that A be predictably measurable .
Theorem 8 ( Doob-Meyer Decomposition: General Case ) . Let Z be a cadlag supermartingale with Z0 = 0 of Class D. Then there exists a unique, increasing, predictable process A with Ao = 0 such that Mt = Zt + At is a
uniformly integrable martingale.
Before proving Theorem 8 let us introduce the concept of a natural process. We introduce two definitions and prove the important properties of natural processes in the next three theorems.
Definition. An FV process A with A0
=
0 is of integrable variation if the
expected total variation is finite: E{fr�' ldA8 I } < oo. A shorthand notation for this is E{IAioo} < oo. An FV process A is of locally integrable variation if there exists a sequence of stopping times (Tn ) n > 1 increasing to oo a.s. such n r that E{J0 ldAs l} = E{IA irn } < oo, for each n.
Definition. Let
A be an (adapted) FV process, A0 = 0, of integrable varia
tion. Then A is a natural process if
E{[M, A] oo} = 0 for all bounded martingales M. Here is the key theorem about natural processes. This use of natural pro cesses was Meyer's original insight that allowed him to prove Doob ' s conjec ture, which became the Doob-Meyer decomposition theorem.
Let A be an FV process, Ao = 0, and E{IAioo} < oo. Then A is natural if and only if
Theorem 9.
112
III Semimartingales and Decomposable Processes
for any bounded martingale M. Proof. By integration by parts we have Then MoAo = 0 and letting Nt = J; As - dMs , we know that N is a local martingale (Theorem 20 of Chap. II). However using integration by parts we see that E{N�} < oo, hence N is a true martingale (Theorem 51 of Chap. I). Therefore E { J000 As - dMs } = E{ N00 } - E{ No} = 0, since N is a martingale, so that the equality holds if and only if E{[M, A] 00 } = 0. D
Let A be an FV process of integrable variation which is natural. If A is a martingale then A is identically zero. Proof. Let T be a finite stopping time and let H be any bounded, nonnegative martingale. Then E{J0T Hs_dA s } = 0, as is easily seen by approximating sums and the Dominated Convergence Theorem, since J: ldA8 I E L 1 and E{Ar} = 0. Using the naturality of A, E{Hr Ar } = E{f0T H8_dAs } = 0, and letting Ht = E{l{A r>O} I Ft} then shows that P(Ar > 0) = 0. Since D E{ Ar } = 0, we conclude Ar 0 a.s., hence A 0. Theorem 11. Let A be an FV process of integrable variation with A0 = 0 . If A is predictable, then A is natural. Theorem 10.
-
The proof of this theorem is quite intuitive and natural provided we accept a result from Chap. IV. (We do not need this theorem to prove the theorem we are using from Chap. IV.) Let M be a bounded martingale. First assume A is bounded. Then the stochastic integral Jr;'' As dMs exists, and it is a martingale by Theo rem 29 in Chap. IV combined with, for example, Corollary 3 of Theorem 27 of Chap. II. Therefore E{f000 AsdMs } = E{AoMo} = 0, since Ao = 0. How ever E{fr�' A s_dMs } = E{Ao - Mo} = 0 as well, since Ao - = 0. Further,
Proof.
1
00
AsdMs -
1
00
A s_dMs = =
=
00
1 (As - As- )dMs 00 1 LlA8dM8
2: LlA8LlMs O<s l Z
Z
S ) i b; < I� Zs,l < 2b; . We define Z0 ( t ) Zt and inductively define Zi+ 1 ( t ) Zi ( t ) and A; ( t ) 0 if Si is totally inaccessible; whereas Ai ( t ) E { � Z ,I.Fs }1{ s, E{Zi+ l (U2 )}. Again letting Sf be an announcing sequence for si we have E{Zi (Ul)} - E{Zi(U2 )} = E{Zi (U1 )} - E{Zi((Ul V Sf) 1\ U2 )} + E{Zi ((Ul V Si) 1\ U2 )} - E{Zi ((Ul V Si ) 1\ U2 ) } + E{Zi ((Ul V Si) 1\ U2 )} - E{Zi (U2 )}
'
'
k
'
where each of t he summands on the right side above are nonnegative. Then let n ----> oo to get E{Zi (U1 )} - E{Zi (U2 )} > 0. Finally we want to show that E{I:; = l Aj (oo)} < C. It is easy to check (see Exercise 1 for example) that the minimum of predictable stopping times is predictable, hence we can order the predictable times so that sl < s2 < . . . on the set where they are all finite. Let Ai be the collection of all j such that j < i and such that the Sj are all predictable. Then ""' E{Z8 k - Zs } -E{" � �Zs } = lim � k A;
1
1
1
< lif L (E{Zs;vs1_ , } + E{Zs1 _ 1 - Zs;vs1 _. } A,
+E{Zs; - Z00 } + E { Zo - Z8k }) = E{Zo - Zoo}, which is bounded by a constant not depending on i.
'
To complete the proof, we note that because the processes Ai are in creasing in t and the expectation of their sum is bounded independently of i, we have that 2::7 1 Ai (t) converges uniformly in t a.s. , as h ----> oo, and we call the limit A = (t). Fatou 's Lemma gives us that A = (t) is integrable. Each A(t) is easily seen to be predictable, and hence A00 is the almost sure limit of predictable measurable processes, so it too is predictable. Next set Z= (t) = Zt + A00 (t) = limi Zi (t). Since each Zi is a supermartingale, cou pled with the uniform convergence in t of the partial sums of the Ai pro cesses, we get that Z00 ( t) is a supermartingale. Moreover since each Zi is cadlag, again using the uniform convergence of the partial sums we obtain
115
3 The Doob-Meyer Decompositions
that Zoo is also cadlag. Since the partial sums are uniformly bounded in ex pectation, Zoo (t) is of Class D. Finally by our construction of the stopping times Si , E{.6.Z00 (T)IFr - } = 0 for all predictable times T. We can then show as before that ,W(J) 0 in probability as J ---> 0 for Z00 , and we obtain Z00 (t) = Mt - A(t). The process At = A 00 (t) + A(t) is then the desired increasing predictable process. D --->
'
The next theorem can also be considered a Doob-Meyer decomposition theorem. It exchanges the uniform integrability for a weakening of the con clusion that M be a martingale to that of M being a local martingale.
Theorem
13
(Doob-Meyer Decomposition: Case Without Class D).
Let Z be a cadlag supermartingale. Then Z has a decomposition Z = Z0 + M - A where M is a local martingale and A is an increasing process which is predictable, and M0 = A0 = 0. Such a decomposition is unique. Moreover if limt _, 00 E{Zt} > -oo, then E{A00 } < oo. Proof. First consider uniqueness. Let Z = Z0 + M - A and Z = Zo + N - C be two decompositions. Then M - N = A - C by subtraction. Hence A - C n is a local martingale. Let M T be a uniformly integrable martingale. Then n
and therefore E{A; } < E{Zo - Zt}, using Theorem 17 of Chap. I. Letting n tend to oo yields E{ At} < E{ Zo - Zt}. Thus A is integrable on [0, to ] , each t0 , as is C. Therefore A - C is of locally integrable variation, predictable, and a local martingale. Since A0 - C0 = 0, by localization and Theorems 10 and 11, A - C = 0. That is, A = C, and hence M = N as well and we have umqueness. in Next we turn to existence. Let = inf { t : IZtl > m } 1\ m. Then crease to oo a.s. and since they are bounded stopping times Zr = E L 1 each m (Theorem 17 of Chap. I) . Moreover the stopped process zr= is dominated by the integrable random variable max(IZr= I , m ) . Hence if X = zr= for fixed m , then 1t = { Xr ; T a stopping time } is uniformly integrable. Let us implicitly stop Z at the stopping time Tm and for n > 0 define Y';;n = Zt - E{Zn i Ft} , with Y';;n = Y';;'J,n = 0 when t > n. Then yn is a positive supermartingale of Class D and hence Y';;n = Y0n + Mtn - Af. Letting N!: = E{Zn iFt}, a martingale, we have on [0, n] that
rm
rm
To conclude, therefore, it suffices to show that Ar' = Af on [0, n], for m > n. This is a consequence of the uniqueness already established. The uniqueness also allows us to remove the assumption that Z is stopped at the time Finally, note that since E{At} < E{Zo - Zt}, and since A is increasing, we have by the Monotone Convergence Theorem
rm.
1 16
III Semimartingales and Decomposable Processes
E{Aoo} = tlim -+ oo E{Zo - Zt} -+oo E{At} < tlim which is finite if limt-+oo E{ Zt} >
D
-oo.
4 Quasimartingales
X be a cacllag, adapted process defined on [0, oo]. 1 Definition. A finite tuple of points T = (t0 , t 1 , . . . , tn+ l ) such that 0 = to < t 1 < · · · < tn + l = oo is a partition of [0, oo ] . Definition. Suppose that is a partition of [0, oo] and that Xti E L 1 , each ti E T. Define n C(X, T) = 2: IE{Xti - Xt;+l i FtJ I. Let
T
The variation of X along
i=O is defined to be
T Var7 (X) = E{C(X, T)}.
The variation of X is defined to be
Var (X) = sup Var7 (X), T
where the supremum is taken over all such partitions.
Definition. An adapted, cadHtg process X is a quasimartingale on
if E{IXtl}
< oo, for each t, and if Var(X) < oo.
[0, oo]
Before stating the next theorem we recall the following notational conven tion. If X is a random variable, then
x+ = max (X, O) , x - = - min(X, 0).
[0, oo ) , we set Xoo = 0. Theorem 14. Let X be a process indexed by [0, oo ) . Then X is a quasi martingale if and only if X has a decomposition X = Y - Z where Y and Z are each positive right continuous supermartingales. Also recall that by convention if X is defined only on
1
It is convenient when discussing quasimartingales to include oo in the index set, thus making it homeomorphic to [0, t] for 0 < t < oo. If a process X is defined
only on
[0, ) we extend it to [0, ] by setting X00 0. oo
oo
=
4 Quasimartingales
117
Proof For given s > 0, let I:(s) denote the set of finite subdivisions of [s, oo] . For each T E I:(s ), set
YST = E{C(X, T) + [ .Fs} and z; = E{C(X, T) - [ .Fs} where C(X, T) + denotes L:t; E E{Xt; - Xt;+1 [ .Ft; }+, and analogously for C (X, T) - . Also let --< denote theT ordering of set containment. Suppose a, T E I: (s) with a --< T. We claim Y8cr < Y87 a.s. To see this let a = (to, . . . , tn ) · It suffices to consider what happens upon adding a subdivision point t before to, after t n , or between ti and ti + l· The first two situations being clear, let us consider the third. Set
A = E{Xt; - Xt [.Ft; } ; B E{Xt - Xt;+1 [.Ft} ; C = E{Xt; - Xt; + 1 [ .FtJ; then C = A + E{B[.Ft; }, hence c + < A+ + E{B[ .Ft; } + < A + + E{B + [.FtJ , =
by Jensen's inequality. Therefore and we conclude Y.cr < Y87 • Since E{Y87} is bounded by Var(X), taking limits in £ 1 along the directed ordered set I: (s) we define
and we can define Z8 analogously. Taking a subdivision with t0 = s and tn + l = oo, we see YST - z; = E{c+ - c- [.J=·s } = Xs , and we deduce that Ys - Z8 = X8• Moreover if s < t it is easily checked that Ys > E{Yt[.Fs} and Z8 > E{Zt[.F8}. Define the right continuous processes yt Yt+ , Zt = Zt + ' with the right limits taken through the rationals. Then Y and Z are positive supermartingales and Ys - Z8 = X8• For the converse, suppose X = Y - Z, where Y and Z are each positive supermartingales. Then for a partition T of [0, t] -
= E{Y0} + E{Z0} - (E{yt} + E{Zt}). Thus X is a quasimartingale on [0, t], each t > 0.
D
118
III Semimartingales and Decomposable Processes
A quasimartingale X has a unique decom position X = M + A, where M is a local martingale and A is a predictable process with paths of locally integmble variation and Ao = 0. 0 Proof This theorem is a combination of Theorems 13 and 14.
Theorem 1 5 (Rao's Theorem) .
5 Compensators
Let A be a process of locally integrable variation, hence a fortiori an FV process. A is then locally a quasimartingale, and hence by Rao's Theorem (Theorem 15), there exists a unique decomposition words, there exists a unique, where A is a predictable-FV process. In other predictable FV process A such that A - A is a local martingale. -
0, _with locally inte!Q"able total variation. The unique FV predictable process A such that A - A is a local martingale is called the compensator of A. The most common examples are when A is an increasing process locally of integrable variation. When A is an increasing process it is of course a submartingale, and thus by the Doob-Meyer Theorem we know that its com pensator A i� also increasing. We also make the obvious observation that E{At} = E{At} for all t, 0 < t < oo Theorem 16. Let A be an increasing process of integmble variation, and let H E L be such that E{f; H8dA8} < oo Then, Definition. Let
A be an FV process with A0
=
.
.
Since A - A is a martingale, so also is has constant expectation equal to 0.
Proof
J H8d(As - A8 ) , and hence it
0
In Chap. IV we develop stochastic integration for integrands which are predictable, and Theorem 16 extends with H E L replaced with H predictable. One of the simplest examples to consider as an illustration is that of the Poisson process N = (Nt)t>O with parameter .X. Recall that Nt - .Xt is a martingale. Since the process At = .Xt_is continuous and obviously adapted, it is predictable ( natural ) . Therefore Nt = >.t, t > 0. A natural extension of the Poisson process case is that of counting processes without explosions. We begin however with a counting process that has only one jump. Let TJ be the counting process 'flt = l{t> T }, where T is a nonnegative random variable. Let lF be the minimal filtration making T a stopping time.
5 Compensators
119
Let P(T = 0) = 0 and P(T > t) > 0, each t > 0. Then the lF compensator A of 7], where 7Jt = l{t>7 } , is given by
Theorem 17.
t r A 1 dF At = J (u) , o 1 - F(u-) where F is the cumulative distribution function of T. If T has a diffuse distribution {that is, if F is continuous}, then A is continuous and A t = - ln( l - F (T 1\ t)) . Before proving Theorem 1 7 we formalize an elementary result as a lemma.
Lemma. Let (n, F, P) be a complete probability space. In addition, suppose
a positive F measurable random variable and no = a {T 1\ t } , where o n = n v N, N are the null sets of :F, and :Ft = nu >t :F2. Then lF so constructed is the smallest filtration making T a stopping time. Let Y E L 1 (:F). Then
T is
By the hypotheses on T the a-algebra n is equal to the Borel a algebra on [0, t] together with the indivisible atom (t, oo). Observe that n = D a{ T 1\ u; u < t } , and the result follows easily.
Proof.
Proof of Theorem 1 7. Fix to > 0 and let be a sequence of partitions of [0, to] with liiDn-Hx' mesh (nn ) = 0. Define Ar = L:1r n E{ 7Jt;+' - 7Jt; 1Ft; } for 0 < t < to. Then 'lrn
} } { E 77t;+, >t; l{T { F. } 1 E 77t;+I I t ; = 7Jti+l {T t l {T>t;}
i)
by the lemma preceding this proof. The first term on the right above is 7Jt;+, l{t;>7} = l{t;+1 >7}l {t;>7} = l {t;>T} · Furthermore, l{7>t;}7Jt;+' = 1 {T > t;} l{t;+1>T } = l{t; t 1 S1 = s 1 , . . . ' si - 1 = Si- 1 ) where the s1 are in [0, oo] , j > 1 . (If one of the s1 takes the value +oo, then
the corresponding cumulative distribution function is concentrated at +oo.) Define
t 1 ¢i (s 1 , . . . , si - 1 ; t) =
Fi ( s 1 , . . . , si-1 ., s-) dFi (s 1 , . . . , s i - ; s) . describe the compensator of N for its minimal -1
1
o
We now have the tools to completed filtration. We omit the proof, since it is simply a more notationally cumbersome version of the proof of Theorem 17.
Let N be a counting process without explosions and let lF be its minimal completed filtration. Then the compensator A of N is given by At = ¢ 1 (Sl ) + c/>2 (81 ; S2 ) + · · · + c/>i -1 (81 , . . . , Si - 2 ; Si - 1 ) + ¢i (S1 , . . . , Si- 1 ; t - Ti ) on the event {Ti < t < Ti+ d · Corollary. Let N be a counting process without explosions and independent interarrival times. Then the functions ¢i defined in equation ( * ) have the Theorem 19.
simplified form
and the compensator
A is given by i-1
At = 2::.: [2::.: ¢J ( SJ ) + cpi (t - Ti )]l{ T; 1 j = 1 Example (hazard rates and censored data) . In applications, an intuitive interpretation often used is that of hazard rates. Let
T be a positive random
122
III Semimartingales and Decomposable Processes
variable with a continuous density for its distribution function F, given by The hazard rate A is defined, when it exists, to be
f.
1
lim P(t < T < t + h i T > t). A(t) = h--+0 h The intuition is that this is the probability the event will happen in the next infinitesimal unit of time, given that it has not yet happened. Another way of viewing this is that P(t < T < t + hiT > t) = Ah + o(h). By Theorem 1 7 we have that the compensator of Nt = l{t> T } is At = J�I\T A(s)ds. Or in language closer to stochastic integration,
Nt -
1
ti\ T
A(s)ds
is a local martingale.
In medical trials and other applications, one is faced with situations where one is studying random arrivals, but some of the data cannot be seen (for example when patients disappear unexpectedly from clinical trials). This is known as arrivals with censored data. A simple example is as follows. Let T be a random time with a density and let U be another time, with an arbitrary distribution. T is considered to be the arrival time and U is the censoring time. Let X = T 1\ U, Nt = l {t> X }l{ u> T} ' and Nf = l {t>x}l{ T > U } with Ft = a { Nu , N;j; u < t}. Let A be the hazard rate function for T. A is known as the net hazard rate. We define A#, known as the crude hazard rate, by
A# = �� � P(t < T < t + hiT > t, U > t)
when the limit exists. We then have that the compensator of N is given by At = f� l{x>u} A#(u)du = J�I\X A#(u)du = J�I\TI\ U A#(u)du, or equivalently
t
Nt - lo l{x> u } A# (u)du If we impose the condition that as
is a local martingale.
A = A#, which can be intuitively interpreted
P(s < T < s + dsiT > s) = P(s < T < s + dsiT > s, U > s),
then we have the satisfying result that
t
Nt - lo l{x> u } A(u )du
is a local martingale.
As we saw in Chap. II, processes of fundamental importance to the the ory of stochastic integration are the quadratic variation processes [X, X] = ([X, X]t)t>o, where X is a semimartingale.
Definition. Let
X
be a semimartingale such that its quadratic variation process [X, X] is locally integrable. Then the conditional quadratic vari ation of X, denoted (X, X ) = ((X, X)t)t>o, exists and it is defined to be the compensator of [X, X] . That is (X, X) = [X, X]. �
5 Compensators
123
If X is a continuous semimartingale then [X, X] is also continuous and hence already predictable; thus [X, X] = (X, X) when X is continuous. In particular for a standard Brownian motion B, [B, B]t = (B, B)t = t, all t > 0. The conditional quadratic variation is also known in the literature by its notation. It is sometimes called the sharp bracket, the angle bracket, or the oblique bracket. It has properties analogous to that of the quadratic variation processes. For example, if X and Y are two semimartingales such that (X, X), ( Y, Y) , and (X + Y, X + Y) all exist, then (X, Y) exists and can be defined by polarization
(X, Y) =
2
1 ((X
+ Y, X + Y)
- (X, X) - ( Y, Y) ) .
However (X, Y) can be defined independently as the compensator of [X, Y ] provided of course that [X, Y] is locally of integrable variation. In other words, there exist stopping times (Tn )n > l increasing to oo a.s. such that n T E{f0 l d [X, Y] s l } < oo for each n. Also, (X, X) is a non-decreasing pro cess by the preceding discussion, since [X, X] is non-decreasing. The condi tional quadratic variation is inconvenient since unlike the quadratic variation it doesn't always exist. Moreover while [X, X], [X, YJ , and [Y, Y] all remain invariant with a change to an equivalent probability measure, the sharp brack
ets in general change with a change to an equivalent probability measure and may even no longer exist. Although the angle bracket is ubiquitous in the
literature it is sometimes unnecessary as one can often use the quadratic vari ation instead, and indeed whenever possible we use the quadratic variation
rather than the conditional quadratic variation (X, X) of a semimartingale X in this book. Nevertheless the process (X, X) occurs naturally in extensions
of Girsanov's theorem for example, and it has become indispensable in many areas of advanced analysis in the theory of stochastic processes. We end this section with several useful observations that we formalize as theorems and a corollary. Note that the second theorem below is a refinement of the first (and see also Exercise 25).
Let A be an increasing process of locally integrable variation, all of who�e jumps occur at totally inaccessible stopping times. Then its com pensator A is continuous. Theorem 21. Let A be an increasing process of locally integrable variation, and let:__T be a jump time of A which is totally inaccessible. Then its compen sator A is continuous at T. 0 Proof. Both theorems are simple consequences of Theorem 7. Corollary. Let A be an increasing predictable process of locally integrable variation and let T be a stopping time. If P(Ar =f. Ar - ) > 0 then T is not Theorem 20.
totally inaccessible.
124 III Semimartingales and Decomposable Processes Proof _§uppose T were totally inaccessible. Let A be the compensator:_of A. Then A is continuous at T. But since A is already predictable, A A, and we have a contradiction by Theorem 21. 0 Theorem 22. Let T be a totally inaccessible stopping time. There exists a martingale M with paths of finite variation and with exactly one jump, of size one, occurring at time T {that is, Mr -/=- Mr- on {T < oo } ). Proof Define Ut l{t> T} · -
=
=
Then U � an increasing, bounded process of integrable variation, and we let A = U be the compensator of U. A is continuous by Theorem 20, and M = U A is the required martingale. 0 -
6 The Fundamental Theorem of Local Martingales
We begin with two preliminary results.
Let T be a stopping time and let H be an integrable random variable such that E{H IFr - } = 0 on {T < oo}. Then the right continuous martingale Ht = E{HIFt} is zero on [O, T) = {( t , w) 0 < t < T (w) } . Proof Since the martingale (Ht)t>o is right continuous it suffices to show that Htl{t < T} = 0 almost surely for all t. Let A E Ft. Then A n {t < T} belongs both to Fr - and also to Ft. Hence 0 = E{H lAn {t< T } } = E{Htl An {t< T } } · Since A E Ft is arbitrary, this implies that Ht l{t < T } = 0 a.s. If H is Fr measurable then Hl { T 0. Then there exist local martingales N, D such that D is an FV process, the jumps of N are bounded by 2{3, and
M = N + D. Proof. We first set
Ct =
I�Msll{I � M. I> ,B } , L O<s n or IMtl > n}.
I�MRn l < I MRn l + IMRn- 1 < IMRn l + n . Moreover eRn < eRn- + I�MRn I < IMRn I + 2n, which is in L 1 . Since Rn increases to oo, we have C is locally integrable. By stopping we further assume that E{ C00} < oo .
Then
Next we define
At = L �Msl{I � M.I > ,B} , O<s 1,
where :Ft = Pi V N, with N all the P-null sets of F. Let Mt = E{XI:Ft}, the right continuous version. Then M is not a locally square integrable martingale. The next example shows another way in which local martingales differ from martingales. A local martingale need not remain a local martingale under a shrinkage of the filtration. They do, however, remain semimartingales and thus they still have an interpretation as a differential.
Y be a symmetric random variable with a continuous distri bution and such that E{ IY I} = oo Let Xt = Yl{t> l }, and define Example. Let
a{IYI}, {a{Y}, .
g� where
=
0 < t < 1, t > 1,
G = Wt)t>o is the completed filtration. Define stopping times Tn by
{
rn = 0,
oo ,
if IY I > n , otherwise.
7 Classical Semimartingales
1 27
Then rn reduce X and show that it is a local martingale. However X is not a local martingale relative to its completed minimal filtration. Note that X is still a semimartingale however. The full power of Theorem 25 will become apparent in Sect. 7. 7 Classical Semimartingales
We have seen that a decomposable process is a semimartingale (Theorem 9 of Chap. II) . We can now show that a classical semimartingale is indeed a semimartingale as well.
A classical semimartingale is a semimartingale. Proof. Let X be a classical semimartingale. Then Xt = Mt + At where M is a local martingale and A is an FV process. The process A is a semimartin gale by Theorem 7 of Chap. II, and M is decomposable by the corollary of
Theorem 26.
Theorem 25, hence also a semimartingale (Theorem 9 of Chap. II) . Since semi martingales form a vector space (Theorem 1 of Chap. II) we conclude X is a semimartingale. 0
Corollary. A cadlag local martingale is a semimartingale.
Proof.
A local martingale is a classical semimartingale.
Theorem 27.
A c adtag quasimartingale is a semimartingale.
D
Proof. By Theorem 15 a quasimartingale is a classical semimartingale. Hence it is a semimartingale by Theorem 2 6 .
Theorem 28.
Proof.
A cadlag supermartingale is a semimartingale.
0
Since a local semimartingale is a semimartingale (corollary to The orem 6 of Chap. II) , it suffices to show that for a supermartingale X, the stopped process x t is a semimartingale. However for a partition T of [0, t],
t; ET t; ET = E{Xo} - E{Xt} ·
Therefore x t is a quasimartingale, hence a semimartingale by Theorem 27.
Corollary. A submartingale is a semimartingale.
0
128
III Semimartingales and Decomposable Processes
We saw in Chap. II that if is a locally square integrable local martingale and H E IL, then the stochastic integral H · is also a locally square inte grable local martingale (Theorem 20 of Chap. II) . Because of the corollary of Theorem 25 we can now improve this result.
X
X
Let M be a local martingale and let H E IL. Then the stochastic integral H · M is again a local martingale. Proof A local martingale is a semimartingale by the corollary of Theorem 25 and Theorem 9 of Chap. II; thus H · M is defined. By the Fundamental Theorem of Local Martingales (Theorem 25) for (3 > 0 we can write M = N + A where N, A are local martingales, the jumps of N are bounded by (3, and A has paths of finite variation on compacts. Since N has bounded jumps, by stopping we can assume N is bounded. Define T by Theorem 29.
T = inf{ t >
t
0 : lo ldAsl > m}.
Then E{ft\T ldAs l } < m + (3 + E{I�MT I} < oo , and thus by stopping A can be assumed to be of integrable variation. Also by replacing H by H81{ S>O} for an appropriate stopping time S we can assume without loss of generality that H is bounded, since H is left continuous. We also assume without loss that Mo - No = Ao = 0. We know H N is a local martingale by Theorem 20 of Chap. II, thus we need show only that H · A is a local martingale. Let O"n be a sequence of random partitions of [0, t] tending to the identity. + 1 ATt ) tends to (H · A in ucp, where a is the sequence Then I: HTn ( ATt n 0 = T0 < Tf < · · < Tt < · · · . Let (nk ) be a subsequence such that the sums converge uniformly a.s. on [0, t]. Then ·
-
•
)t
·
by Lebesgue's Dominated Convergence Theorem. Since the last limit above equals (H · A)s , we conclude that H · A is indeed a local martingale. D
A note of caution is in order here. Theorem 29 does not extend completely to processes that are not in IL but are only predictably measurable, as we will see in Emery's example of a stochastic integral behaving badly on page 176 in Chap. IV. = X0 + M + A be a be a classical semimartingale, and let Let decomposition where M0 = A0 = 0, M is a local martingale, and A is an FV
X
Xt
t t
7 Classical Semimartingales process. Then if the space (!1, :F, (:Ft)t>o, P) supports a Poisson process we can write
1 29
N,
Xt Xo + {Mt + Nt - t} + {At - Nt + t} =
as another decomposition of X. In other words, the decomposition of a classi cal semimartingale need not be unique. This problem can often be solved by choosing a certain canonical decomposition which is unique. be a semimartingale. If X has a decomposition Xt = Xo + Mt + At with Mo = Ao = 0, M a local martingale, A an FV process, and with A predictable, then X is said to be a special semimartingale.
Definition. Let
X
= 0. Theorem 30. If X is a special semimartingale, then its decomposition X = M + A with A predictable is unique. Proof. Let X = N + B be another such decomposition. Then M -N = B -A, hence B - A is an FV process which is a local martingale. Moreover, B - A is predictable, and hence constant by Theorem 12. Since Bo - Ao = 0, we conclude B = A. 0 Definition. If X is a special semimartingale, then the unique decomposition X = M + A with Mo = Xo and A0 = 0 and A predictable is called the To simplify notation we henceforth assume X0
canonical decomposition.
Theorem 1 5 shows that any quasimartingale is special. A useful sufficient condition for a semimartingale X to be special is that X be a classical semi martingale, or equivalently decomposable, and also have bounded jumps.
Let X be a classical semimartingale with bounded jumps. Then X is a special semimartingale. Proof. Let Xt = Xo + Mt + At be a decomposition of X with Mo = Ao = 0, M a local martingale, and A an FV process. By Theorem 25 we can then also
Theorem 31.
write
Xt = Xo + Nt + Bt
where N is a local martingale with bounded jumps and B is an FV process. Since X and N each have bounded jumps, so also does B. Consequently, it is locally a quasimartingale and therefore decomposes
where L is a local martingale and B is a predictable FV process (Theorem 1 5 ) . Therefore �
Xt = Xo + {Nt + Lt} + Bt is the canonical decomposition of X and hence X is special. �
0
130
III Semimartingales and Decomposable Processes
Corollary. Let X be a classical semimartingale with continuous paths. Then
X is special and in its canonical decomposition the local martingale
M and the FV process A have continuous paths.
Proof. X is continuous hence trivially has bounded jumps, so it is special by Theorem 31. Since X is continuous we must have flMT = -flAT for any stopping time T (flAT = 0 by convention on {T = oo } ). Suppose A jumps at a stopping time T. By Theorem 3, T = TA 1\ Ts , where TA
is accessible and Ts is totally inaccessible. By Theorem 21 it follows that P( l flATB I > 0) = 0. Hence without loss of generality we can assume T is accessible. It then suffices to consider T predictable since countably many predictable times cover the stopping time T. Let Sn be a sequence of stopping times announcing T. Since A is predictable, we know by Theorem 24 that flAT is :FT- measurable. Therefore flMT is also FT - measurable. Stop M so that it is a uniformly integrable martingale. Then
flMT = E{flMT I:FT- } = 0,
M, and hence A, are continuous, using Theorem 7 of Chap. I. Theorem 31 can be strengthened, as the next two theorems show.
and
criteria given in these theorems are quite useful.
D
The
Let X be a semimartingale. X is special if and only if the process Jt = SUP s inf > 0 : bounded. Let Tn , and let be defined by From now on we assume Then is still < and the stopping times Tn an FV process and is still predictable, increase to oo a.s. Integration by parts gives us also
[A, i]i ] = EQ{[A, T } i
· If we show that is also a Q local martingale, we will be done. It suffices to show that 0 for every stopping time T. Since is FV and predictable, by stopping at T- if necessary, we can assume the jumps of are bounded. And since is a Q local martingale, its supremum process is locally integrable (Theorem 34). This is enough to give us that is Q locally of integrable variation. By stopping we assume it is integrable. We then have that is Q natural, so if T is a stopping time, we replace with and we still have a natural process, and thus we have 0. Since T was arbitrary, this implies that is a martingale. Of course, the process is implicitly stopped at several stopping times, so what we have actually shown is that is a Q local martingale. We now have that
A
[A, i]
[A, i]
A
[A, i] A AT EQ{[A, i ]T } =
A
1 At = Q-local martingale + 1t 1 dAs t o Zs which in turn gives us, recalling that A = (X, Z), Xt = Z1t (ZtXt ) = Z1t (Mt + (X, Z) t ) t 1 1 1 d(X, Z)s · = -M + Q-local martingale + t Zt 0 sSince Z ( i M) = M = P local martingale, we have that i M is also a Q lo cal martingale. Thus finally X - I� z!_ d(X, Z) s is a Q local martingale, and the t z
z
proof is done.
D
Let us next consider the interesting case where Q is absolutely continuous with respect to but the two probability measures are not equivalent. (That is, Q.) We begin with a simple result, and we will assume a.s.
P, we no longer assume that P is also absolutely continuous with respect to :F0 = { 0,!1} Theorem 37. Let X be a P local martingale with X0 = 0. Let Q be another probability absolutely continuous with respect to P, and let Zt = E { � I :Ft } .
8
13 5
Girsanov's Theorem
Assume that {X, Z) exists for P. Then At = J; z!_ d {X, Z)s exists a.s. for the probability Q, and Xt - J; z!_ d{X, Z) s is a Q local martingale. Proof. Zo = E{Z} = 1 , so if we let Rn = inf{t > 0 : Zt < 1 /n} , then Rn increase to oo , Q-a.s., and the process z:_ is bounded on [0, Rn ] · By Theorem 36, x[ln - J; Z�- d{X, z):-n is a Q local martingale, each n. Since a D
local, local martingale is a local martingale, we are done.
We now turn to the general case, where we no longer assume the existence of {X, Z), calculated with (As before, we take X to be a local martingale. ) We begin by defining a key stopping time: R = inf{ > 0 : Zt = 0, Zt - > 0 }. Note that Q ( R < oo ) = 0, but it is entirely possible that R < oo ) > 0. We further define Ut = �XR 1{t> R} · Then U is an FV process, and moreover U is locally integrable Let Tn increase to oo and be such that XTn is a uniformly integrable martingale. Then
P.
P t P(
(dP).
Thus U has a compensator local martingale.
U, and of course U is predictable and U - U is a �
�
�
P Theorem 38 (Lenglart-Girsanov Theorem) . Let X be a P local martin gale with Xo = 0. Let Q be a probability absolutely continuous with respect to P, and let Zt = Ep{ � IFt}, R = inf{t > 0 : Zt = 0, Zt- > 0}, and Ut = �XR I{t> R} · Then t 1 1 Xt - -d[X, Z]s + Ut 0 Zs
is a Q local martingale. Proof. Let Rn = inf {t > 0 : Zt < �}. (Recall thatRn Zo = 1R, nand also note that it is possible that Rn = R . ) Then both X and Z are P local n R Rn Rn martingales. Also note that A� = J; zfn 1 {z:Zn >O} d[X , z ]s , y Rn = x Rn - A Rn + URn are all ?-well-defined. We can define
At =
u n , and
lo �s l{z. >o} d[X, Z]s t
on [O , R) , since d[X Rn , zRn ]s does not charge (R, oo), and z.An 1 { z Rn > o} = 0 at R. Thus we need only to show Y Rn is a Q local martingale for each fixed which is the same as showing that zRn y Rn is a p local martingale. Let us assume all these processes are stopped at Rn to simplify notation. We have 8
n,
ZY = ZX - ZA + ZU. �
136
III
Semi martingales and Decomposable Processes
Hence,
d(ZX) = Z�dX + X�dZ + d[Z, X] = local martingale + d[Z, X] d(AZ) = A�dZ + ZdA = local martingale + ZdA = local martingale + l { z >o}d[X, Z] d(ZU) = Z�dU + UdZ local martingale + Z�dU = local martingale + Z�dU �
�
�
�
where the last equality uses that U-U is a local martingale ( dP). Summarizing we have
ZY = ZX - ZA + ZU = local martingale + [Z, X] - local martingale + l{z>o} d[X, Z]) + local martingale + Z�dU �
J
which we want to be a local martingale under case if
J
dP. This will certainly be the
d[Z, X] - l { z > o}d[X, Z] + Z�dU = 0.
However ( *) equals
�ZR �XR l{tA Rn > R} + ZR� �XR l{tA Rn > R} · But �ZR = ZR - ZR � = 0 - ZR� = -ZR� , and this implies that equation (*)
D
is indeed zero, and thus the Lenglart-Girsanov Theorem holds.
Corollary. Let X be a continuous local martingale under P. Let Q be abso
(X, Z) = [Z, X] = [Z, X] c exists, t f Xt - Jo s� d[Z, X]� = Xt - Jo asd[X , X]s
lutely continuous with respect to and
P.
Then
t zl
which is a Q local martingale. Proof By the Kunita-Watanabe inequality we have
which shows that it is absolutely continuous with respect to whence the result.
d[X, X]s
a.s. ,
D
8
137
Girsanov's Theorem
is the solution of a stochastic exponential equation Z dZ8 Z8_H8dX8 (which it often is) , then = H8 • Example. A problem that arises often in mathematical finance theory is that one has a semimartingale S = M + A defined on a filtered probability space (0, :F, IF, P) satisfying the usual hypotheses, and one wants to find an equivalent probability measure Q such that under Q the semimartingale X is a local martingale, or better, a martingale. In essence this amounts to finding a probability measure that "removes the drift." To be concrete, let us suppose S is the solution of a stochastic differential equation3 dS8 = h(s, S8)dBs + b(s; Sr ; r < s)ds, where B is a standard Wiener process (Brownian motion) under P. Let us postulate the existence of a Q and let Z = �� and Zt = E{Z I :Ft }, which is clearly a cadlag martingale. By Girsanov's Theorem rJot h(s, Ss)dBs - ro t 1 d[Z, Jro h(r, Sr )dBr ]s J Zs is a Q local martingale. We want to find the martingale Z. In Chap. IV We remark that if of the form =
0! 8
we will study martingale representation and show in particular that every local martingale on a Brownian space is a stochastic integral with respect to Brownian motion. Thus we can write = for some predictable process If we assume is well behaved enough to define = t, then which gives us a linear stochastic differential = we have equation to solve for Thus if we let = = we get that It remains to determine We do this by observing from our previous Girsanov calculation that
J. Z Zt 1 + J� H8Z8dB8, Z. H.
Zt 1 + J� J8dB8
H8
Nt J� H8dB8,
Zt
t:(N) t .4 t{ h(s, S8)dBs t ..l_zsHsh(s, Ss)ds = {o t h s, S8)dBs - {o t Hsh(s, Ss)ds - Jo Zs J ( J Jo -b(s; Sr ; 0 there exists a c > 0 such that Q(( > c)
=
III
146
Semimartingales and Decomposable Processes
A
Let E F such that Q(A) > 0. Then there exists a constant d such that Q(( > d) < Q(A)/2, for all ( E /3, by assumption. Using this constant let c = 2d, and we have that 0 < clA tt (3, and moreover if B+ denotes all bounded, positive r.v. , then clA is not in the (dQ) closure of /3-B+, denoted is L00 , and f3 - B+ is f3 - B+ . That is, clA tt f3 - B+ . Since the dual of convex, by a version of the Hahn-Banach Theorem (see, e.g. , Treves [223, page 190] ) there exists a bounded random variable Y such that
d,
L1
sup
(Ef3,TJEB+
EQ {Y(( - 17) }
L1
< cEQ {Y1A}.
Replacing 17 by a l {Y 0 a.s., since otherwise the expectation on the left side above would get arbitrarily large. Next suppose 17 = 0. Then the inequality above gives sup EQ {Y(} ( E (3
< cEQ{YlA } < +oo.
Now set H = {Y E B+ : sup(E (3 EQ {Y(} < oo } Since 0 E B+ , we know H is not empty. Let A = {all sets of the form {Z = O}, Z E H}. We wish to show that there exists a Z E H such that Q(Z = 0) = infAEA Q(A). Suppose, then, that Zn is a sequence of elements of H. Let Cn = sup( E f3 E{ Zn (} and dn = II Zn llv"' · (Since 0 E /3, we have Cn > 0). Choose bn such that 'L, bn Cn < oo and 'L, bn dn < oo, and set Z = 'L, bn Zn . Then clearly Z E H. Moreover, { Z 0 } = nn { Zn = 0}. Thus A is stable under countable intersections, and so there exists a Z such that Q( Z = 0) = infAEA Q(A). We now wish to show Z > 0 a.s. Suppose not. That is, suppose Q(Z = 0) > 0. Let Y satisfy ( ) (we have seen that there exists such a Y and that it hence is in H). Further we take for our set A in ( ) the set = { Z = 0}, for which we are assuming Q(A) > 0. Since 0 E f3 and 0 E B+ , we have from Lemma 2 that 0 < E{Y1A } = E{Y1 {z=o} } · Since each of Y and Z are in H, their sum is in H as well. But then the above implies .
=
*
*
A
Q{Y + Z = 0 }
= Q{Z = 0} - Q({Z = 0 } n {Y > 0}) < Q(Z = 0). This, then, is a contradiction, since Q(Z = 0) is minimal for Z E H. Therefore we conclude Z > 0 a.s., and since Z E B+ , it is bounded as well, and Lemma 3 is proved; thus also, Theorem 43 is proved. 0 We state again, for emphasis, that Theorems 26 and 43 together allow us to conclude that semimartingales (as we have defined them) and classical semimartingales are the same.
Exercises for Chapter III
147
Bibliographic Notes
The material of Chap. III comprises a large part of the core of the "general theory of processes" as presented, for example in Dellacherie [41], or alterna tively Dellacherie and Meyer [45, 46, 44] . We have tried once again to keep the proofs non-technical, but instead of relying on the concept of a natural process as we did in the first edition, we have used the approach of R. Bass [11] , which uses the P. A. Meyer classification of stopping times and Doob's quadratic in equality to prove the Doob-Meyer Theorem (Theorem 13) , the key result of the whole theory. The Doob decomposition is from Doob [55], and the Doob-Meyer decom position (Theorem 13) is originally due to Meyer [163, 164] . The theory of quasimartingales was developed by Fisk [73] , Orey [187] , K. M. Rao [207] , Stricker [218] , and Metivier-Pellaumail [159] . The treatment of compensators is new to this edition. The simple example of the compensator of a process with one jump of size one (Theorem 17), dates back to 1970 with the now classic paper of Dellacherie [40] . The case of many jumps (Theorem 19) is due to C. S . Chou and P. A. Meyer [30] , and can be found in many texts on point processes, such as [24] or [139] . The example of hazard rates comes from Fleming and Harrington [74] . The Fundamental Theorem of Local Martingales is due to J. A. Yan and appears in an article of Meyer [1 72] ; it was also proved independently by Doleans-Dade [51]. Le Jan's Theorem is from [112] . The notion of special semimartingales and canonical decompositions is due to Meyer [171] ; see also Yoeurp [234] . The Girsanov-Meyer theorems (The orems 35 and 36) trace their origin to the 1954 work of Maruyama [152] , followed by Girsanov [83] , who considered the Brownian case only. The two versions presented here are due to Meyer [ 171] , and the cases where the two measures are not equivalent (Theorems 37 and 38) are due to Lenglart [142] . The example from finance theory (starting on page 137) is inspired by [203] . Kazamaki's criterion was published originally in 1977 [1 24] ; see also [125] , whereas Novikov's condition dates to 1972 [184] . The Bichteler-Dellacherie Theorem (Theorem 43) is due independently to Bichteler [13, 14] and Dellacherie [42] . It was proved in the late 1970's, but the first time it appeared in print was when J. Jacod included it in his 1979 tome [103] . Many people have made contributions to this theorem, which had at least some of its origins in the works of Metivier-Pellaumail, Mokobodzki, Nikishin, Letta, and Lenglart . Our treatment was inspired by Meyer [176] and by Yan [233] . Exercises for Chapter III
Exercise 1 . Show that the maximum and the minimum of a finite number of predictable stopping times is still a predictable stopping time.
148
Exercises for Chapter III
S be a totally inaccessible stopping time, and let T = S + 1 . Show that T is a predictable stopping time. Exercise 3. Let S, T be predictable stopping times. Let A = {S = T}. Show that SA is a predictable stopping time. Exercise 4. Let (0, F, lF, P) be a filtered probability space satisfying the usual hypotheses. Show that the predictable a-algebra (on JR.+ 0) is con Exercise 2. Let
x
tained in the optional a-algebra. ( Hint: Show that a cadlag, adapted process can be approximated by processes in lL.)
T be stopping times with S < T. Show that ( S, T] = S, t < T(w)} is a predictable set. Show further that [S, T) {andt (S, T) are optional sets. Last, show that if T is predictable, then (S, T) is a predictable set, and if both S and T are predictable, then [S, T) is a predictable set. Exercise 6. Let T be a predictable stopping time, and let (Sn ) n >l be a sequence of stopping times announcing T. Show that Fr - = Vn Fsn . Exercise 7. A filtration lF is called quasi left continuous if for every pre dictable stopping time T one has Fr Fr- .7 Show that if lF is a quasi left continuous filtration, then whenever one has a non-decreasing sequence of stopping times (Sn ) n > l , with limn oo Sn = T, it follows that Fr = Vn Fsn . filtration such that there exists ( Note: One can have a quasi left continuous .)
Exercise 5. Let ( , w) : S(w)
T } ·
and suppose the filtration is constant: :Ft = :F0 for all t > 0. Show that the compensator A of N is At = l {t> T} · (This illustrates the importance of the filtration when calculating the compensator.) Let
:Fo
Nt
=
a{T}
N be a compound Poisson process, Nt = Li> l Ui 1{t> T,} where the times Ti are the arrival times of a standard Poisson process with parameter A and the Ui are i.i.d. and independent of the arrival times. Suppose E{JUiJ} < oo and E{Ui } = 11 · Show that the compensator A of N for the natural filtration is given by At = AJ-lt. Exercise 15. Show that if T is exponential with parameter A, its hazard rate is constant and equal to A. Show also that if R is Weibull with parameters (o:, {3), then its hazard rate is A(t) = o:{3cx t cx- l . Exercise 16. Let T be exponential with parameter A and have joint distri bution with U given by P(T > t, U > s) = exp{- At - J1S - Bts } for t > 0, s > 0, where A, J1, and B are all positive constants and also B < AJ-l. Show that the crude hazard rate of (T, U) is given by A# (t) = A + Bt. *Exercise 17. Let M be a martingale on a filtered space where the filtra tion is quasi left continuous. Show that (M, M) is continuous. (Hint: See the discussion on quasi left continuous filtrations on page 1 89 of Chap. IV.) Exercise 18. Let X be a semimartingale such that the process Dt = SUPs I of stopping times increasing to Mn + An , with An a predictable finite variation process, for each n. Show that X is special. (That is, a semimartingale which is locally special is also special.) Exercise 20. Prove the Lemma on page 1 32. Let Q rv P, and Zt = Ep { �� J:Ft } · Show that an adapted, dtdlag process M is a Q local martingale if and only if M is a P local martingale. Exercise 14. Let
00
Z
*Exercise 21. Let lF
C G be filtrations satisfying the usual hypotheses, and
let X be a G quasimartingale. Suppose that Yt = E{Xtl:Ft}, and that Y can be taken cadlag (this can be proved to be true, although it is a little hard to
150
Exercises for Chapter III
prove) . Show that Y is also a quasimartingale for the filtration JF. Rao's Theorem .)
(Hint: Use
*Exercise 22. Suppose that A is a predictable finite variation process with E{[A, A]l } < oo , and that M is a bounded martingale. Show that [A, M] is 1
a uniformly integrable martingale.
Exercise 23. Let T be a strictly positive random variable, and let :Ft = a{T 1\ s; s < t}. Show that (Ft)t>O is the smallest filtration making T a stopping time.
Z be a cadlag supermartingale of Class D with Z0 = 0 and suppose for all predictable stopping times T one has E{�ZriFr- } = 0, a.s. Show that if Z = M - A is the unique Doob-Meyer decomposition of Z, then A has continuous paths almost surely. Exercise 25. Let A be an increasing process of integrable variation, and let T be a predictable jump time of A such that E{6.Ar iFr - } = 0. Then its compensator A is continuous at T. (This exercise complements Theorem 21.) *Exercise 26. A supermartingale Z is said to be regular if whenever a sequence of stopping times (Tn ) n> 1 increases to T, then limn-+ oo E{ Zrn } = E{Zr }. Let Z be a cadlag supermartingale of Class D with Doob-Meyer decomposition Z = M - A. Show that A is continuous if and only if Z is Exercise 24. Let
regular.
*Exercise 27 (approximation of the compensator by Laplacians) . Let Z be a dtdlag positive supermartingale of Class D with limt-+ oo E{Zt} = 0. (Such a supermartingale is called a potential.) Let Z = M - A be its Doob Meyer decomposition and assume further that
A is continuous. Define
Show that for any stopping time T, limh-+0 A�
=
Ar with convergence in £1 .
Z be a dtdlag positive supermartingale of Class D withh limt-+oo E{ Zt} = 0. Let Z = M- A be its Doob-Meyer decomposition. Let A be as given in Exercise 27. Show that for any stopping time T, limh-+O A� = Ar , but in this case the convergence is weak for £ 1 ; that is, the convergence *Exercise 28. Let
is in the topology a( £1 , £ 00 ). 8
Z be a cadlag positive supermartingale of Class D with limt-+oo E{ Zt} = 0. Let Z = M - A be its Doob-Meyer decomposition and assume further that A is continuous. Exercise 29 (discrete Laplacian approximations) . Let
Define 8
xn converges to X in cr (L 1 , U)() ) if xn, X are in L 1 and for any a.s. bounded random variable Y, E(XnY) -+ E(XY) .
Exercises for Chapter III
00
151
E{Z k A� = """' L-J
k=O
k k+ z , IF 2ft" � 2ft" }
Show that limn-+oo A� = A00 with convergence in
L1 .
Exercise 30. Use Meyer's Theorem (Theorem 4) to show that if X is a strong
(Markov) Feller process for its natural completed filtration ]]N-' , and if X has continuous paths, then the filtration JFIL has no totally inaccessible stopping times. (This implies that the natural filtration of Brownian motion does not have any totally inaccessible stopping times.)
*Exercise 31. Let (0, F,
lF, P) be the standard Brownian space. Show that
the optional a-algebra and the predictable a-algebra coincide. Meyer's Theorem (Theorem 4) and Exercise 30.)
(Hint:
Use
P) be a filtered probability space satisfying the usual hypotheses. Let X be a (not necessarily adapted) cadlag stochas *Exercise 32. Let (0, F, JF,
tic process such that for A > 0, E{f;' e - .At iXtldt} < oo. Let E{f000 e - >.s Xt+ sdsiFt}, the right continuous version. Show that
is an
lF martingale.
R.A(Xt)
*Exercise 33 (Knight's compensator calculation method) . Let
X be
a cadlag semimartingale. In the framework of Exercise 32 suppose the limits below exist both pathwise a.s. and are in L 1 , and are of finite variation in finite time intervals:
Show that X is a special semimartingale, and A is the predictable term in its semimartingale decomposition.
IV General Stochastic Integration and Local Times
1 Introduction
We defined a semimartingale as a "good integrator" in Chap. II, and this led naturally to defining the stochastic integral as a limit of sums. To express an integral as a limit of sums requires some path smoothness of the integrands and we limited our attention to processes in lL, the space of adapted processes with paths that are left continuous and have right limits. The space lL is sufficient to prove Ito's formula, the Girsanov-Meyer Theorem, and it also suffices in some applications such as stochastic differential equations. But other uses, such as martingale representation theory or local times, require a larger space of integrands. In this chapter we define stochastic integration for predictable processes. Our extension from Chap. II is very roughly analogous to how the Lebesgue integral extends the Riemann integral. We first define stochastic integration for bounded , predictable processes and a subclass of semimartingales known as H.2 . We then extend the definition to arbitrary semimartingales and to locally bounded predictable integrands. We also treat the issue of when a stochastic integral with respect to a martingale or a local martingale is still a local martingale, which is not always the case. In this respect we treat the subject of sigma martingales, which has recently been shown to be important for the theory of mathematical finance. 2 Stochastic Integrat ion for Predictable Integrands
In this section, we will weaken the restriction that an integrand H must be in JL. We will show our definition of stochastic integrals can be extended to a class of predictably measurable integrands.
Throughout this section X will denote a semimartingale such that X0 = 0. This is a convenience involving no loss of generality. If Y is any semimartin gale we can set Yt = yt Y0, and if we have defined stochastic integrals for -
154
IV General Stochastic Integration and Local Times
semimartingales that are zero at 0, we can next define
1t HsdYs 1t Hsd� + Ho Yo . When Yo =f. 0, recall that we write J�+ HsdYs to denote integration on (0, t], and J� H8dY8 denotes integration on the closed interval [0, t]. We recall for convenience the definition of the predictable a-algebra, al =
ready defined in Chap. III.
p on JR.+
n is the smallest a E lL }. algebra making all processes in lL measurable. That is, P = a{ We let bP denote bounded processes that are P measurable.
Definition. The predictable a-algebra
X
H:H
X = M + A be a decomposition of a semimartingale X, with X0 = Mo = A0 = 0. Here M is a local martingale and A is an FV process (such Let
(
a decomposition exists by the Bichteler-Dellacherie Theorem Theorem 43 of Chap. III)) . We will first consider special semimartingales. Recall that a semimartingale X is called special if it has a decomposition
A
where N is a local martingale and is a predictable FV process. This decom position is unique by Theorem 30 in Chap. III and it is called the canonical
decomposition.
Definition. Let X be a special semimartingale with canonical decomposition The 7t2 norm of X is defined to be X=N
+ A.
The space of semimartingales rf2 consists of all special semimartingales with finite 1{2 norm. In Chap. V we define an equivalent norm which we denote
Theorem 1 .
Proof.
I · IIH2 .
The space of 7t2 semimartingales is a Banach space.
The space is clearly a normed linear space and it is easy to check 2 that · 1i'H2 is a norm recall that E{N 00 } = E{[N, N] 00 }, and therefore II X IIH2 = 0 implies that E{N!, } = 0 which implies, since N is a martingale, that N = 0). To show completeness we treat the terms N and separately. Consider first N. Since E{N!, } = II [N, N] � 2 I I i2 , it suffices to show that the space of L2 martingales is complete. However an L 2 martingale M can be identified with Moo E L 2 , and thus the space is complete since L 2 is complete. Next suppose is a Cauchy sequence of predictable FV processes in p > 1 . To show converges it suffices II · where II A II v = II f000
I
(
A
l2
(An ) dA , l sii i LP
(An )
2 Stochastic Integration for Predictable Integrands
155
to show a subsequence converges. Therefore without loss of generality we can assume L n II An ll 2 < oo . Then L A n converges in · 11 1 to a limit A. Moreover
I
oo
r I dA� I = 0 L m-+ oo n >m Jo in L 1 and is dominated in L2 by Ln f000 IdA� I · Therefore L An converges to the limit A in II · ll 2 as well, and there is a subsequence converging almost surely. To see that the limit A is predictable, note that since each term in the sequence (An ) n >l is predictable, the limit A is the limit of predictably lim
D
measurable processes and hence also predictable.
For convenience we recall here the definition of JL.
Definition. lL (resp. blL) denotes the space of adapted processes with caglad 1 (resp. bounded, caglad) paths.
We first establish a useful technical lemma.
A be a predictable FV process, and let H be in lL such that E{f000 I Hs l l dAs l } < oo. Then the FV process (J� H8dA8)t>O is also pre Lemma. Let
dictable.
Proof.
We need only to write the integral J� H8dA8 as the limit of Riemann sums, each one of which is predictable, and which converge in ucp to J� H8 dA8 , showing that it too is predictable. D The results that follow will enable us to extend the class of stochastic integrands from blL to bP, with X E rt.2 (and Xo = 0). First we observe that if H E blL and X E H.2 , then the stochastic integral H · X E 7t 2 . Also if X = N + A is the canonical decomposition of X, then H N + H · A is the canonical decomposition of H · X by the preceding lemma. Moreover, ·
The key idea in extending our integral is to notice that [N, N] and A are FV processes, and therefore w-by-w the integrals J� H;(w)d[N, N]8(w) and J� IHs l l dAs l make sense for any H E bP and not just H E lL.
Definition. Let X E 7t 2 with X = N + A its canonical decomposition, and let H, J E bP. We define dx(H, J) by
1
"caglad" is the French acronym for left continuous with right limits.
156
IV General Stochastic Integration and Local Times
Theorem 2.
For X E H2 the space blL is dense in bP under dx (·, ·).
Proof. We use the Monotone Class Theorem. Define A = {H E bP : for any c > 0, there exists J E blL such that dx(H, J) < c}. Trivially A contains blL. If Hn E A and Hn increases to H with H bounded, then H E bP, and by the Dominated Convergence Theorem if J > 0 then for some N(J), n > N(J) implies dx(H, Hn ) < J. Since each Hn E A, we choose no > N ( J) and there exists J E blL such that dx ( J, Hno ) < J. Therefore given c > 0, by taking J = c/2 we can find J E blL such that dx(J, H) < c, and therefore H E A. An application of the Monotone Class Theorem yields the D
result.
Let X E H2 and Hn E blL such that Hn is Cauchy under dx . Then Hn X is Cauchy in 1{2. Proof. Since IIHn . x - Hm · X II H2 = dx(Hn , Hm ), the theorem is immediate. Theorem 3. ·
D
Let X E 1{2 and H E bP. Suppose Hn E blL and Jm E blL are two sequences such that limn dx ( Hn , H ) = limm dx(Jm , H) = 0. Then Hn · X and Jm · X tend to the same limit in 1{2. Proof Let Y = lim n-+oo Hn · X and Z = limm-+oo Jm · X, where the limits are taken in H2. For r:; > 0, by taking n and large enough we have II Y - Z IIH2 < II Y - H n · XIIH2 + II Hn · X - Jm · X ll w + IIJm X - ZIIH2 < 2c + IIHn · X - Jm · X II H2 < 2r:: + dx(Hn , Jm ) < 2r:: + dx(Hn , H) + dx(H, Jm ) < 4r::, Theorem 4.
m
0
and the result follows. We are now in a position to define the stochastic integral for H
X E H2) .
D
E bP (and
and let H E bP. Let Hn E blL Definition. Let X be a semimartingale in be such that limn -+ oo dx ( Hn , H) = 0. The stochastic integral H · X is the
(unique) semimartingale Y E write H o X = (J� H8dX8 ) t>O ·
H2
H2 such that limn-+oo Hn · X
=
Y in
H2. We
We have defined our stochastic integral for predictable integrands and semimartingales in as limits of our (previously defined) stochastic integrals. In order to investigate the properties of this more general integral, we need to have approximations converging uniformly. The next theorem and its corollary give us this.
1{2
2 Stochastic Integration for Predictable lntegrands
Theorem 5.
157
Let X be a semimartingale in H 2 • Then
Proof
For a process H, let H* = supt IHt l· Let X = N + A be the canonical decomposition of X. Then
Doob's maximal quadratic inequality (Theorem 20 of Chap. I) yields
and using
(a + b) 2 < 2a2 + 2b2 we have E{(X *) 2}
l < 8E{[N , N] oo } + 2 ii fooo jdAs l ll i2 < 2E{ (N * ) 2 } + 2E{ ( oo ldAs l ) 2 } < 8 II X II�2 ·
0
Corollary. Let ( xn ) be a sequence of semimartingales converging to X in H2. Then there exists a subsequence ( ) such that limnk -> (Xnk - X)* = 0
nk
a.s.
oo
Proof
By Theorem 5 we know that (X n - X)* = supt IX;' - Xt l converges 0 to 0 in L 2 • Therefore there exists a subsequence converging a.s. We next investigate some of the properties of this generalized stochastic integral. Almost all of the properties established in Chap. II (Sect. 5) still
hold. 2
Theorem 6.
Let X, Y E H 2 and H, K E bP. Then (H + K) · X = H · X + K · X,
and
H · (X + Y) = H · X + H
·
Y.
One need only check that it is possible to take a sequence Hn E blL that approximates H in both dx and dy. 0
Proof
Theorem 7. H · (XT ).
Let T be a stopping time. Then (H · Xf = Hl (o, TJ · X =
2 Indeed, it is an open question whether or not Theorem
integrands in
bP.
16 of Chap. II extends to
See the discussion at the end of this section.
158 Proof.
IV General Stochastic Integration and Local Times
Note that l [o , T] E b!L, so H l ro ,T] E bP. Also, X T is clearly still in H2 . Since we know this result is true for H E b!L (Theorem 12 of Chap. II) , the result follows by uniform approximation, using the corollary of Theorem 5. 0
Theorem 8.
The jump process (tl(H · X)s)s > O is indistinguishable from
(Hs(LlXs))s> O · Proof. Recall that for a process J , Ll Jt = Jt - Jt - , the jump of J at time t. (Note that H · X and X are dtdUtg semimartingales, so Theorem 8 makes sense.) By Theorem 13 of Chap. II we know the result is true for H E b!L. Let H E bP, and let Hn E b!L such that limn --->oo dx (Hn , H) = 0. By the
corollary of Theorem 5, there exists a subsequence (
limoo ( Hnk · X - H · X)* nk -+ This implies that, considered as processes,
=
nk ) such that
0 a.s.
lim D.(Hnk · X) = D.(H · X), nk -+OO outside of an evanescent set.3 Since each Hn k E b!L, we have tl(Hnk · X ) = Hnk (tlX), outside of another evanescent set . Combining these, we have limoo Hnk (t::J. X )l { b. X �O } = lim b. (Hnk · X)l { b. X �O} nk -+OO nk -+ = D.(H . X)l { b.X�O } , and therefore
In particular, the above implies that limnk ---> oo Hf' k (w) exists for all (t, w) in { tlX =1- 0}, a.s. We next form
k (w) =f-Ht(w) and LlXt(w) =f-0}. A = {w : there exists t > O such that lim Hf' -+00 Suppose
nk
P(A) > 0. Then
00 { (LlNu ) 2 )}1!£2 dx (Hnk , H) > lil A { Jo (H'; k - Hs) 2 d( O< u <s 00 + I I 1 A { 0 I H'; k - Hs l d( L I!::J. Aul )}ll£2 , O< u <s and if LlXs =1- 0, then jtlNs l + jtlA l > 0. The left side of (*) tends to 0 and the right side of ( * ) does not. Therefore P(A) = 0, and we conclude tl(H · X) = limn k ---> oo Hn k t::J.X = HtlX. 0
L
1
as
nk
s
----> oo ,
3 A set
A C
JR+ x
the zero process.
n
is evanescent if
lA
is a process that is indistinguishable from
2 Stochastic Integration for Predictable lntegrands
159
X E rt2 , H E bP, and T a finite stopping time. Then H (xT- ) = (H · Proof By Theorem 8, (H · X f - = (H · X) T - Hr fl.Xrl{t> T} · On the other hand, x r - = X T - fl.Xrl{t> T } · Let At = fl.Xr l {t> T }· By the bilinearity (Theorem 6), H · (x T- ) = H · (XT ) - H · A. Since H · (XT ) = (H · X) T by Theorem 7, and H · A = Hr t:. Xr l { t> T } the result follows. D Corollary. Let
xf-.
·
•
The next three theorems all involve the same simple proofs. The result is known to be true for processes in blL; let (H n ) E blL approximate H E bP in dx ( · , ·), and by the corollary of Theorem 5 let n k be a subsequence such that nk lim (H nk -+00
· X H · X) * = O -
a.s.
Then use the uniform convergence to obtain the desired result. We state these theorems, therefore, without proofs.
H2
Theorem 9. Let X E have paths of finite variation on compacts, and H E bP. Then H · X agrees with a path-by-path Lebesgue-Stieltjes integral.
Theorem 10 (Associativity) .
H2 and H · (K
Let X E H2 and H, K E bP . Then K · X E
X) = ( HK) · X. Theorem 11. Let X E be a (square integrable) martingale, and H E bP. Then H X is a square integrable martingale. Theorem 12. Let X, Y E and H, K E bP. Then ·
H2
·
H2
and in particular [H · X , H · X]t =
Proof
t
1 H;d [X, X] s,
(t > 0 ).
As in the proof of Theorem 29 of Chap. II, it suffices to show
t
[H · X, Y]t = lo Hsd[X, Y]s · Let (Hn ) E blL such that limn -+oo dx(Hn , H ) = 0. Let rm = inf{t > 0 : IY't l > m}. Then (rm) are stopping times increasing to oo a.s. and j YT I < m. 4 Since it suffices to show the result holds on [0, Tm ) , each m, we can assume without loss of generality that Y_ is in blL. Moreover, the Dominated Convergence Theorem gives limn dx (Hn y_ , HY_ ) = 0. By Theorem 29 of Chap. II, we have "'
..... oo
4
Recall that
y_
denotes the left continuous version of Y .
160
IV General Stochastic Integration and Local Times [Hn
· X, Y]t =
lo H-;d[X, Y]s t
(t > 0),
and again by dominated convergence
t
· X, Y] = { Hsd[X, Y]s (t > 0). lo It remains only to show limn _,oo [Hn · X, Y] = [H · X, Y] . Let zn Hn · X , be a subsequence such that limnk _, 00 ( zn k - Z)* = 0 a.s. , where and let Z = H · X (by the corollary to Theorem 5). Integration by parts yields [znk Y] = znk y (y_ ) znk ( znk ) . y = znk y - (Y_ Hn k ) X - (Znk ) . Y, limoo [Hn n -+
=
nk
,
_
.
_
.
where we have used associativity (Theorem 10). We take limits so that lim [znk , Y] nk -+OO
= ZY - y_ · (H · X) - z_ · Y = ZY - Y_ · (Z) - Z_ · Y = [Z, Y] = [H · X, Y].
0
At this point the reader may wonder how to calculate in practice a canon ical decomposition of a semimartingale X in order to verify that X E 7t2 . Fortunately Theorem 13 will show that 7t2 is merely a mathematical conve nience. Lemma. Let be an FV process with Ao = 0 and f000 ldAs l E L 2 . Then A E 7t 2 . Moreover IIAI!?-£2 < 6 11 f000 ldAs l ll £2 .
A
Proof
If we can prove the result for A increasing then the general result will follow by decomposing A = A+ - A- . Therefore we assume without loss of generality that A is increasing. Hence as we noted in Sect. 5 of Chap. III, the compensator A of A is also increasing and E{Aoo} = E{Aoo} < oo. Let M be a martingale bounded by a constant k. Since A - A is a local martingale, Corollary 2 to Theorem 27 of Chap. II shows that -
-
L = M(A - A) - [M, A - A] is a local martingale. Moreover sup ILt l
t
< k(Aoo + Aoo) + 2k L I�(A - A)sl s < 3k(A oo + Aoo) E L . -
1
Therefore L is a uniformly integrable martingale (Theorem 51 of Chap. I) and E{Loo} = E{Lo} = 0. Hence
2 Stochastic Integration for Predictable Integrands
161
E{Moo (A - A)oo} = E{[M, A - A]oo} = E{ [M, A]oo} - E{ [M, A]oo} = E{ [M, A] 00 }, because A is natural. By the K unit a-Watanabe inequality (the corollary to Theorem 25 of Chap. II)
E{l [M, A] ool} < (E{[M, M]oo} E{[A, A]oo} ) 1 12 < 21 E{[M, M]oo} + 21 E{[A, A]oo},
where the second inequality uses 2ab < a 2 + b2 . However
E{[M, M]oo} = E{M!} (Corollary 4 of Theorem 27 of Chap. II) and also [A, A] oo < A� a.s. Therefore Since
M is an arbitrary bounded martingale we are free to choose
and we obtain
and using the Monotone Convergence Theorem we conclude
E{(A - A)�} < E{A�,}. Consequently
E{ A� } < 2E{A � } + 2E{(A - A)�} < 4E{A� } < oo, A - A is a square integrable martingale, and IIA117-t 2 = II (A - A)ooll £2 + II Aoo ll £2 < 3IIAooll £2 for A increasing. 0 Remarks. The constant 6 can be improved to 1+VS < 4 by not decomposing A into A + and A-. This lemma can also be proved using the Burkholder Gundy inequalities (see Meyer [171 , page 347]). and
�
�
�
162
IV General Stochastic Integration and Local Times
In Chap. V we use an alternative norm for semimartingales which we denote II · II HP , 1 < p < oo . The preceding lemma shows that the norms II · 117-£2 and I I · II H2 are equivalent. The restrictions of integrands to bP and semimartingales to 1t 2 are mathematically convenient but not necessary. A standard method of re laxing such hypothesis is to consider cases where they hold locally. Recall from Sect. 6 of Chap. I that a property is said to hold locally for a process X if there exists a sequence of stopping times (Tn ) n >O such that 0 = T0 < T 1 < T2 < · · · < Tn < · · and limn oo Tn = oo a.s. , and such that xr n 1 { T >O } has property for each n. Since we are assuming our semi n martingales X satisfy Xo = 0, we could as well require only that xrn has property for each n. A related condition is that a property hold prelocally. 1r
_,
·
1f
1r
Definition. A property is said to hold prelocally for a process X with Xo = 0 if there exists a sequence of stopping times (Tn ) n > l increasing to oo a.s. such that xrn _ has property for each n > 1 . Recall that x r- = Xt 1 { o oo Tn = oo a.s., such that xrn - E H 2 for each n.
Proof
Recall that xrn - = Xt 1 [o,T ) + Xrn _ 1 [T ,oo) · By the Bichteler Dellacherie Theorem (Theorem 43 of nChap. III) we ncan write X = M + A , where M is a local martingale and A is an FV process. By Theorem 25 of Chap. III we can further take M to have bounded jumps. Let f3 be the bound for the jumps of M. We define
Tn = inf{t > 0 : [M, M]t > n
t
lo JdAs l > n}
or
and let Y = xrn- Then Y has bounded jumps and hence it is a special semimartingale (Theorem 31 of Chap. III). Moreover .
or
Y = L + C,
where L = MTn and C = A Tn - - (�MT )1 [T ,oo ) · Then [L, L] < n + (32 , so n n L is a martingale in H 2 (Corollary 4 to Theorem 27 of Chap. II) , and also
hence C E H 2 by the lemma. Therefore xrn -
=
L+C
E
H2 .
0
2 Stochastic Integration for Predictable lntegrands
163
We are now in a position to define the stochastic integral for an arbi trary semimartingale, as well as for predictable processes which need not be bounded. Let X be a semimartingale in 1t2 . To define a stochastic integral for pre dictable processes H which are not necessarily bounded ( written H E P), we approximate them with Hn E b P . Definition. Let X E H2 with canonical decomposition X = N + A. We say H E P is (1t 2 , X) integrable if
Theorem 14. Let X be a semimartingale and let H E P be (H 2 , X) inte grable. Let Hn = H1 { 1HI O } ) t>o is bounded. Note that any process in L is locally bounded. The next example is suffi ciently important that we state it as a theorem.
Let X be a semimartingale and let H E P be locally bounded. Then H E L(X). That is, the stochastic integral H · X exists. Proof Let ( sm ) m > l , (Tn ) n > be two sequences of stopping times, each in and creasing to oo a.s., such that H8"' 1 { S"'>O } is bounded fo r each xrn _ E H2 for each n. Define Rn = min (sn , Tn ) nThen H = HRn 1 { Rn> O } on (O, Rn) and hence it is bounded there. Since X R - charges only (O, Rn), we have that H is (H 2 , X Rn - ) integrable for each n > 1 . Therefore using the Theorem 15.
1
m,
.
sequence Rn which increases to
oo
a.s. , we are done.
0
We now turn our attention to the properties of this more general integral. Many of the properties are simple extensions of earlier theorems and we omit their proofs. Note that trivially the stochastic integral H · X, for H E L(X), is also a semimartingale. Theorem 16. Let X be a semimartingale and let H, J E L(X). Then o:H + (3J E L(X) and (o:H + (3J) · X = o:H · X + (3J · X. That is, L(X) is a linear
space. Proof Let (Rm) and (Tn ) be sequences of stopping times such that H is (H 2 , X R"' - ) integrable, each and J is (H2 , xrn - ) integrable,n each n. Tak ing sn = Rn 1\ Tn , it is easy to check that o:H + (3J is (1t2 , X s - ) integrable m,
for each
0
n.
Let X, Y be semimartingales and suppose H E L(X) and H E L(Y). Then H E L(X + Y) and H · (X + Y) = H · X + H Y.
Theorem 17.
·
2 Stochastic Integration for Predictable lntegrands
165
Let X be a semimartingale and H E L(X). The jump process (b.(H X)s) s > O is indistinguishable from (Hs(b.Xs )) s > O · Theorem 19. Let T be a stopping time, X a semimartingale, and H E L(X). Then (H X f = Hl[o, T] · X = H · (X T ). Moreover, letting oo - equal oo , we have moreover
Theorem 18. ·
·
. Let X be a semimartingale with paths of finite variation on compacts. Let H E L(X) be such that the Stieltjes integral J� I Hs lldX8 I exists a.s., each t > 0. Then the stochastic integral H · X agrees with a path-by-path Stieltjes integral. Theorem 21 (Associativity) . Let X be a semimartingale with K E L(X). Then H E L(K · X) if and only if HK E L(X), in which case H · (K · X) = (HK) · X. Theorem 22. Let X, Y be semimartingales and let H E L(X), K E L(Y). Then [H · X, K · Y]t = lot H.Ksd[X, Y]s (t > 0). Note that in Theorem 22 since H · X and H · Y are semimartingales, the
Theorem 20 .
quadratic covariation exists and the content of the theorem is the formula. Indeed, Theorem 22 gives a necessary condition for H to be in L(X), namely that J� H;d[X, X]s exists and is finite for all t > 0. The next theorem (The orem 23) is a special case of Theorem 25, but we include it because of the simplicity of its proof. Theorem 23. Let X be a semimartingale, let H E L(X), and suppose Q is another probability with Q � P. If HQ · X exists, it is Q indistinguishable
from Hp · X. Proof HQ · X denotes the stochastic integral computed under Q. By Theo rem 14 of Chap. II, we know that HQ · X = Hp · X for H E L, and therefore if X E H2 for both P and Q, they are equal for H E bP by the corollary of Theorem 5 . Let (Rf)£> 1 , (Tn ) n > l be two sequences of stopping times in creasing to oo a.s. such that H is (H2 , X Rt - ) integrable under Q, and H is n (H 2 , x r - ) integrable under P, each £ and n . Let sm = Rm 1\ Tm , so that H is (H2, x s = - ) integrable under both P and Q. Then H · X = limn _,oo Hn · X on [0 , sm ) in both dx(P) and dx(Q), where Hn = H l { I I n } E bP. Since D H? · X = H'Q · X, each n, the limits are also equal. H
0. Note that E{I [N, A]ool} < (E{[N, N]oo}) 112 (E{ [A, A]oo}) 1 12 < oo, by the Kunita-Watanabe inequalities. Also E{ [M, A]oo} = 0 for all bounded martingales because A is natural. Since bounded martingales are dense in the space of L 2 martingales, there exists a sequence (Mn ) n > l of bounded martingales such that limn _, 00 E{[Mn - N , M n - N] oo } = 0. Again using the Kunita-Watanabe inequalities we have
E{I [N - M n , A] ool} < (E{[N - Mn , N - Mn]oo}) 112 (E{[A, A]oo}) 112 and therefore limn -> oo E{[Mn , A]oo} = E{[N, A] 00 }. Since E{[Mn , A]oo } = 0, each n, it follows that E{[N, A]oo} = 0. 0 Note that in the preceding proof we established the useful equality E{[X, X] t} = E{[N, N]t} + E{[A, A] t} for a semimartingale X E 1t2 with canonical decomposition X = N + A. Theorem 24. For X a semimartingale in H 2 ,
II X I !?- t2 Proof
(H · X) :"x, ll £2 + 2II [X, X] � 2 II £2 < 5IIXI!?- t2· II HJ < l
< sup
I
By Theorem 5 for
!H I < 1
Since
oo
2 II [M , M] � 2 II £2 + 211 fo ldAs l ll £2 = 2IIXII 7-t 2 , where X = M + A is the canonical decomposition of X, we have the right inequality. For the left inequality we have II [M, M] � 2 II £2 < II [X, X] � 2 II £2 by the lemma preceding this theorem. Moreover if IHI < 1, then 2II [X, X] �2 II £2
0 and let 0 = To < T1 < · · · < Tn = t be a random partition of [0, t] . Choose c: 1 1 , En non-random and each equal to 1 or - 1 and let H = 2::� Ei 1 (T,_, ,Ti ] . Then H is a simple predictable process and •
•
•
1
n (H · X)oo = L Ei (Xr, - Xr,_, ) . i= l Let o: = supi H I < I II (H · X ) � ll£ 2 · We then have 5
However the corollary does give some insight into the relationship between The orems 12 and 14 in Chap. V.
IV General Stochastic Integration and Local Times
168
n o: 2 > E{( H · X) �} = I: Ei E1 E{(Xr, - Xr,_1 )(Xrj - Xrj _ 1 ) } . i ,j=l
If we next average over all sequences c- 1 , . . { ±1 } n , we deduce
.
, En
taking values in the space
n n o:2 > I: E{ (Xr, - Xr,_1 )2 } = E{ I: (Xr, - Xr,_ 1 )2 } . i=l
i=l
Next let crm = {Tim } be a sequence of random partitions of [O, t] tending to the identity. Then Li (Xrt• - Xr;:'-. 1 )2 converges in probability to [X, X]t . Let { mk } be a subsequence so that L i (Xr,"'k - Xr;."_1 )2 converges to [X, X]t a.s. Finally by Fatou's Lemma we have
)2 } < o: 2 . "'k - Xr.1:nk E{""'(X E{[X, X] t} < limk ---7inf 1 r . 1 L...t 00 m i Letting t tend to
oo
we conclude that
E{[X, X]oo} < o:2 =
sup II (H · X) � lli2 . I HI < l
It remains to show that if sup i H I < I I I (H · X)� ll£2 < oo , then X E 1f2 . We will show the contrapositive. If X f/- 1i2 , then sup i H I < I II (H · X)� ll£2 = oo . Indeed, let Tn be stopping times increasing to oo such that x rn - is in 1i 2 for each (cf. , Theorem 13). Then
n
Letting
II XTn - IIH2 < 3
n tend to
oo
sup I I ( H · X Tn -) � 11£2 IHI < l
< 9II XTn - IIH2 ·
gives the result.
0
Before proving Theorem 25 we need two technical lemmas.
A be a non-negative increasing FV process and let Z be a positive uniformly integrable martingale. Let T be a stopping time such that A = AT- (that is, A00 = Ar- ) and let k be a constant such that Z < k on [0, T) . Then Lemma 1. Let
Proof.
Since
Ao - = Zo - = 0, by integration by parts
2 Stochastic Integration for Predictable Integrands
169
where the second integral in the preceding is a path-by-path Stieltjes integral. Let Rn be stopping times increasing to oo a.s. that reduce the local martingale (J� A8_dZs)t >o · Since dAs charges only [0, T) we have
t
t
1 Z8dA8 < 1 kdAs < kA00 for every
t > 0. Therefore E{(AZ) Rn } = E{
1
Rn
A8_dZs} + E{
1
Rn
Z8dAs}
< 0 + E{kAoo } · The result follows by Fatou's Lemma.
0
X be a semimartingale with Xo = 0, let Q be another prob ability with Q � P, and let Zt = Ep {� I .Ft}· If T is a stopping time such that Zt < k on [0, T) for a constant k, then Lemma 2. L et
Proof By Theorem 24 we have II Xr - ll ?t2(Ql < sup EQ {((H · x r -) * )2} 1 + 2EQ{[xr - , x r - ] 4 } IHI < l
n} and set Rn = sn 1\ rn. Then X Rn - E 1i2(P) n 1i2 ( Q ) by Lemma 2, and H is (1i2, X Rn -) integrable under P. We need to show H is (1i2 , X Rn - ) integrable under Q, which will in turn imply that H E L(X) under Q. Let X Rn - = N + C be the canonical decomposition under Q. Let Hm = H1 { IHI<m } · Then Theorem 25.
/
J I H';'I / dCsi i P (Q)
(Eq{ (Hr;' ) 2 d [N, N]s }) 112 + 1/ = IIHm . x R " - II?t2(Q) < 5 vfnii Hm . x Rn - ll?t 2( P) < 5y niiH · X Rn - �� ?t2 (P) < oo, and then by monotone convergence we see that H is (1i2, X Rn - ) integrable under Q. Thus H E L(X) under Q, and it follows that Hq · X = Hp . X, r::;
0
Q-a.s.
Theorem 25 can be used to extend Theorem 20 in a way analogous to the extension of Theorem 1 7 by Theorem 18 in Chap. II.
Let X, X be two semimartingales, and let H E L(X), H E L(X). Let A = {w : H. (w) = H. (w) and X (w) = X. (w)}, and let B = {w : t 1--4 Xt (w) is of finite variation on compacts}. Then H · X = H · X on A, and H · X is equal to a path-by-path Lebesgue-Stieltjes integral on B . Proof Without loss of generality assume P(A) > 0. Define Q by Q(A) = P(AIA). Then Q � P and therefore H E L(X), H E L(X) under Q as well as under P by Theorem 25. However under Q the processes H and H as well as X and X are indistinguishable. Thus Hq ·X = Hq X and hence E-X = H · X P-a.s. on A by Theorem 25, since Q � P. Theorem 26.
.
·
The second assertion has an analogous proof (see the proof of Theorem 1 8 of Chap. II) . o
Note that one can use stopping times to localize the result of Theorem 26. The proof of the following corollary is analogous to the proof of the corollary of Theorem 1 8 of Chap. II.
Corollary. With the notation of Theorem 26, let S, T be two stopping times
S < T. Define C = {w : Ht(w) = Ht (w); Xt (w) = Xt (w); S(w) < t < T(w)}, D = {w : t 1--4 Xt(w) is of finite variation on S(w) < t < T(w)}. T Then H x r - H · X8 = H · X - H · X8 on C and H · xr - H · X8 equals a path-by-path Lebesgue-Stieltjes integral on D. with
·
2 Stochastic Integration for Predictable Integrands
171
Let Pk be a sequence of probabilities such that X is a Pk semimartingale for each k. Let R = I:;;' 1 >.. k Pk where A k > 0, each k, and I:%" Ak = 1 . Let H E L(X) under R. Then H E L(X) under Pk and HR · X = Hp• · X, Pk -a.s., for all k such that Ak > 0. Proof. If Ak > 0 then Pk « R. Moreover since Pk (A) < }. R(A), it follows that H E L(X) under Pk . The result then follows by Theorem 25. 0 Theorem 27. 1
We now turn to the relationship of stochastic integration to martingales and local martingales. In Theorem 11 we saw that if M is a square integrable martingale and H E bP , then H M is also a square integrable martingale. When M is locally square integrable we have a simple sufficient condition for H to be in L(M). ·
M be a square integrable martingale and let H E P be such 00 that E{f0 H'1d[M, M]s } < oo. Then H · M is a square integrable martingale. Proof. If H k E bP , then H k M is a square integrable martingale by The orem 11. Taking H k = H1{1 H I 1 increasing to oo a.s. such that E{f0Tn H';d[M, M]s } < oo.
Proof. We assume that M is a square integrable martingale stopped at the time Tn . The result follows by applying the lemma. o Theorem 29. Let M be a local martingale, and let H E P be locally bounded. Then the stochastic integral H · M is a local martingale. Proof. By stopping we may, as in the proof of Theorem 29 of Chap. III, assume that H is bounded, M is uniformly integrable, and that M = N + A where N is a bounded martingale and A is of integrable variation. We know that • k R there exists R increasing to oo a.s. such that M - E 1i 2 , and since H E b P • • there exist processes Hl E blL such that IIH£ MR - - H M R - II1t 2 tends • • to zero. In particular, Hl MR - te nds to H MR - in ucp. Therefore we • can take Hl such that Hl · MR - tends to H M in ucp, with Hl E blL . Finally without loss of generality we assume Hl · M converges to H · M i n ucp. Since Hl · M = Hl N + Hl A and Hl · N converges to H · N in ucp, we deduce Hl A converges to H · A in ucp as well. Let 0 < s < t and assume Y E b.F8• Therefore, since A is of integrable total variation, and since Hl · A is a martingale for Hl E blL ( see Theorem 29 of Chap. III), we have ·
·
·
·
·
·
·
·
172
IV General Stochastic Integration and Local Times
1
t t lim H�dAu} = E{Y lim E{Y HudAu} = E{Y l->oo t->oo s+ s+ = lim E{Y
1
l->oo
= 0,
t H�dAu} s+ t H�dAu} s+
1 j
where we have used Lebesgue's Dominated Convergence Theorem both for the Stieltjes integral w-by-w ( taking a subsequence if necessary to have a.s. convergence) and for the expectation. We conclude that (f� H8dAs)t>O is a martingale, hence H · M = H N + H · A is also a martingale under the assumptions made; therefore it is a local martingale. D ·
In the proof of Theorem 29 the hypothesis that H E P was locally bounded was used to imply that if M = N + A with N having locally bounded jumps and A an FV local martingale, then the two processes
are locally integrable. Thus one could weaken the hypothesis that H is locally bounded, but it would lead to an awkward statement. The general result, that
M a local martingale and H E L(M) implies that H · M is a local martingale, is not true!
See Emery's example, which precedes Theorem 34 for a stochastic integral with respect to an H 2 martingale which is not even a local martingale! Emery's counterexample has lead to the development of what are now known as sigma martingales, a class of processes which are not local martingales, but can be thought to be "morally local martingales." This is treated in Section 9 of this chapter.
Corollary. Let M be a local martingale,
Mo = 0, and let T be a predictable
MT - is a local martingale. Proof. The notation MT- means
stopping time. Then
Mr- = Mt 1 {t < T } + Mr _ 1 {t > T} > where M0 _ = M0 = 0. Let (Sn ) n > l be a sequence of stopping times announc ing T. On {T > 0}, lim =1 n---+cx:> 1 [0' sn ] [0 T) , and since 1 [o , s ] is a left continuous, adapted process it is predictable; hence n l [o , T) is predictable. But f� 1[o ,r) (s)dMs = Mt - by Theorem 18, and hence it is a local martingale by Theorem 29. 0 '
2 Stochastic Integration for Predictable Integrands
173
Note that if M is a local martingale and T is an arbitrary stopping time, it is not true in general that MT- is still a local martingale. Since a continuous local martingale is locally square integrable, the theory is particularly nice.
Let M be a continuous local martingale and let H E P be such that I; H;d[M, M]s < oo a.s., each t > 0. Then the stochastic integral H · M exists (i.e., H E L(M)) and it is a continuous local martingale. Proof Since M is continuous, we can take Rk = inf{t > 0 : I Mt l > k}. Then I Mti\ Rk I < k and therefore M is locally bounded, hence locally square integrable. Also M continuous implies [M, M] is continuous, whence if t yk = inf{t > 0 : lo H_;d[M, M ] s > k},
Theorem 30.
we see that u; H_;d[M, M]s)t >o is also locally bounded. Then H·M is a locally square integrable local martingale by Theorem 28. The stochastic integral H · M is continuous because !l(H·M) = H(!lM) and 11M = 0 by hypothesis. D
In the classical case where the continuous local martingale M equals B, a standard Brownian motion, Theorem 30 yields that if H E P and I H; ds < 00 a.s., each t > 0, then the stochastic integral (H . Bt)t >O = HsdBs)t>O exists, since [B, B]t = t.
u�
�
X be a continuous semimartingale with ( unique) decomposi tion X = M + A. Let H E P be such that t t H'1d[M, M]s + I Hs l ldAs l < 00 a.s. Corollary. Let
lo
lo
each t > 0. Then the stochastic integral continuous.
(H · X)t =
I� H8dXs exists and it is
Proof.
By the corollary of Theorem 31 of Chap. III, we know that M and A have continuous paths. The integral H · M exists by Theorem 30. Since H · A exists as a Stieltjes integral, it is easy to check that H E L(A) , since A is D continuous, and the result follows from Theorem 20. In the preceding corollary the semimartingale X is continuous, hence [X, X] = [M, M] and the hypothesis can be written equivalently as
t
t
lo H'1d[X, X]s + lo I Hsl l dAs l < 00
a.s.
each t > 0. We end our treatment of martingales with a special case that yields a particularly simple condition for H to be in L (M) .
174
IV General Stochastic Integration and Local Times
Let M be a local martingale with jumps bounded by a constant (3. Let H E P be such that I� H,?d[M, M]s < oo a.s., t > 0, and E{H:j.} < oo for any bounded stopping time T. Then the stochastic integral (J� H8dMs) t >O exists and it is a local martingale. Proof Let Rn = inf{t > 0 : I� H,?d[M, M]s > n }, and let rn = min(Rn , n). Theorem 31.
Then rn are bounded stopping times increasing to oo a.s. Note that
E{
fo
Tn
H,?d[M, M] s} < n + E{ Hfn (LlMr n )2} < n + !P E{Hf } < oo, n
and the result follows from Theorem 28.
0
The next theorem is, of course, an especially important theorem, the Dom
inated Convergence Theorem for stochastic integrals.
Theorem 32 (Dominated Convergence Theorem). Let X be a semi martingale, and let Hm E P be a sequence converging a.s. to a limit H. If there exists a process G E L(X) such that I H m l < G, all m, then Hm , H are
in L(X) and Hm · X converges to H · X in ucp. Proof. First note that if I J I < G with J E P, then J E L(X). Indeed, let (Tn ) n > l increase to oo a.s. such that G is (1i2, X Tn -) integrable for each n.
Then clearly
E{
1
00
1
00
J;d[N, N]s} + E{( I Js l ldAs l )2 } 00 00 < E{ a;d[N, N]s} + E{ ( I Gs l ldAs i )2} < oo ,
fo
fo
and thus J is (1i 2, x rn - ) integrable for each n. (Here N + A is the canonical decomposition of xrn - .) To show convergence in ucp, it suffices to show uniform convergence in probability on intervals of the form [0, to] for to fixed. Let E > 0 be given, and choose n such that P ( Tn < to) < c, where x rn - E 1i2 and G is (1i 2, x rn - ) integrable. Let x rn - = N + A, the canonical decomposition. Then E {sup ! H m . x rn
t < to
_ - H . x rn - 1 2}
The second term tends to zero by Lebesgue's Dominated Convergence Theo rem. Since IHm - HI < 2G, the integral (Hm - H) · N is a square integrable martingale (by the lemma preceding Theorem 28) . Therefore using Doob's maximal quadratic inequality, we have
2 Stochastic Integration for Predictable Integrands
E { sup I (Hm - H) · Nj2 }
t < to
175
< 4E{ j (Hm - H) · Nto l 2 }
and again this tends to zero by the Dominated Convergence Theorem. Since convergence in L2 implies convergence in probability, we conclude for c5 > 0, limsupP{ sup I Hm · Xt - H · Xt l > c5} t < to m -+<XJ < lim sup P{sup i ( (Hm - H) · xr n - )ti t c5} + P (Tn < to) 0
We use the Dominated Convergence Theorem to prove a seemingly in nocuous result . Generalizations, however, are delicate as we indicate following the proof.
Let lF = (Ft)t>O and G = Wt ) t >o be two filtrations satisfying the usual hypotheses and suppose Ft C 9t , each t > 0, and that X is a semimartingale for both IF and G. Let H be locally bounded and predictable for lF. Then the stochastic integrals Hw · X and HG · X both exist, and they are equal.6 Proof. It is trivial that H is locally bounded and predictable for (Qt)t >o as Theorem 33.
well. By stopping, we can assume without loss of generality that H is bounded. Let
1i = {all bounded, :F predictable H such that Hw · X = HG · X } . Then 1i is clearly a monotone vector space, and 1i contains the multiplicative class blL by Theorem 16 of Chap. II. Thus using Theorem 32 and the Monotone Class Theorem we are done.
0
It is surprising that the assumption that H be locally bounded is impor tant. Indeed, Jeulin [114, pages 46, 47] has exhibited an example which shows that Theorem 33 is false in general. Theorem 33 is not an exact generalization of Theorem 1 6 of Chap. II. Indeed, suppose IF and G are two arbitrary filtrations such that X is a semi martingale for both IF and G, and H is bounded and predictable for both of them. If It = Ft n 9t , then X is still an (It)t >o semimartingale by Stricker's Theorem, but it is not true in general that H is (It ) t >o predictable. It is an open question as to whether or not HF · X = Hg · X in this situation. For a partial result, see Zheng [248] . 6
HF · X and HG · X denote the stochastic integrals computed with the filtrations
(.Ft)t>o and Wt)t� o, respectively.
IV General Stochastic Integration and Local Times
176
Example (Emery's example of a stochastic integral behaving badly) .
The following simple example is due to M. Emery, and it has given rise to the study of sigma martingales, whose need in mathematical finance has become apparent. Let X = (Xt) t >o be a stochastic process given by the following description. Let T be an exponential random variable with parameter ,\ = 1 , let U be an independent random variable such that P { U = 1 } = P { U = 1 } = 1/2, and set X = U1 {t >T } . Then X together with its minimal filtration satisfying the usual hypotheses is a martingale in 1i2. That is, X is a stopped compound Poisson process with mean zero and is an L2 martingale. Let Ht = ! 1 {t>O} · Therefore H is a deterministic integrand, continuous on ( O, oo ) , and hence predictable. Consequently the path-by-path Lebesgue-Stieltj es integral Zt = J� H8 dX8 exists a.s. However H · X is not locally in JiP for any p > 1 . ( However since it is still a semimartingale7 , it is prelocally in 1i2.) Moreover , even though X is an L2 martingale and H is a predictable integrand, the stochastic integral H · X is not a local martingale because E{ I Zs l } = oo for every stopping time S such that P(S > 0) > 0. -
The next theorem is useful, since it allows one to work in the convenient space 1i2 through a change of measure. It is due originally to Bichteler and Dellacherie, and the proof here is due to Lenglart. We remark that the use of the exponent 2 is not important, and that the theorem is true for JiP for any p > l.
Let X be a semimartingale on a filtered complete probability space (n, :F, lF, P) satisfying the usual hypotheses. Then there exists a proba bility Q which is equivalent to P such that under Q, X is a semimartingale in 1i2. Moreover, �� can be taken to be bounded.
Theorem 34.
Before we begin the proof we establish a useful lemma.
(D, :F, P)
be a complete probability space and let Xn be a se quence of a.s. finite valued random variables. There exists another probability Q, equivalent to P and with a bounded density, such that every Xn is in L2( dQ ) .
Lemma. Let
Proof.
Assume without loss that each of the Xn is positive. For a single ran dom variable X we take Ak = { k < X < k + 1 } and Y = L k > I T k 1 Ak · Then Y is bounded and
EQ{X2} = E{X2Y} < 2:.::: ( k ;k1)2 < oo . k> l For the general case, choose constants O:n such that P(Xn > o:n ) < 2- n . The Borel-Cantelli Lemma implies that a.s. Xn < O:n for all n sufficiently large. Next choose constants Cn such that Ln > l Cn O:n < oo. Let Yn be the bounded 7
H X is still a semimartingale since it is of finite variation a.s. on compact time ·
sets.
2 Stochastic Integration for Predic table Integrands
177
density chosen for Xn individually as done in the first part of the proof, and take Y = Ln >l cnYn so that we have the result. D Proof of Theorem 34. Using the lemma we make a first change of measure making all of the random variables [X, X]n integrable. Recall that [X, X] is invariant under a change to an equivalent probability measure. By abusing notation, we will still denote this new measure by P. This implies that if Jt = supso, P) satisfying the usual hypotheses. We begin by considering only L 2 martingales. Later we indicate how to extend these results to locally square integrable local martingales. of L 2 martingales is all martingales M such that supt E { M[} < oo, and Mo = 0 a.s. Notice that if M E M2 , then lim t---. oo E{Ml } = E{M!} < oo , and Mt = E{Moo i :Ft } . Thus each M E M2 can be identified with its terminal value M00 • We can endow M2 with a norm
Definition. The space
M2
IIM II = E{ M�j 1/ 2 = E{[M, M] 00 } 1 1 2 ,
and also with an inner product
(M, N) = E{MooNoo}, for M, N E M2 . It is evident that M2 is a Hilbert space and that its dual space is also M2 . If E{M[} < oo for each t, we call M a square integrable martingale. If in addition supt E{M[ } = E{M!} < oo, then we call M an L 2 martingale.
The next definition is a key idea in the theory of martingale representation. It differs slightly from the customary definition because we are assuming all martingales are zero at time t = 0. If we did not have this hypothesis, we would have to add the condition that for any event A E :F0 , any martingale M in the subspace F, then Ml11 E F.
Definition. A closed subspace F of M2 is called a stable subspace if it is stable under stopping (that is, if M E F and if T is a stopping time, then MT E F).8 Theorem 35. Let F be a closed subspace of M2 . Then the following are
equivalent. (a) F is closed under the following operation. For M E F, (M - Mt )lA E F for A E :Ft, any t > 0. (b) F is a stable subspace. 8
Recall that M[ = Mti\T, and Mo = 0.
1 79 (c) If M E F and H is bounded, predictable, then (f� H8dMs)t> o = H · M E F. (d) If M E F and H is predictable with E{f000 H;d[M, M]s} < oo, then H · M E F. Proof. Property (d) implies (c) , and it is simple that (c) implies (b) . To get 3 Martingale Representation
(b) implies (a) , let T = t11 , where t11
=
{
t, oo ,
if w E A, if w fj_ A.
Then T = t 11 is a stopping time when A E :Ft, and (M - Mt )lA = M - MT ; since F is assumed stable, both M and MT are in F. It remains to show only that (a) implies (d) . Note that if H is simple predictable of the special form n
Ht = 1Aa 1 {o} + L 1 Ai 1(t, ,t,+tl i=l
with Ai E :Ft, , 0 < i < n, 0 = to < t 1 < · · · < tn+ I < oo, then H · M E F whenever M E F. Linear combinations of such processes are dense in blL which in turn is dense in bP under dM( ·, ·) by Theorem 2. But then bP is dense in the space of predictable processes such that E{f000 H;d[M, M]s} < oo, as is easily seen (cf., Theorem 14) . Therefore (a) implies (d) and the theorem is proved. D
1{
Since the arbitrary intersection of closed, stable subspaces is still closed and stable, we can make the following definition.
Definition. Let A be a subset of M 2 . The stable subspace generated by
A, denoted S(A), is the intersection of all closed, stable subspaces containing A. As already noted on page 178, we can identify a martingale M E M2 with its terminal value Moo E L2 . Therefore another martingale N E M 2 is (weakly) orthogonal to M if E{N00 M00 } = 0. There is however another, stronger notion of orthogonality for martingales in M2 .
Definition. Two martingales N, M E M2 are said to be strongly orthog onal if their product L = NM is a (uniformly integrable) martingale.
Note that if N, M E M 2 are strongly orthogonal, then NM being a (uniformly integrable) martingale implies that [N, M] is also a local martin gale by Corollary 2 of Theorem 27 of Chap. II. It is a uniformly integrable martingale by the Kunita-Watanabe inequality (Theorem 25 of Chap. II) . Thus M, N E M2 are strongly orthogonal if and only if [M, N] is a uni formly integrable martingale. If N and M are strongly orthogonal then E{N00 M00 } = E{Loo} = E{Lo} = 0, so strong orthogonality implies or thogonality. The converse is not true however. For example let M E M 2 , and
180
IV General Stochastic Integration and Local Times
let Y E :Fo, independent of M, with Nt = YMt , t > 0. Then N E M2 and
P(Y
=
1)
=
P(Y
= -1)
�- Let
E{Noo Moo} = E{YM�J = E{Y}E{M!} = 0, so M and N are orthogonal. However MN = YM2 is not a martingale (unless M = 0) because E{YM[ I:Fo} = YE{M[ I :Fo} =f- o = YM;?.
A of M2 we let A_!_
(resp. Ax ) denote the set of all elements of M 2 orthogonal (resp. strongly orthogonal) to each element of A.
Definition. For a subset
Lemma 1. If A is any subset of M2 , then Ax is (closed and) stable.
Proof Let Mn be a sequence of elements of Ax converging to M, and let N E A. Then Mn N is a martingale for each n and Ax will be shown to be closed if MN is also one, or equivalently that [M, N] is a martingale. However E{I [Mn , N] - [M, N] t l } = E{I [Mn - M, N] tl } < (E{[Mn - M, Mn - M] t } ) 11 2 (E{[N, N] t } ) 11 2 by the Kunita-Watanabe inequalities. It follows that [Mn , N] t converges to [M, N]t in L 1 , and therefore [M, N] is a martingale, and Ax is closed. Also Ax is stable because M E Ax , N E A implies [MT , N] = [M, Nj T is a martingale and thus MT is strongly orthogonal to N. 0 Lemma 2. Let N, M be in M2 . Then the following are equivalent. (a) M and N are strongly orthogonal. (b) S(M) and N are strongly orthogonal. (c) S(M) and S(N) are strongly orthogonal. (d) S(M) and N are weakly orthogonal. (e) S(M) and S(N) are weakly orthogonal.
Proof
If M and N are strongly orthogonal, let A = { N} and then M E Ax . Since Ax is a closed stable subspace by Lemma 1 , S(M) C { Ny . Therefore (b) holds and hence (a) implies (b) . The same argument yields that (b) implies (c) . That (c) implies (e) which implies (d) is obvious. It remains to show that (d) implies (a) . Suppose N is weakly orthogonal to S(M) . It suffices to show that [N, M] is a martingale. By Theorem 21 of Chap. I it suffices to show E{ [N, M] T } = 0 for any stopping T. However E{ [N, M] T } = E{ [N, MT ] oo} = 0, since N is orthogonal to MT which is in S(M ). 0
Let M 1 , . . . , Mn E M2 , and suppose Mi , MJ are strongly or thogonal for i =f- j. Then S(M1, . . . , Mn ) consists of the set of stochastic integrals n H 1 . M 1 + . . . + Hn . Mn = 2: Hi . Mi ' i= 1
Theorem 36.
3 Martingale Representation
181
where Hi is predictable and 1 < -
.
n
-
n.
Let I denote the space of processes L i=l H' · M' , where H' satisfy the hypotheses of the theorem. By Theorem 35 any closed, stable subspace must contain I. It is simple to check that I is stable, so we need to show only that I is closed. Let
Proof.
.
.
Then the mapping (H 1 , H2, . . . , Hn ) --> L� 1 Hi Mi is an isometry from Llr1 EB · · · EB Llr n into M2. Since it is a Hilbert space isometry its image I is complete, and therefore closed. D ·
Let A be a subset of M2 which is stable. Then AJ.. is a stable subspace, and if M E AJ.. then M is strongly orthogonal to A. That is, AJ.. = Ax , and S(A ) = AJ..J.. = Ax J.. = Ax x . Proof We first show that AJ.. = Ax . Let M E A and N E AJ... Since N is or Theorem 37.
thogonal to S(M) , by Lemma 2, N and M are strongly orthogonal. Therefore AJ.. C Ax . However clearly Ax C AJ.., whence Ax = AJ.., and thus AJ.. = Ax is a stable subspace by Lemma 1. By the above applied to AJ.. , we have that (AJ.. ) J.. = (AJ..)x . It remains to show that S( A) = AJ..J... Since AJ..J.. = A, the closure of A in M2, it suffices to show that A is stable. However it is simple to check that condition (a) of Theorem 35 is satisfied for A, since it already is satisfied for D A, and we conclude A is a stable subspace.
A be a stable subspace of M2 . Then each M E M2 has a unique decom position M = A + B, with A E A and B E Ax . Proof A is a closed subspace of M2 , so each M E M2 has a unique decom position into M = A + B with A E A and B E AJ... However AJ.. = Ax by Corollary 1. Let
Theorem
37.
D
Corollary 2. Let M, N E
M2 , and let L be the projection of N onto S(M) ,
the stable subspace generated by M. Then there exists a predictable process H such that L = H M. ·
We know that such an L exists by Corollary 1 . Since { M} consists of 0 just one element we can apply Theorem 36 to obtain the result.
Proof
182
IV General Stochastic Integration and Local Times
Definition. Let A be finite set of martingales in M2 . We say that A has the (predictable) representation property if I = M2 , where I
= {X : X = L Hi . M i ' Mi E A} ' n
i =l
each Hi predictable such that
Corollary 3. Let A = { M l , . . . , Mn }
M 2 , and suppose Mi , M1 are strongly orthogonal for i =f- j. Suppose further that if N E M2 , N _l A in the strong sense implies that N = 0. Then A has the predictable representation property. C
Proof. By Theorem 36 we have S(A) = I. The hypotheses imply that D S(A)J. = {0}, hence S(A) = M2 •
Stable subspaces and predictable representation can be considered from an alternative perspective. Up to this point we have assumed as given and fixed an underlying space (n, :F, (:Ft ) t >O, P) , and a set of martingales A in M2 . We will see that the property that S(A) = M2, intimately related to predictable representation (cf. , Theorem 36), is actually a property of the probability measure P, considered as one element among the collection of probability measures that make L2 (:Ft h>o martingales of all the elements of A. Our first observation is that since the filtration IF = (:Ft ) t >O is assumed to be P complete, it is reasonable to consider only probability measures that are absolutely continuous with respect to P. C
M2 . The set of M2 martingale measures for A, denoted M 2 (A), is the set of all probability measures Q defined on Vo< t< oo :Ft such that
Definition. Let A
(i) Q « P, (ii) Q = P on :Fo , (iii) if X E A then X E M2 (Q),
where M2 (Q) denotes all (Q, IF) L2 martingales.9
Lemma. The set M2 is a convex set. 9
N ote that while lF satisfies the usual hypotheses under P, :F0 does not contain all
of the Q-null sets, and thus lF does not completely satisfy the usual hypotheses under Q. This does not create any problems, however.
3 Martingale Representation
183
Proof Let Q and R E M 2 (A) , and let S = AQ + (1 - A)R, 0 < A < 1 . Then
for X E M 2 (A) , < (1 A)ER{M } = sup Eq {M oo , sup Es{M } } [A ( ( + ] ( t t since Q, R E M 2 (A) . Also if H E b.F8, s < t, then Es{MtH} = AEq {MtH} + ( 1 - A)ER{MtH} = AEq {MsH} + (1 - A)ER{MsH} = Es {MsH} ,
D
Definition. A measure Q E M2 (A) is an extremal point of M 2 (A) if whenever Q = AR + (1 - A)S with R, S E M 2 (A), R -=/:- S, 0 < A < 1 , then A = 0 or 1 . Theorem 38. Let A C M2 . If S(A) = M2 then P is an extremal point of M 2 (A) .
Proof. Suppose P is not extremal. We will show that S(A) -=/:- M2 . Since P is not extremal, there exist Q, R E M 2 (A) , Q -=/:- R, such that P = AQ + (l - A)R, 0 < A < 1 . Let dQ Loo = dP ' and let Lt = E{ � I.Ft}. Then 1 = �� = AL00 + (1 - A) �� > AL00 a.s., so that L00 < i a.s. Therefore L is a bounded martingale with L0 = 1 (since Q = P on .F0), and thus L - L 0 is a nonconstant martingale in M2 (P). However, if X E A and H E b.F8, then X is a Q martingale and for s < t, Ep{XtLtH} = Ep{XtLooH} = Eq {XtH} = Eq {XsH} = Ep{ XsLoo H} = Ep{XsLsH}, and XL is a P martingale. Therefore X( L - L0) is a P martingale, and L - L 0 E M2 and it is strongly orthogonal to A. By Theorem 37 we cannot D have S(A) = M2 .
Let A C M2 . If P is an extremal point of M 2 (A) , then every bounded P martingale strongly orthogonal to A is null. Proof. Let L be a bounded nonconstant martingale strongly orthogonal to A. Let c be a bound for ILl, and set Loo Loo 1 ( )dP and dR = (1 dQ = + 2c )dP. 2c We have Q, R E M 2 (A), and P = �Q + �R is a decomposition that shows that P is not extremal which is a contradiction. 0 Theorem 39.
184
IV General Stochastic Integration and Local Times
Let A = { M 1 , . . . , Mn } C M 2 , with Mi continuous and Mi, MJ strongly orthogonal for i -1- j. Suppose P is an extremal point of M 2 (A) . Then (a) every stopping time is accessible; (b) every bounded martingale is continuous; (c) every uniformly integrable martingale is continuous; and (d) A has the predictable representation property. Proof. (a) Suppose T is a totally inaccessible stopping time and P(T < oo ) > 0. By Theorem 22 of Chap. III, there exists a martingale M with l:!.MT = l {T 0 : I Mt l > n } , I M Rn I < n + 1 and thus M is locally bounded. Indeed, we then have MRn are bounded martingales strongly orthogonal to Mi . By Theorem 39 we have MRn = 0 for each n. Since limn _.oo Rn = oo a.s., we conclude M = 0, a Theorem 40.
contradiction. (b) Let M be a bounded martingale which is not continuous, and assume Mo = 0. Let ye: = inf{t > 0 : l llMt l > c}. Then there exists c > 0 such that for T = re: , P{ I L:!.MT I > 0 } > 0. By part (a) the stopping time T is accessible, hence without loss we may assume that T is predictable. Therefore MT - is a bounded martingale by the corollary to Theorem 29, whence N = MT - MT - = LlMT l {t > T } is also a bounded martingale. However N is a finite variation bounded martingale, hence [N, Mi] = 0 each i. That is, N is a bounded martingale strongly orthogonal to A. Hence N = MT - MT- = 0 by Theorem 39, and we conclude that M is continuous. (c) Let M be a uniformly integrable martingale closed by M00• Define
Then Mn are bounded martingales and therefore continuous by part (b). However 1 < E {I M� - Moo ! } M c} P{sup I M;' - t l >
t
c
by an inequality of Doob 1 0 , and the right side tends to 0 as n tends to oo . Therefore there exists a subsequence (n k ) such that lim k _.oo M;'k = Mt a.s., uniformly in t. Thus M is continuous. (d) By Corollary 3 of Theorem 37 it suffices to show that if N E Ax then N = 0. Suppose N E Ax . Then N is continuous by (c) . Therefore N is locally D bounded, and hence by stopping, N must be 0 by Theorem 39. 10
See, for example, Breiman [23, page 88] .
185
3 Martingale Representation
The next theorem allows us to consider subspaces generated by countably infinite collections of martingales.
Let M E M2, yn E M2, n > 1, and suppose Y� converges to Y00 in L2, and that there exists a sequence Hn E L(M ) such that �n = I� H�dM8, n > 1. Then there exists a predictable process H E L(M) such that yt = I; H8dM8 • Proof If Y� converges to Y00 in L2, then yn converges to Y in M2. By Theorem 36 we have that S(M) = I(M), the stochastic integrals with respect to M. Moreover yn E S(M), each n. Therefore Y is in the closure of S(M) ; but S(M) is closed, so Y E S(M) = I(M), and the theorem is proved. D Theorem 42. Let A = { M 1 , M2 , . . . , Mn , . . . } , with Mi E M2, and suppose there exist disjoint predictable sets Ai such that 1A , d[Mi , Mi ] = d[Mi , Mi ], i > 1. Let At = L� J; 1A, (s)d[M i , Mi ]s · Suppose that (a) E{Aoo } < oo; and (b) for :P c :Foo such that for any Xi E b:Fi , we have Xf = E{X i \:Ft } = I� H! dM;, t > 0, for some predictable process Hi . Then M = L� Mi exists and is in M2, and for any Y E b V i P , if yt = E{Y\:Ft}, we have that yt = J� H8dM8, for the martingale M = L� M i and for some H E L(M). Proof. Let Nn = L� Mi . Then [Nn , Nn ] t = L� 1 f; 1A, (s)d[Mi , M i ]s, hence E{(N:;_, )2} = E{[Nn , Nn ]oo} < E{A00}, and Nn is Cauchy in M2 with limit equal to M. By hypothesis we have that if Xi E b:Fi then t t E{X i \:Ft} = Xf = lo H;dM; = lo 1A' H;dM; t = { 1 A' H; d(M; + 2: MJ ) i #J t = lo 1A,H!dM8• Theorem 41.
1
1
1
1
lo
Therefore if i
-=/=- j we have that
t
lo 1A'H! 1Aj HJd [M, M]s t = 1 1 A'nAj H;H1 d [M, M]s
[Xi , X1 ] =
= 0,
since Ai n AJ
= 0, by hypothesis. However using integration by parts we have
186
IV General Stochastic Integration and Local Times
t t x;x; = lo x;_ dxt + fo xt_dx; + [Xi , xJ ] t t t J dx i i dXJs + 1 xs= j xss 0 0 t t = lo x;_ lJ1j HtdMs + lo XL lAiH�dMs t = 1 Hi,J dM 0
S
Sl
where Hi ,j is defined in the obvious way. By iteration we have predictable representation for all finite products ftO be an adapted, right continuous increasing process, which need not always be finite-valued. The change of time (also known as a time change) associated to A is the process Tt = inf{s > 0 : As > t}. Some observations are in order. We have that t f---> Tt is non-decreasing and hence Tt - exists. Also, since {At > s} = U,, > 0 {At > s + c}, we have that t f---> Tt is right continuous. It is continuous if A has strictly increasing paths. Moreover Ar, > A7, _ > t, and A(r._) - < A(r.) - < t. (Here we use the convention To - = 0.) Finally note that { Ts - < t} = { s < At}, which implies that Tt is a stopping time, and since Tt = lim"_,0 T t E:) we conclude that Tt Definition. Let
-
(+ -
is also a stopping time.
Let a be a positive, finite, right continuous, increasing function on [0, oo). Let c denote its right continuous inverse (change of time). Let f be a positive Borel function on [0, oo) . If G is any positive, finite, right continuous function on [0, oo) with G(O-) = 0, then Theorem 45 (Lebesgue's Change of Time Formula) .
00 00 1 f(s)dG(a(s)) = 1 f(c(s-) )l {c( s - )
that N is not a priori absolutely continuous. This is to explain our hypotheses in the next theorem:
Let (O, F, lF, P) be a filtered probability space satisfying the usual hypotheses. Assume that the a-algebra :F = :F= and that it is separable. 1 3 Assume further that lF is quasi left continuous. To prevent trivialities assume also that it is not an absolutely continuous space. Then there exists a change of time Tt and a new filtration G given by gt = Fr, , such that if N is a G adapted counting - process with totally inaccessible jump times and without explosions, then dNt � dt. Proof. Let (Mi ) i> l be the countable sequence constructed in Theorem 44. Define the strictly increasing process At = t + I: i> I fMi , Mi ]t, and note that E{ At} < oo for 0 < t < oo. Let C denote the compensator of A. One can check that Ct = t + I:i> 1 {M i ,M i ) t. Then C is both strictly increasing and continuous, since lF is quasi left continuous. Let Tt = inf{s > 0 : C8 > t} and then Ct = inf{s > 0 : T8 > t}, Theorem 47 (Absolutely Continuous Compensators) .
T
where is the time change of the theorem statement. be a G adapted counting process and let Let = As sume, by stopping if necessary, that We write G to de E note the sharp bracket taken using the G filtration. Note that = Nc, is also a counting process, and since continuous implies that Qc, C (see Exercise 38 of Chap. II) , we conclude is adapted to lF. We then have = = hence = But also = N and we have
N
X= L2 . T L [L, L]t I: s l J; H!dM!
and by the uniqueness of the compensator of L we have we know that On the other hand, by Theorem = for some predictable processes and thus
44i (H ),
for some (predictable) nonnegative processes ci whose existence is assured « because a.s. The process J8 = by Fubini's Theorem. Next, note that since )r = we have (L - L, L - L)�, = Jd"' which by the corollary of Lebesgue's change of time formula, implies = J� Jrsds. 0
d{M i , Mi ) t dCt Li> l (H!)2c�, {L-L, L L = {X, X)g,, {X, X)f J8dC8, {X, X)f -
4 Martingale Duality and the J acod-Yor Theorem on
Martingale Representation
'H2
We have already defined the space for semimartingales; we use the same definition for martingales, which in fact historically came before the definition for semimartingales. We make the definition for all p, p > 1.
M be a local martingale with cadlag paths. We define the JiP norm of M to be J I M li P = E{[M, M] � 2 P IP . If I M I ·w < we say that M is in JiP. Finally we call JiP the space of all cadlag martingales with finite Definition. Let
oo,
JiP norm.
It is simple to verify that II 111-lv is a norm if we identify martingales that are almost surely equal. Indeed, the only property that is not obvi ous is the subadditivity. For the case p = for example, we have that + < + since by the Kunita-Watanabe + < inequality which gives the subadditivity. It is also simple to verify that JiP is a linear space. We state without proof the important martingale inequalities known as the Burkholder-Davis-Gundy in equalities, and often referred to by the acronym BDG. (For a proof one can consult many different sources, including and ·
1 11 2 [M N,M Nj ll2 [M,M]1 2112 [N1 ,N] [M,N] [M,M] 1 [N,N] 12 ,
[44].) Theorem 48 (Burkholder-Davis-Gundy Inequalities) . Let M be a mar Then tingale with cadlag paths and let p > 1 be fixed. Let Mt = SUPs l is a sequence of stopping times increasing to a.s., then 1 / MTn / ·r£1 < 1 / M / 1 ?-t l , and also MTn converges to M in 'H 1 . Choose Tn to be the first time [U, U]t has total variation larger than n, and also I [N, Nj Tn / < n - 1; that is, Tn = inf{ t > 0 J� [U, U]t > n, or [N, N] t > n - 1} with Tn increasing to Then [N, N] Tn < n and thus NTn is in 'H 1 for each n. Therefore we turn out attention to urn , and since [U, U] Tn < n + / .6. Urn I , we need consider only the one jump of U at time Tn .6.Urn . Let � denote .6.Urn . Letting �k = l;1 {1 � 1 < k} > since �k is bounded we can compensate �k 1 {t > Tn } > and call the resulting martingale Vt. The compensator of �kl{t >Tn } is in 'H2 because �k is bounded. We have then that v; is in 'H2 and converges to Vn as k in total variation norm, and hence also in 'H 1 . This proves that 'H 2 is dense in 'H 1 , and since bounded martingales are dense in 'H 2 , they too are dense in 'H 1 . Theorem 51. Let M be a local martingale. Then M is locally in 'H 1 . Proof. By the Fundamental Theorem of Local Martingales we know that M = + U, where N has jumps bounded by a constant /3 and U is locally of Nintegrable variation. By stopping, we thus assume that N is bounded and U has paths of integrable variation. The result then follows by the previous theorem (Theorem 49) . c
Theorem 49.
0
oo
:
oo .
:
---> oo
0
0
One can further show, by identifying JiP with LP through the terminal value of the martingale, that JiP is a Banach space (in particular it is complete) for each p > One can further show that the dual space of continuous linear p oo. This continues nicely functional on JiP is where � + ! = It can be shown (see the analogy with LP except for the case p = is not that the dual of It turns out that a better analogy than LP is that of Hardy spaces in complex analysis, where C. Fefferman showed that can be identified with the space of functions of bounded mean the dual of oscillation, known by its acronym as With this in mind we define the space of martingales.
1.
'H 1 BMO
'H 1
'Hq ,
'H00
1, 1 < < 1.
.
BMO.
[47] )
4 M art ingale Duality and Jacod-Yor Theorem
1 95
Definition. Let M be a local martingale. M is said to be in BMO if M is
'H2
in and if there exists a constant have
c
such that for any stopping time
E{(Moo - Mr - ) 2 1 Fr} < c2
T we
a.s.,
where M0 _ = 0 by convention. The smallest such c is defined to be the BMO norm of M, and it is written IIM II BMO · If the constant c does not exist, or if M is not in then we set II MIIB MO = oo.
'H2,
c2
Note that in the above definition E {M! I Fo } < (with the convention that Mo- = 0) and therefore IIMII'H2 < II MII BMO· Note in particular that , II M II BM O = 0 implies that M = 0. Let T be a stopping time and A E :Fr . Replacing T with TA shows that the above definition is equivalent to the statement
for every stopping time the BMO norm:
T. This in turn gives us an equivalent description of
- Mr - )2} E{(Moo I IM II BMO = sup P(T < oo) where the supremum is taken over all stopping times T. Note that this second T
characterization gives that II · IIBMO is a semi-norm, since it is the supremum of quadratic semi-norms. An elementary property of BMO martingales is that M E BMO if and only if all of the jumps of M are uniformly bounded. Thus trivially, continuous L2 martingales and bounded martingales are in BMO . The next inequality is quite powerful.
Let M and N be two local mar tingales. Then there exists a constant c such that
Theorem 52 (Fefferman's Inequality) .
Fefferman's inequality is a special case of the following more general result.
Theorem 53 (Strengthened Fefferman Inequality) . There exists a con stant c such that for all local martingales M and N, and U an optional process,
Proof.
Let
Ct = f; u;d[M, M] s and define H and K by
2
Ht =
y'lJ; +
u? -/Ct l
#t 2 = t. C K >O}, l{ t C t {t > O}
IV General Stochastic Integration and Local Times
196
Using integration by parts yields
Ht2 d[M, M] t = l{ c,>o } v'(J; dCt t + y'Ct - 1 {t>O} From the definitions of H and K we have Ht2 Kt2 > 21 ut2 1 { c, >o} ·
=
1 {t>o} d Vr;::y Ct .
The Kunita-Watanabe inequality implies
1
00
00
/Us/1 { cs =o } / d[M, N] s / < ( 1 u;1 {Cs =o } d[M, M] s ) 1 12 ([N, N]oo ) 1 12 00 = ( 1 1 { Cs=O} dC8 ) 1 12 ([N, N] 00 ) 1/ 2 = 0 a.s.,
/
and since /d[M, N] 8 is absolutely continuous with respect to consequence of the Kunita-Watanabe inequality, we have
d[M, M] s as a
oo oo 1 r r y12 E{ j /Us l / d[M, N] s/ } = yl2 E{ j / 1{ Cs >o} Us l / d[M, N]s /} 0 00 0 < E{ 1 / HsKs/ ld[M, N]s/} 00 00 < E{ 1 Hid[M, M]s} E{ 1 Kid[N, N]s} · 1
,.-----
But
00
00
E{ 1 H;d[M, M]s} < E{ 1 d �} = E{ JC:} 00 = E {( 1 U]d[M, M] s) 1/ 2 } and
oo
oo
E{ 1 K; d[N, N] s } = E{ 1 ([N, N]oo - [N, N]s - )dK;} 00 = E{ 1 (E{[N, N]oo/Fs } - [N, N] s - )dK;}. But E{E{[N, N]oo / Fs} - [N, N]s- } is bounded by we have that
00
//N I/ �MO on (0, oo), hence
00
E{ 1 K; d[N, N]s} = E{ 1 (E{[N, N]oo / Fs} - [N, N]s - )dK; } < / N / I � M E{ JC:}, o
and the result follows.
0
4 Martingale Duality and Jacod-Yor Theorem
Remark. The constant c in Theorems
197
52 and 53 can be taken to be J2, as
can be seen from an analysis of the preceding proof.
Let N E H 2 • Then N is in BMO if and only if there is a constant c > 0 such that for all M E H2 , jE{[M, N]oo}l < c jj M II w · Moreover IINII B MO < V6 c. Proof If N is in BMO, then we can take c = V2 11N II B MO from Fefferman ' s inequality (Theorem 52) and the remark following the proof of Theorem 53. Now suppose that M is in H2 and that jE{[M, N]oo} l < cjjMII w ; we want to show that N is in BMO. We do this by first showing that I No l < c a.s. , and then showing that N has bounded jumps. Let A = {INol > c} . Suppose P(A) > 0. Let � = si�/�o) 1A. Then E{l �l} = 1 , and if we define the trivial martingale Mt = � for all t > 0, then M E }{2 and I I M I Iw = E{l�l} 1, whence 1 { l ol > c jN j E{[M, N]oo}l = E{MoNo} = E{ P N0 > c } > c = c i! M IIw · This of course is a contradiction and we conclude jNo l < c a.s. We next show j�Nj < 2c. Since every stopping time T can be decomposed Theorem 54.
�i � ;
=
into its accessible and totally inaccessible parts, and since each accessible time can be covered by a countable collection of predictable times with disjoint graphs, we can assume without loss of generality that T is either totally inac cessible or predictable. We further assume P(T > 0) = 1. Suppose then that P(j�Nrl > 2c) > 0, and set sign( �Nr)
� = P(j�Nr l > 2c) l{ I�Nr l > 2c } · Let M be the martingale consisting of � 1 {t >T } minus its compensator. Then M is in H2 and has at most one jump, which occurs at T. The jump is given
by
�MT =
{ � - E{�IFr- },
T is predictable, T is totally inaccessible.
� Note that we also have II M II w is less than the expected total variation of M, which in turn is less than 2E{I�I } = 2. If T is totally inaccessible then 1 { 1 �Nrl > 2c} �N r j j t: E{[M ' N] oo } = E{�MT�NT } = E{ �NT } = E{ P( �N 2c) } j rj > > 2c > c IIMII w , '>
which is a contradiction. On the other hand, if T is predictable, then we know that E{�Nr iFr - } = 0 and thus we are reduced to the same calculation and
198
IV General Stochastic Integration and Local Times
the same contradiction. We conclude that P([.6.Nr[ > 2c) = 0, and thus N has jumps bounded by 2c. Last let T be any stopping time. Let M = N - NT and TJ = [N, N] 00 [N, N] r - Then M is in 7t2 and [M, M] 00 = [M, N]oo = TJ · By our hypotheses it now follows that
E{ry} = E{[M, N]oo} < c [[ M [[ rtl = cE{ �} = c E{J77 l{ T 0 : [Nt[ > n }. Then [NTn [ < n + c, and N is locally bounded. D
Proof
The key result concerning H 1 and BMO is the Duality Theorem which is Theorem 55 that follows. First let us lay the foundation. For N chosen and fixed in BMO we define the operator L N from H 1 to lR by
L N (M) = E{[M, N]oo}
for all M in 1t 1 . Then one can easily check to see that L N is linear, and Feffer man's inequality proves that it is bounded as well, and therefore continuous. If BMO is the Banach space dual of H 1 then it is also complete, a fact that is apparently not easy to verify directly.
4 Martingale Duality and Jacod-Yor Theorem }[1 is r£l
1 99
BMO).
The Banach space dual of all (bounded) linear junctionals on can be identified with B M0 . Moreover if LN is such a functional then the norms II LN II and II N IIBMO are equivalent. Proof. Let N be in BMO. By Fefferman 's inequality we have
Theorem 55 (The Dual of
M in }[1 . This shows that LN is in the dual of }[1 and also that Il LN il < c II N IIBMO · Note further that LN cannot be trivial since LN (N) = LN > 0 unless is identically 0. Therefore the mapping is an injective linear mapping from BM 0 into Hh, the dual of }{1 . Let be an arbitrary linear functional in the dual of H 1 . We have I L(M) I < II L II II M II rtl < II L II II M II w · This means that L is also a bounded linear functional on H2. Since H2 is isomorphic as a Hilbert space to the L2 space of the terminal random variables of the martingales in }{2, we have that there must exist a unique martingale in }{2 such that for any M in }{1 we have: for all
E{[N, N] oo } L
N
F(N)
=
N
Clearly LN = L on }{2 , and since }{2 is dense in H1 by Theorem 50, we have that and L N are the same functional on }{1 . This shows that B M0 equipped with the norm II LN II is isomorphic to }{1* and thus, being the dual of a Banach space, it is itself a Banach space and in particular it is complete. Combining equation ( * ) with Theorem 54 we have that I I LN II and II N II BMO are equivalent norms. This completes the proof. D
L
While Fefferman's inequality, the space of BMO martingales, and the du ality of H00 and BMO are all of interest in their own right, we were motivated to present the material in order to prove the important Jacod-Yor Theorem on martingale representation, which we now present, after we recall the version of the Hahn-Banach Theorem we will use.
Let X be a Banach space and let Y be a closed linear subspace. Then Y = X if and only if the only bounded linear functional L which has the property that L(Y) = 0 is the functional which is identically zero.
Theorem 56 (Hahn-Banach Theorem) .
Theorem 57 (Jacod-Yor Theorem on Martingale Representation) .
Let A be a subset of H2 containing constant martingales. Then S(A), the stable subspace of stochastic integrals generated by A, equals H2 if and only if the probability measure P is an extremal point of M 2 (A), the space of probability measures making all elements of A square integrable martingales.
IV General Stochastic Integration and Local Times
200
Proof
The necessity has already been proved in Theorem 38. By the Hahn Banach Theorem, r£1 = S(A) if and only if L (S(A)) = 0 implies L is identi cally zero, where L is a bounded linear functional. Let L be a bounded linear functional which is such that L(S(A)) = 0. Then there exists a martingale such that L = LN . The local martingale N is locally bounded, N in so by stopping we can assume it is bounded and that N0 = 0. ( See the two corollaries of Theorem Let us also assume it is not identically zero, and let c be a bound for N. We can then define two new probability measures Q and R by
BMO
54.)
Noo Noo dQ = (1 - 2c )dP, dR = (1 + 2c )dP. R are both in M 2 (A), and P = �Q + �R
Then Q and shows that P not extremal in M 2 (A) , a contradiction. Therefore we must have that L identically zero and we have that r£1 = S(A) . As far as H2 is concerned, is a subspace of }{ 1 , hence }{ 2 C S(A) . But by construction of S(A) , it contained in H 2 , and we have martingale representation.
is is it is D
5 Examples of Martingale Representation
In Sect. 3 we have already seen the most important example of martingale representation, that of Brownian motion. In this section we give a method to generate a family of examples which are local martingales with jumps, and which have the martingale representation property. The limitations are that the family of examples is one dimensional (so that we exclude vector valued local martingales such as n-dimensional Brownian motion) , and that the descriptions of the jumps are all of the same rather simple kind. The idea is to construct a class of local martingales H such that for X E H we have both that X0 = 0 and the compensator of [X, X]t is At = t. If X has the martingale representation property, then there must exist a predictable process H such that
The above equation is called Emery's structure equation, and it is written (in a formal sense) in differential notation as
In order to establish that solutions to Emery's structure equation actually exists we write it in a form resembling a differential equation:
d[X, X]t = dt + ¢(Xt - )dXt
Equation ( ** ) is unusual and different from the stochastic differential equa tions considered later in Chap. V, since while the unknown is of course the
5 Examples of Martingale Representation
201
local martingale X (and part of the structure equation is to require that any solution X be a local martingale) , no a priori stochastic process is given in the equation. That is, it is lacking the presence of a given stochastic driving term such as, for example, a Brownian motion, a compensated Poisson pro cess, or more generally a Levy process. Since therefore no probability space is specified, the only reasonable interpretation of equation ( **) is that of a weak solution. That is, we want to show there exists a filtered probabil ity space (0, F, JF, P) satisfying the usual hypotheses, and a local martingale X, such that X verifies equation ( * *). It would also be nice to have weak uniqueness which means that if X and Y are solutions of ( **) for a given ¢, possibly defined on different filtered probability spaces, then X and Y have the same distribution as processes. That means that for every A, a Borel set on the function space of dtdlag functions mapping JR.+ to JR., we have P(w : t f---7 Xt (w) E A) = Q(w : t f---7 Xt(w) E A), where P and Q are the probability measures where X and Y are respectively defined. Inspired by knowledge of stochastic differential equations, it is natural to conjecture that such weak solutions exist and are unique if the coefficient ¢ is Lipschitz continuous. 14 This is true for existence and was proven by P. A. Meyer [179] ; see alternatively [136] . Since the proof uses weak conver gence techniques which are not within the scope of this book, we omit it.
Theorem 58 (Existence of Solutions of the Structure Equation) . ¢ : JR. IR be Lipschitz continuous. Then Emery 's structure equation
Let
---+
has a weak solution with both (Xt ) t >o and (J� rj;(Xs - )dXs) t >O local martin gales. The issue of uniqueness is intriguing. Emery has shown that one has uniqueness when ¢ is linear, but uniqueness for others ¢'s, including the Lip schitz case, is open. The next theorem collects some elementary properties of a solution X.
Let X be a (weak) solution of (* * *). Then the following hold. (i) E{Xf} = E{[X, X]t} = t, and X is a square integrable martingale on compact time sets. (ii) All jumps of X are of the form f:l.Xt = ¢ (Xt ) . (iii) X has continuous paths if and only if ¢ is identically 0, in which case X is standard Brownian motion. (iv) If a stopping time T is a jump time of X, then it is totally inaccessible. Proof We prove the statements in the order given. Since a solution X
Theorem 59.
-
and the integral term are both required to be local martingales, we know 14
Lipschitz continuity is defined and discussed in Chap. V.
202
IV General Stochastic Integration and Local Times
(T1 n ) n> l L.
of stopping times increasing to oo such that there exists a sequence J;I\Tn ¢(X8_)dX8 is in Therefore E{[X, X]t/\ Tn } = E{t i\ } , and apply ing the Monotone Convergence Theorem to each side of the equality in this equation yields E{[X, X]t} = t which further implies that X is a martingale on [0, t] for each t < oo and that E{Xf} = E{ [X, X]t} = t. For the second statement, recall that
Tn
[X, X]t = [X, X]f + �) �Xs) 2 , s oo �nk (a) = yt uniformly in t on compacts, a.s. ·
·
206
IV General Stochastic Integration and Local Times
I
Let s� ,i , j = SUPt < u Yi (a, t, ·) - Y1 (a, t, ·)1. Since yi is cadlag in t the function (a, w) 1---t S�, i ,J is A 0 F measurable. By hypothesis we have lim i ,J ->oo S� i J = in probability. Let no( a) = 1 , and define inductively ,,
Proof.
0
nk (a) = inf{m > max(k, nk - 1 (a)) : sup P(Sf:'i 1 > T k ) < T k }. i ,j > m '
We then define Since each
zk (a ' t ' w)
= yn a (a t w) . k( )
' '
a 1---t nk (a) is measurable, so also is zk . Define 1 r::'i ' = sup iZi ( a, t, w) - z (a, t, w)i; t u 1·
2- k ) < 2- k for any m > 1. k,k The Borel-Cantelli Lemma then implies that limi,J ->oo r:;, i J = 0 almost surely, ,, which in turn implies that lim zi (a, t, ) exists a.s., t -> 00 with convergence uniform in t. Let Aa be the set where zi converges uniformly ·
(note that
Aa E A 0 F and P(Aa) = 1, each fixed a), and define
Y ( a, t, w ) =
{
limi ---->00 zi (a, t , w), 0,
Then Y is cadlag thanks to the uniform convergence, and it is jointly mea surable. D
Theorem 63. Let X be a semimartingale with X0 = 0 a.s. and let H(a, t, w) = H't (w) be A 0 P measurable1 6 and bounded. Then there is a function Z (a, t, w ) in A 0 B(JR + ) 0 F such that for each a E A, Z(a, t, w) is a cadlag, adapted
version of the stochastic integral J; H�dX8• Proof. Let 1t = {H E bA0P such that the conclusion of the theorem holds}. If K = K (t , w) E bP and f = f(a) E bA, and if H(a, t, w) = f(a)K(t, w), then
lo
t
H(a, s, ·)dX8
= lot f(a)K(s, ·)dX8 = f(a) lo
t
K(s, ·)dX8,
and thus clearly H = fK is in H. Also note that 1t is trivially a vector space, and that of the form H = fK generate bA 0 P. Next let Hn E 1t and suppose that Hn converges boundedly to a process H E bA 0 P. By Theorem 32 (for example) we have that Hn · X converges uniformly in t in probability on compacts, for each a. Therefore H E H, and an application of the Monotone Class Theorem yields the result. D
H
16 Recall that P denotes the predictable u-algebra.
6
Stochastic Integration Depending on a Parameter
207
Corollary. Let X be a semimartingale (Xo = 0 a.s.) , and let H(a, t, w) = Hf(w) E A 0 P be such that for each a the process Ha E L(X). Then there exists a function Z (a, t, w) = Zf E A 0 B(JR+ ) 0 :F such that for each a, Zf is an a.s. cadlag version of f� H�dX8
•
Proof By Theorem 32 the bounded processes za, k Ha l{ I Ha i < k} "X converge to Ha ·X in ucp, each a. But za, k can be chosen cadlag and jointly measurable by Theorem 63. The result now follows by Theorem 43. 0 Theorem 64 (Fubini's Theorem) . Let X be a semimartingale, Hf H (a, t, w) be a bounded A 0 P measurable function, and let J.l be a finite mea sure on A. Let Zf f� H�dX8 be A 0 B(JR + ) 0 :F measurable such that for each a, za is a cadlag version of Ha · X. Then yt f Zf'J.l (da) is a cadlag version of H · X, where Ht f HfJ.l(da) . Proof. By pre-stopping we may assume without loss of generality that X E =
=
=
=
=
A
A
rf.2 , and because the result holds for the finite variation part of the canonical
decomposition of X by the ordinary Stieltjes Fubini Theorem, we may further assume that X is a martingale with E{[X, X]oo} < oo. Next suppose Hf is of the form H(a, t, w) = K(t, w)f(a) where K E bP and f is bounded, measurable. Then K E L(X) and f l f(a)IJ.l (da) < oo . In this case we have Zf = f(a)K · X, and moreover
j Zff.l (da) j f(a)K · Xf.l (da) =
= = =
K · X j f(a)J.l(da)
(! f(a)J.l(da)K) · X
H · X.
By linearity the same result holds for the vector space V generated by pro cesses of the form K(t, w)f(a) with K E bP and f bounded, measurable. By the Monotone Class Theorem it now suffices to show that if Hn E V and limn oo Hn = H, then the result holds for H. Let z�, t = H� · X, the cadlag version. Then by Jensen's and the Cauchy-Schwarz inequalities, .....
� L
(E{ ll ll
s�p IZ� ,t
< E{ r sup IZ� t
- Zfl f.l (da)} ) 2 - Zfi 2 J.l (da)}
r E{sup IZ� t - Zfi 2 }J.l(da) '
t } } t < 4 E{(Z� , oo - Z� ) 2 }J.l(da) 4 E{[Z� - z a , z� - za ] oo }f.l (da) by Doob ' s quadratic inequality for the martingales Z� and za , and by Corol lary 3 of Theorem 2 7 of Chap. II. Continuing, the preceding equals
L
A
'
=
=
A
L
208
IV General Stochastic Integration and Local Times
and the above tends to 0 by three applications of the Dominated Convergence Theorem. We conclude from the preceding that < a.s. t A } and therefore fA IZflp,(da) < oo for all t, a.s. Moreover
r sup IZ� ,t - Z�lp,(da)
()()
r Zfp,(da) l} < E{ r sup IZ�,t - Zflp,(da)} fl(da) z� t t }A ' }A }A t which tends to 0. Therefore taking Hn ,t = f H� ,t p,(da) we have Hn · Xt = fA Z�, t fl(da) converges in ucp to f Zfp,(da). Since Hn · X converges to H · X by Theorem 32, we conclude H · X = J Zfp,(da). 0 E {sup I r
The version of Fubini's Theorem given in Theorem 64 suffices for the applications of it used in this book. Nevertheless it is interesting to determine under what more general conditions a Fubini-type theorem holds.
Theorem 65 (Fubini's Theorem: Second Version) . martingale, let Hf = H (a, t, w) be A 0 P measurable, let p,
measure on A, and assume
Let X be a semi be a finite positive
Letting Zf = f� H:dxs be A 0 B(lR+ ) ® F measurable and za cadlag for each a, then yt = fA Zfp,(da) exists and is a cadlag version of H · X, where Ht = fA Hfp,(da) . Proof. By pre-stopping we may assume without loss of generality that X E 7t2 and that II Ha I I L 2 ( dl-' ) is ( 7t2 , X) integrable. Let X = N + A be the canonical decomposition of X. Then Next observe that
00
00
E{ 1 IIH� II L 2 (dl-' ) l dAsl } > E{ll 1 IH� I IdA siii L2 ( d!-') } oo > cE{ll fo I H�II dAsiii L 1 (di-L) } 00 = c E{ 1 IH� I IdAsl }p,(da). Also
L
E{1
00
6
Stochastic Integration Depending on a Parameter
L (H� )2 jj(da)d[N, N] s} = L E{1
00
209
(H�) 2 d[N, N]s}f.l ( da),
and therefore E{j000 I H� I I d As l} < oo and E{j000 (H�)2d[N, N]8} < oo for jj-almost all a E A. Whence Ha E L(X) for Jl almost all a E A. Next define Hn = H1 { IH I< n } ' and the proof of Theorem 64 works as well 0 h�. The hypotheses of Theorem 65 are slightly unnatural, since they are not invariant under the transformation
where i.p is any positive function such that f �.p(a)Jl ( da) < oo . This can be alleviated by replacing the assumption ( fA (Ha )2jj(da)) 1 12 E L(X) with a)2 ( H (j cp (a) Jl (da)) 1 12 E L(X) for some positive i.p E L 1 (djj). One can also relax the assumption on Jl to be a-finite rather than finite.
Example. The hypotheses of Theorem
65
are a bit strange, however they are in some sense best possible. We give here an example of a parameterized process (Ha ) a E A , a positive, finite measure Jl on A, and a semimartingale X such that
(i ) (a, t) Hf is A 0 P measurable, (ii ) Ha E L(X), each a E A, and ( iii ) fA I Hf lf-l(da) E L(X) -+
but such that if Zf = f; H�dX8 then fA Zfjj(da) does not exist as a Lebesgue integral. Thus a straightforward extension of the classical Fubini Theorem for Lebesgue integration does not hold.
Indeed, let A = N = { 1 , 2, 3 , . . . } and let Jl be any finite positive measure on A such that /-l( {a}) > 0 for all a E A. Let X be standard Brownian motion, let t0 < t1 < t 2 < be an increasing sequence in [0, 1], and define · · ·
Then
is in
L2(dt), whence Ht = fA IHflf-l ( da) E L(X) , and moreover if t > 1, 00 1 H Xt = """ 0 - (ta - t a - t) 1/ 2 (Xta - Xt a-I ) , a=l a ·
where the sum converges in
L2 • However if t > 1 then
210
IV General Stochastic Integration and Local Times
and
because (ta - ta - t) -1 12 (Xta - Xt ) is an i.i.d. sequence and 2::;:' 1 a - 1 = oo . Note that this example can be modified to show that we also cannot replace a
the assumption that
_
,
(L (H:)2Jl(da)) 112 E L(X)
with the weaker assumption that
(jA ( Hf)Pjj(da)) 1 1P E L(X) for some p < 2.
7 Local Times
32
In Chap. II we established Ito's formula (Theorem of Chap. II) which showed that if f : IR --> IR is C2 and X is a semimartingale, then f(X) is again a semimartingale. That is, semimartingales are preserved under C2 transfor mations. This property extends slightly: semimartingales are preserved under convex transformations, as Theorem below shows. (Indeed, this is the best one can do in general. If B = (Bt)t>o is standard Brownian motion aqd yt = f(Bt) is a semimartingale, then f must be the difference of convex func tions. (See Qinlar-Jacod-Protter-Sharpe We establish a related result in Theorem 71 , later in this section.) Local times for semimartingales appear in the extension of Ito's formula from C2 functions to convex functions (Theo rem 70) .
66
[34]. )
Let f lR --> IR be convex and let X be a semimartingale. Then f(X) is a semimartingale and one has
Theorem 66.
:
f(Xt) - f(Xo) Jt f'(Xs_)dXs + At o+ where f ' is the left derivative of f and A is an adapted, right continuous, increasing process. Moreover flAt = f(Xt) - f(Xt - ) - f'(Xt_ )LlXt. Proof First suppose lXI is bounded by n, and in rt2, and that X0 0. Let g be a positive coo function with compact support in ( OJ such that 00 00 j 00 g(s)ds 1 . Let fn (t) = n j 00 f(t + s)g(ns)ds. Then fn is convex and C 2 and moreover f� increases to f' as n tends to oo. By Ito's formula =
=
-oo,
=
7 Local Times where
�1
211
t
A� = L {fn (Xs) - fn (Xs - ) - f�(Xs_)boXs} + J;: (Xs - )d[X, X]� . 0 O<s 0, if X < 0.
Note that our definition of sign is not symmetric. We further define •
ho(x) = lxl and ha (x) = lx - al . Then sign(x) is the left derivative of h0 (x ) , and sign(x - a ) is the left derivative of ha (x). Since ha (x ) is convex by Theorem 66 we have for a semimartingale X ha (Xt) = IXt - ai = IXo - ai + t sign(Xs - - a)dX8 + A� , ( * * ) lo+ where Af is the increasing process of Theorem 66. Using ( * ) and ( * * ) as defined above we can define the local time of an arbitrary semimartingale. Definition. Let X be a semimartingale, and let ha and Aa be as defined in ( * ) and ( * * ) above. The local time at a of X, denoted Lf = La (X)t, is
defined to be the process given by
L� = A�
-
L {ha (X8) - ha (X8_ ) - h� (X8_ ) LlX8 }. O<so, and finally so too does the local time Lf. We always choose this jointly measurable, cadlag version of the local time, without any special mention. We further observe that the jumps of the process Aa defined in ( ** ) are precisely l:sO is continuous in t. Indeed, the local time L a is the continuous part of the increasing process Aa . The next theorem is quite simple yet crucial to proving the properties of La that justify its name.
Theorem 68.
Then
Let X be a semimartingale and let La be its local time at a.
t (Xt - a) + - (Xo - a) + = l{ X ,_ > a } dXs + L l { X ,_ > a } (Xs - a) 0+ O<s a.
{
Then define
Tr(w) = inf {t > Sr(w) : Xt_ (w) > a}. Then [Sr, Tr) C {X_ < a}, and moreover the interior of the set {X_ < a} equals Ur E IQ!, r>O ( Sr, Tr). As we have now seen, dLa does not charge the interior of the set {X_ < a}. This set is open on the left, hence it differs from its interior by an at mo �t countable set, and since dLa is diffuse it doesn't charge countable sets. Thus dLa does not charge the set {X_ < a} itself. Analogously one can show dLa does not charge {X_ > a} . Hence its 0 support is contained in the set {X_ = a}, and we are done. The next theorem gives a very satisfying generalization of Ito's formula (Theorem 32 of Chap. II) .
Let f be the difference of two convex functions, let f' be its left derivative, and let J.l be the signed measure (when restricted to compacts) which is the second derivative of f in the generalized function sense. Then the following equation holds: t f(Xt) - f(Xo ) = 1 f'(Xs_)dXs + L {f(Xs) - f(Xs - ) - f'(Xs_) Ll Xs} O+ O<s 2. Theorem 71.
=
We then have
t
t ( 1) IXt I = ypt f3 10 y8/3 - l d Y. + /3 /32 10 y8/3 - 2 d[Y Y] =
S
l
Sl
since Y has continuous paths. Using Theorem 69 and the Meyer-Tanaka for mula ( Corollary 3 of Theorem 70 ) we have, where Lt = L� is the local time of X at 0,
t
Lt = 1 1 { Xs=O} dLs t t = - 1 1{xs =O} sign (X8)dX8 + 1 1 {xs=o} d 1 XIs
yn
is a sequence The integral J; 1 { Xs =O} sign (X8)dX8 is identically zero. For if n of stopping times increasing to 00 a.s. such that xr is a bounded martingale, then
E{(
1
t/\Tn
1 { X s=O} sign (X8)dX8 ) 2 } E{ 1 =
t /\Tn
1 { Xs=O} d[X, X]8}
= E{1: Lf/\Tn 1 {o} (a)da} = 0.
218
IV General Stochastic Integration and Local Times
Therefore Lt = L� = becomes
J� 1{x.=o} djXj8• Since {Xs = 0} equals {Ys = 0}, this
Using the Meyer-Tanaka formula again we conclude that
Since X is a continuous local martingale, so also is the stochastic integral on the right side above (Theorem 30) ; it is also non-negative, and equal to zero at 0. Such a local martingale must be identically zero. This completes the ��
0
It is worth noting that if X is a bounded, continuous martingale with X0 = 0, then yt = !Xt!", 0 < a < 1 /2 is an example of an asymptotic martingale, or AMART, which is not a semimartingale.18 We next wish to determine when there exists a version of the local time Lf which is jointly continuous in (a, t) f-+ Lf a.s., or jointly right continuous in (a, t) f-+ Lf . We begin with the classical result of Kolmogorov which gives a sufficient condition for joint continuity. There are several versions of Kol mogorov's Lemma. We give here a quite general one because we will use it often in Chap. V. In this section we use only its corollary, which can also be proved directly in a fairly simple way. Before the statement of the theorem we establish some notation. Let � denote the dyadic rational points of the unit cube [0, l] n in R.n , and let �m denote all x E � whose coordinates are of the form k2-m, 0 < k < 2m.
Let (E, d) be a complete metric space, and let ux be an E-valued random variable for all x dyadic rationals in R.n . Suppose that for all x, y, we have d(Ux , UY ) is a random variable and that there exist strictly positive constants C, f3 such that
Theorem 72 (Kolmogorov's Lemma) .
E,
Then for almost all w the function x ...... u x can be extended uniquely to a continuous function from R.n to E. 18
For a more elementary example of an asymptotic martingale that is not a semi martingale, see Gut [86, page 7].
7
Local Times
219
Proof We prove the theorem for the unit cube [0, l] n and leave the extension to !Rn to the reader. Two points x and y in Ll m are neighbors if supi j xi yi j = 2- m . We use Chebyshev's inequality on the inequality hypothesized to get P{ d(U x , UY) > T "m } < C2""'m T m (n+t3) . Let
Am =
{w :
Since each x E 2 mn we have
:3
neighbors x, y E
Llm
'
Llm with
d(Ux (w), UY(w)) > T"m }.
has at most 3n neighbors, and the cardinality of Llm is
P Am ) < c2m (ac - t3)
(
c = 3n c. Take a sufficiently small so that ac < {3. Then P(Am ) < cT m 8 where (j = {3-ac > 0. The Borel-Cantelli Lemma then implies P(Am infinitely often) = 0. That is, there exists an m0 such that for m > mo and every pair u, v ) of points of Llm that are neighbors,
where the constant
(
ux
We now use the preceding to show that x f--+ is uniformly continuous on Ll and hence extendable uniquely to a continuous function on [0, l] n . To this end, let x, y E Ll be such that ll x - Yll < 2- k - I . We will show that < c2-ak for a constant c, and this will complete the proof. Without loss of generality assume k > m0 . Then x = (xl , . . , xn and y = (y 1 , . . . , yn ) in Ll with ll x - Y ll < 2- k -1 have dyadic expansions of the form
d(Ux , UY )
.
)
ui + 2:= ajT 1 j>k y i = v i + 'L b; TJ j> k where aj, bj are each 0 or 1, and u, v are points of Ll k which are either equal or neighbors. Next set uo = u, u1 = uo + ak+ 12- k - 1 , u2 = u1 + ak+ 22- k - 2 , . . . . We also make analogous definitions for vo, v1, v2, . . . . Then ui - 1 and Ui are equal or neighbors in Ll k+i> each i, and analogously for vi_1 and vi. Hence xi
=
d(Ux (w), uu (w)) < Lk T"J j= d(UY(w), Uv (w)) < 2:==k T"J 00
00
j
220
IV General Stochastic Integration and Local Times
and moreover D
The result now follows by the triangle inequality.
Comment. If the complete metric space (E, d) in Theorem 72 is separable, then the hypothesis that d(Ux , UY ) be measurable is satisfied. Often the metric spaces chosen are one of IR, JRd , or the function space C with the sup norm and these are separable. A complete metric space that arises often in Chap. V is the space E = vn of cadlag functions mapping [0, oo ) into !Rn, topologized by uniform conver gence on compacts. While this is a complete metric space, it is not separable. Indeed, a compatible metric is d( f,
00
g) = �
1 ( 1 1\ 0 ��� l f( s ) - ( s ) l ). n 2 n
g
However if fa (t) 1 [a ,oo ) (t), then d(fa, ff3 ) = 1 /2 for all a, /3 with 0 < a < f3 < 1 , and since there are uncountably many such a , {3, the space is not separable. Fortunately, however, the condition that d(Ux, UY ) be measurable is nevertheless satisfied in this case, due to the path regularity of the functions in the function space vn. (Note that in many other contexts the space vn is endowed with the Skorohod topology, and with this topology vn is a complete metric space which is also separable; see for example Ethier-Kurtz [71] or Jacod-Shiryaev [1 10] .)
=
We state as a corollary the form of Kolmogorov's Lemma (also known as Kolmogorov's continuity criterion ) that we will use in our study of local times.
Corollary 1 (Kolmogorov's Continuity Criterion) . Let be a parameterized family of stochastic processes such that t a.s., each a E !Rn. Suppose that
f---+
E { sup IX� - X�l"' }
sO, a E JRn Xf is cadlag
< C(t)lla - b ll n+ f3
Xf
for some c, f3 > 0, C(t) > 0. Then there exists a version of Xf which is B(IR+) 181 B ( !Rn ) 181 :F measurable and which is cadlag in t and uniformly continuous in a on compacts and is such that for all a E JRn , t > 0, Xf
= Xf
a.s.
(The null set {Xf =1- xn can be chosen independently of t.) In this sec tion we will use the above corollary for parameterized processes xa which are continuous in t. In this case the process obtained from the corollary of Kolmogorov's Lemma, _X a , will be jointly continuous in (a, t) almost surely. In particular, Kolomogorov's Lemma can be used to prove that the paths of standard Brownian motion are continuous.
7 Local Times
221
Corollary 2. Let B be standard Brownian motion. Then there is a version of B with continuous paths, a.s.
Proof Since Bt - Bs is Gaussian with mean zero and variance t - s, we know that E{ I Bt - Bs l 4} < c(t - s) 2 . (One can give a cute proof of this moment
estimate using the scaling property of Brownian motion.) If we think of time as the parameter and the process as being constant in time, we see that the exponent 4 is strictly positive, and that the exponent on the right, 2 , is strictly bigger than the dimension, which is of course 1. Corollary 2 now follows from D Corollary 1. Hypothesis A. For the remainder of this section we let X denote a semi martingale with the restriction that Lo < s 0.
Observe that if (n, :F, (:Ft ) t >o, P) is a probability space where (:Ft ) t >o is the completed minimal filtration of a Brownian motion B = (Bt)t >o, then all semimartingales on this Brownian space verify Hypothesis A. Indeed, by Corollary 1 of Theorem 43 all the local martingales are continuous. Thus if X is a semimartingale, let X = M + A be a decomposition with M a local martingale and A an process. Then the jump processes dX and dA are equal, hence
FV
O< s< t since A is of finite variation on compacts. Let X be a semimartingale satisfying Hypothesis A, and let
Jt = L dX8 ; O<s< t that is, J is the process which is the sum of the jumps. Because of our hypoth process and thus also a semimartingale. Moreover yt = Xt - Jt esis J is an is a continuous semimartingale with Yo = 0, and we let
FV
be its (unique) decomposition, where M is a continuous local martingale and process with Mo = Ao = 0. Such a process M is then A is a continuous uniquely determined and we can write
FV
M = Xc the continuous local martingale part of X.
'
Notation. We assume given a semimartingale X satisfying Hypothesis A. If Z is any other semimartingale we write
(a E IR) .
IV General Stochastic Integration and Local Times
222
This notation should not be confused with that of Kolmogorov's Lemma (The orem 72). It is always assumed that we take the B(IR) ®B(IR+ ) ®F measurable, cadlag version of za (cf. , Theorem 63). Before we can prove our primary regularity property (Theorem 75), we need two preliminary results. The first is a very special case of a family of martingale inequalities known as the Burkholder-Davis-Gundy inequalities. Theorem 73 (Burkholder 's Inequality). Let be a continuous local mar tingale with = 0, 2 < p < oo, and T a finite stopping time. Then
X X0 E{(Xr)P} < CpE{[X, X]j"2 } where Cp = {qP( p(p; l))}P/2, with ; + ! = 1 .
X and [X, X] are T T ) p(p 1 1 J Xr JP = p l{o sign(Xs) JXs JP dXs + 2 l{o JXsJP- 2d[X, X]s·
Proof
By stopping, it suffices to consider the case where bounded. By Ito's formula we have
By Doob's inequalities (Theorem
20 of Chap.
I) we have (with
! + ! = 1)
E{(Xr)P} < qPE{ J XrJ P } (p ) T = qP E{ p ; 1 1 JXs J P- 2d[X ,X ]s} < qP ( p(p; 1 ) ) E {( Xr)P - 2[X , X] r } < qP ( p(p ; 1 ) ) E{(Xr)P} l - %E{[X ,X]j"2 }2/p , with the last inequality by Holder's inequality. Since E{(X:T)P} 1 - % < divide both sides by it to obtain and raising both sides to the power p/2 gives the result.
oo ,
we
D
Actually much more is true. Indeed for any local martingale (continuous or not) it is known that there exist constants c , such that for a finite stopping time T
p Cp E{ (Xr )P } l/p < cpE{[x, xJ!f' 2 } 1/P < cpE{ (Xr )P} 1/P for 1 < p < See Sect. 3 of Chap. VII of Dellacherie--Meyer [46] for these and related results. When the local martingales are continuous some results even hold for 0 < p < 1 (see, e.g., Barlow-Jacka-Yor [9 , Table 4.1 page 1 62 ]) . oo .
223 Theorem 74. Let X be a semimartingale satisfying Hypothesis A. There ex ists a version of (X c )'t such that (a, t, w) (Xc )'t(w) is B(IR) ® P measurable, and everywhere jointly continuous in (a, t). Proof Without loss of generality we may assume X - X0 E '}-{2 . If it is not, n we can stop X - Xo at rn - . The continuous local martingale part of x r is then just (Xc ) Tn . Suppose - oo < a < b < oo , and let t O:t(a, b) = E{( l { b > Xs- > a } d[X, X] � ) 2 } . 7 Local Times
f---+
By Corollary 1 of Theorem
O:t (a, b) = E{(
fo
70 we have
1 Lfdu)2 } = (b - a)2 E{( b � a 1b Lfdu)2 } b < (b - a) 2 E{ b � a 1 (Lf)2du}, b
by the Cauchy-Schwarz inequality. The above implies
O:t (a, b) < (b - a) 2 u sup E{(Lf) 2 }. E(a,b) By the definition,
t { Lf < Af < I Xt - Xo l - l sign(Xs - - u)dX8 o+
and therefore
t E{(Lf) 2 } < 2E{ I Xt - Xol 2} + 2E{( l{ sign(Xs - - u)dXs)2} o+ < 4II X - Xo ll�2 + 411 sign(Xs - - u) · (X - Xo) ll�2 < BII X - Xo ll�2 < oo, and the bound is independent of u. Therefore for a constant r < oo, and independent of t. Next using Burkholder's inequal ity (Theorem 73) we have E{sup I (Xc ) �
s
r=
- (Xc ) �l 4 } < C4E{( o l {b> Xs- > a} d[X, X] � )2} J < c4 sup at (a, b) t < c4 r(b - a)2 .
The result now follows by applying Kolmogorov's Lemma (the corollary of Theorem 72). D
224
IV General Stochastic Integration and Local Times
We can now establish our primary result. Theorem 75. Let X be a semimartingale satisfying Hypothesis A . Then there exists a B(IR) Q9 P measurable version of (a, t, w) f-+ L�(w) which is everywhere
jointly right continuous in a and continuous in t. Moreover a.s. the limits Lta - = l"lmb-->a, b< a Ltb exts· t. Proof Since X satisfies Hypothesis A, the process Jt = Lo<s< t LlX8 is an FV semimartingale, and Y = X - J is a continuous semimartingale. We let Y = M + A be the (unique) decomposition of Y, with M0 = A0 = 0. Then X = M + A + J. Further define Sf = L l{x.->a} (Xs - a) - + L l{x._ 0 1 1t 1 { a<X 0 - 10 1 {a-E <X 0, X < 0.
A symmetrized result follows trivially, namely
Lta + Lta- lim -1 1t 1{ \ X. -a\ < c} d[X, X] �, a.s. 2 c---> 0 2c 0 Corollary 3 is intuitively very appealing, and justifies thinking of local time as an occupation time density. Also if X = B, a Brownian motion, then Corollary 3 becomes the classical result t 1 Lf = E--->lim0 -2 10 1 (a-E,a+c) (Bs)ds, a.s. =
C
Local times have interesting properties when viewed as processes with "time" fixed, and the space variable "a" as the parameter. The Ray-Knight the Brownian local time Theorem giving the distribution of sampled at the hitting time of as the square of a two dimensional Bessel diffusion, is one example.
1,
(L}- a )o 0. This hypothesis was needed to prove Theorems 74 through 77 and their corollaries. We present here a counterex ample, known as Azema's martingale, that shows that these results do not hold for general semimartingales. In particular, Theorems 81 and 82 show that Azema ' s martingale has a local time that is zero at every level a except a = 0, and therefore there is no regular version in the space variable. Let (n, F, IF, (Bt)t>o , P) be a standard Brownian motion, Eo = 0, with the filtration IF = (Ft)t>o completed (and hence right continuous) . We define
A
228
IV General Stochastic Integration and Local Times
sign(x) =
{ -11 '
if X > 0, if X < 0.
,
Set 9� = o- sign ( Bs ) ; s < t}, and let O denote the filtration (9�) t> O completed. (It is then right continuous as a consequence of the strong Markov property of Brownian motion.) Let Mt = E{Btl9t}, t > 0. Then for s < t,
{
and M is a 0. Then P(sup B8 > s c) .
An elementary conditioning argument provides a useful corollary. Corollary. Let B be standard Brownian motion, B0 = 0, and 0 < s < Then P( sup Bu > 0, Bs < 0) = 2P( Bt > 0, Bs < 0). s s) = 2P(gt > s, Bs < 0) Bu > 0, B8 < 0) = 2P( s sup 0} = {Y > - .#-s X}. To calculate P( {Y > - #s X, X < 0} ) , use polar coordinates:
1 7r-arcsin 1 oo
J"f 1 - r2 12 rdrd() P (Bt > 0, Bs < 0) = e 2w ,. /2 w - arcsin fs) = ...!_ ( 2w 2 Vt � - ...!.._ arcsin fs_ 4 2w Vt Therefore 4P ( Bt > 0, Bs < 0) = 1 - ; arcsin v1 and P(gt < s) = o
=
; arcsin v't.
0
Theorem 80.
Azema 's martingale M is given by Mt sign(Bt) �Jt - gt > =
t > 0. Proof. By definition,
However E{IBtll9t} = E{ I Btll9ti sign(Bs), s < t }. But the process sign(Bs) is independent of !Btl for < 9t, and sign(Bs) = sign(Bt) for 9t < s < t. Hence sign( B ) is constant after 9t. It follows that 8
8
Therefore if we can show E{IBtl l9t = s} = ��' the proof will be complete. Given 0 < < t we define 8
T = t !\ inf{ u > s Bu = 0}. :
Note that T is a bounded stopping time. Then
230
IV General Stochastic Integration and Local Times
E{IBt l i 9t
< s} = 2E{Bt i 9t < s = 2E{Bt ; T = t
and Bs > 0} and Bs > O}
= 2E{Br ; Bs > 0}
= 2E{Bs ; Bs > 0},
=
since Br 0 on {T of t}, and the last step uses that B is a martingale, and Therefore s < T. The last term above equals E{IBs l } =
{f.;s.
E{IBt I I 9t
= s} = fs E{P(I Bt9lt; gtO
Corollary. The two filtrations lHI and 0. Hence 1it Theorem 80, sign(Mt) sign(Bt), so that 9t C 1it.
Proof.
c
9t ·
However, by
= 0 Next we turn our attention to the local times of Azema's martingale. For a semimartingale (Xt)t >o, let (Lf(X)) t > O = La (X) denote its local time at the level a. Theorem 81.
zero. Proof.
The local time at zero of Azema 's martingale is not identically
=
By the Meyer-Ito formula (Theorem 70) for f(x) lxl we have t sign(Ms_ )dMs + L� (M) + L { I Ms i - I Ms - l - sign(Ms - )6.Ms} · 0
=1 O<s l n l(Tn -t,Tn ] is strictly positive and predictable. Moreover ¢ · X is an 7t1 martingale, whence X = � · ¢ X, and 0 the corollary is established.
n,
·
234
IV General Stochastic Integration and Local Times
Corollary 2. A local martingale is a sigma martingale. This is simply a consequence of the fact that a local martingale is locally a martingale, and trivially a martingale is a sigma martingale. 0
Proof
The next theorem gives a satisfying stability result, showing that sigma martingales are stable with respect to stochastic integration. Theorem 89. If X is a local martingale and H E L(X), then the stochastic
integral H · X is a sigma martingale. Moreover if X is a sigma martingale (and a fortiori a semimartingale) and H E L(X), then H · X is a sigma martingale. Proof Clearly it suffices to prove the second statement, since a local mar
tingale is already a sigma martingale. But the second statement is simple. Since X is a sigma martingale we know an equivalent condition is that there exists a strictly positive predictable process ¢> and a local martingale M such that X = ¢> · M. Since H is predictable, the processes H 1 = H1 {H>O} + 1 and H2 = -H1 {H are both strictly positive for i = 1, 2 we have by Theorem 88 that Hi ¢> · M is a sigma martingale for each i = 1 , 2. Sigma martingales form 0 a vector space, and we are done. We also would like to have sufficient conditions for a sigma martingale to be a local martingale. Theorem 90. A sigma martingale which is also a special semimartingale is
a local martingale. Proof Any sigma martingale is a semimartingale by definition; here we also assume it is special. Thus it has a canonical decomposition X = M + A where M is a local martingale and A is a process of finite variation on compacts which is also predictable. We want to show A 0. We assume A0 = 0. Choose H predictable, H > 0 everywhere, such that H · X is a martingale, which we can easily do using Theorem 88. Then (H 1\ 1) · X = ( Hfi 1 ) . (H . X) is a local martingale, and hence without loss of generality we can assume H is bounded. Then H · A is a finite variation predictable process, and we have H · A = H · X - H · M which is a local martingale. Thus we conclude that H · A = 0. If we replace H with JH where J is predictable, IJI < 1, and is such that J� J8dA8 = J� l dA8 I, then we conclude that J� H8 ldA8 I = 0, and since H > 0 everywhere, we conclude that A = 0 establishing the result. 0 We now have that any criterion that ensures that the semimartingale X is special, will also imply that the sigma martingale X is a local martingale. =
Bibliographic Notes
235
X has continuous paths, then it is a local Corollary 2. If X is a sigma martingale and if either x; SUPs 0. Show there exists a pair of functions y on such that =y y is positive, and is increasing, continuous, = 0, and the induced by is carried by the set {s : y(s) = 0}. Show further measure sup that the function is given by s V 0) . 0. Let Exercise 30. Let M be a continuous local martingale with M M) . Show that is a continuous local martingale with = = sup = 0, and that * Exercise 31. Let be a continuous semimartingale, and suppose satisfies . Show that the equation = sign where = Use Tanaka's formula and Exercise 29.) Exercise 32. Suppose that and Y are continuous semimartingales such = 0. Show that that
J; l [o,r)(s)dM8
l [o,r) (s)
X [X, X]. (Hint: B I Bt l - "' dt oo X x + = max(X,
[ l X I, X]
(Hint:
L� (X)
B
t
x
Bs s)
XX .rJ I X Lf(X)
(I Xtl)t> O ·
L� (X) �L� (X + ) X L� .rJ X I , t)
oo) a) [ O , oo) ( x(O) + a , a (O) a x , da8 a a a(t) = s O}
=
I_2
}
·
k=l Use the fact that s (BsA T, Lsi\ T, ) and s (-Br, _s,L r, -L r, _J have the same joint distribution to show that D satisfies (i) , (ii), and (iii). *Exercise 36. With the notation of Exercise 35, let Y I B I + cL. Show that :FY :FI B I , for all c. n-->oo
t-2
�---->
t-2
�---->
c
=
*Exercise 37. With the notation of Exercise 36, Pitman's Theorem states that Y is a Bessel process of order 3, which in turn implies that for :FY , with
C-1 -
'
yt f3t + Jot Y1s ds for an :FY -Brownian motion (3. Use this to show that (Note: One can show that :FY :FIBI for all c 1.) =
=
=!=
:FY
-::f. :FIB I for
c 1. =
*Exercise 38 (Barlow's Theorem) . Let X solve the equation
B
where is a standard Brownian motion. The above equation is said to have and Y are two solutions, and if a unique weak solution if whenever �----> = /l Y · The is defined by !lx (A) = E A}, then above equation is said to have a unique strong solution if whenever both and are two solutions on the probability space on which the Brownian = 1. Assume that motion is defined, then �----> = t �----> are the equation has a unique weak solution with the property that if any two solutions defined on the same probability space, then for all > 0, = 0. Show that the equation has a unique strong solution.
11x X Y B
L�(X - Y)
X P{w : t Xt(w)
P{w : t Xt(w)
11x
yt(w)}
X, tY
Exercises for Chapter IV
241
Exercise 39. Consider the situation of Exercise 38, and suppose that and b are bounded Borel. Further, suppose that < for all where [0, oo ) [0, oo ) is increasing with J;+ p(�) + oo for every E > 0. Show that if Y are two solutions, then Y) 0.
x, y,
p:
(u (x) - u(y))2 p( l x - Y l) du = L�(X - =
-+
u
X, *Exercise 40. Let B be a standard Brownian motion and consider the equation t Xt = 1 sign(Xs)dB5• Show that the equation has a unique weak solution, but that if X is a solution then so too is -X. (See Exercise 38 for a definition of weak and strong solu tions.) Hence, the equation does not have a unique strong solution. Finally, (Hint: show that if X is a solution, then X is not adapted to the filtration Show first that Bt = J; 1{1 Xsi #O} d 1 Xsl · ) Exercise 41. Let X be a sigma martingale and suppose X is bounded below. Show that X is in fact a local martingale. :FE .
*Exercise 42. Show that any Levy process which is a sigma martingale is actually a martingale (and not just a local martingale) .
be i.i.d. random variables with P(U1 = 1) = (U >l ) n n P(Ul = -1) = 1/2, and let X = En> l 2-n un 1{t> qn} where (qn ) n> l is an enumeration of the rationals in (0, 1). Let Ht = En > l � 1 {t > qn } and show that X E 1{2 and H E L(X), but also that X is of finite variation and Y = H · X has infinite variation. Thus the s ace of finite variation processes Exercise 43. Let
is not closed under stochastic integration!
p
xn P) P, 1{2. xn Lx B x. A = {w : dL�(w) is singular as a measure with respect to ds} where of course ds denotes Lebesgue measure on JR. + . Show that P(A) = 1. Conclude that almost surely the paths s L'; are not absolutely continuous. (Hint: Use Theorem 69.) Exercise 46. Let B be a standard Brownian motion and let N be a Poisson process with parameter >. = 1, with B and N independent as processes. Let L be the local time process for B at level 0, and let X be the vector process Xtway.= (Bt, NLJ. Let IF be the minimal filtration of X, completed in the usual (a) Show that X is a strong Markov process.
*Exercise 44. Let be a sequence of semimartingales on a filtered complete probability space (D, :F, IF, satisfying the usual hypotheses. Show there ex ists one probability Q, which is equivalent to with a bounded density such is a semimartingale in that under Q each Exercise 45. Let be a standard Brownian motion and let be its local time at the level Let
f---+
242
Exercises for Chapter IV
(b) Let T be the first jump time of X. Show that the compensator of the process Ct = l{ t > T} is the process At = L t/\ T ·
Note that this gives an example of a compensator of the first jump time of a strong Markov process which has paths that are almost surely not absolutely continuous. **Exercise 4 7. Let Z be a Levy process. Show that if Z is a sigma martingale, then Z is a martingale. (Note: This exercise is closely related to Exercise 29 of Chap. 1.)
v Stochastic Differential Equations
1 Introduction
A diffusion can be thought of as a strong Markov process ( in JR.n ) with contin uous paths. Before the development of Ito's theory of stochastic integration for Brownian motion, the primary method of studying diffusions was to study their transition semigroups. This was equivalent to studying the infinitesimal generators of their semigroups, which are partial differential operators. Thus Feller's investigations of diffusions (for example) were actually investigations of partial differential equations, inspired by diffusions. The primary tool to study diffusions was Kolmogorov ' s differential equa tions and Feller's extensions of them. Such approaches did not permit an analysis of the paths of diffusions and their properties. Inspired by Levy's in vestigations of sample paths, Ito studied diffusions that could be represented as solutions of stochastic differential equations 1 of the form
where B is a Brownian motion in JR.n , u is an n x n matrix, and b is an n-vector of appropriately smooth functions to ensure the existence and uniqueness of solutions. This gives immediate intuitive meaning. If (.Ft)t >O is the underlying filtration for the Brownian motion B, then for small E > 0
E{Xf+c: - Xfi.Ft} = bi (Xt)E + o (c) E{ (Xf+c: Ebi (Xt))(Xl+c: - xt - c&i (Xt))I.Ft} = (uu ') i1 (Xt)E + o(c), where u ' denotes the transpose of the matrix u. ItO's differential "dB" was found to have other interpretations as well. In particular, "dB" can be thought of as "white noise" in statistical communi cation theory. Thus if �t is white noise at time t, Bt = J; �5ds, an equation
x: -
1
More properly (but less often) called "stochastic integral equations."
244
V Stochastic Differential Equations
which can be given a rigorous meaning using the theory of generalized func tions (cf., e.g. , Arnold [2] ). Here the Markov nature of the solutions is not as important, and coefficients that are functionals of the paths of the solutions can be considered. Finally it is now possible to consider semimartingale driving terms (or "semimartingale noise" ), and to study stochastic differential equations in full generality. Since "dB" and "dt" are semimartingale differentials, they are always included in our results as special cases. While our treatment is very general, it is not always the most general available. We have at times preferred to keep proofs simple and non-technical rather than to achieve maximum generality. The study of stochastic differential equations (SDEs) driven by general semimartingales (rather than just by dB, dt, dN, and combinations thereof, where N is a Poisson process) allows one to see which properties of the so lutions are due to certain special properties of Brownian motion, and which are true in general. For example, in Sect. 6 we see that the Markov nature of the solutions is due to the independence of the increments of the differentials. In Sects. 8 and 10 we see precisely how the homeomorphic and diffeomorphic nature of the flow of the solution is a consequence of path continuity. In Sect. 5 we study Fisk-Stratonovich equations which reveal that the "correction" term is due to the continuous part of the quadratic variation of the differentials. In Sect. 11 we illustrate when standard moment estimates on solutions of SDEs driven by Brownian motion and dt can be extended to solutions of SDEs driven by Levy processes. 2 The HP Norms for Semimartingales
We defined an 1{2 norm for semimartingales in Chap. IV as follows. If X is a special semimartingale with X0 = 0 and canonical decomposition X = N + A, then We now use an equivalent norm. To avoid confusion, we write H2 instead of 1{2 . Moreover we will define HP, 1 < p < oo. We begin, however, with a different norm on the space lDl (i.e. , the space of adapted cadla,g processes) . For a process H E lDl we define IIHII sv
=
IIH * II LP ·
Occasionally if H is in lL (adapted and caglad) we write IIHII-sv as well, where the meaning is clear.
2 The HP Norms for Semim ti gal ar
A
n
es
245
If is a semimartingale with paths of finite variation, a natural definition where of a norm would be denotes the total variation measure on JR. + induced by s �----> Since semimartingales do not in general have such nice paths, however, such a norm is not appropriate. 0, Throughout this chapter, we will let denote a semimartingale with 0). By the Bichteler a.s. Let be an arbitrary semimartingale (with Dellacherie Theorem (Theorem 43 of Chap. III) we know there exists at least one decomposition with a local martingale and an adapted, cadlag process, with paths of finite variation (also 0 a.s. ) . For 1 < p < oo we set
Z
I A I P = I j000 l dAsii i Lv, As (w). ldAs(w)l Z Z = 0 Z0 = Z = N +A, N A N0 = A0 =
Z be a semimartingale. For 1 < p < oo define I Z I =Hv = inf jp( N, A) where the infimum is taken over all possible decompositions Z = N +A where martingale, A E with paths of finite variation on compacts, NandisAoa local = No = 0. Definition. Let
Z=N+A
IDl
The corollary of Theorem 1 below shows that this norm generalizes the norm for local martingales, which has given rise to a martingale theory analogous to the theory of Hardy spaces in complex analysis. We do not pursue this topic cf. , e.g., Dellacherie-Meyer
HP
(
[46]).
(Zo = 0) . Then I [Z, Z]�2 1 Lv < I ZI I HP , Proof Let Z = M + A, Mo = Ao = 0, be a decomposition of Z. Then [Z, Z] �2 < [M, M] �2 + [A, A] ;b 2 = [M,M] �z + (ls ).6.As) 2 ) l/ 2 s Let Z be a semimartingale (1 < p < oo ) .
Theorem 1.
A is a quadratic pure jump semi LP I [Z,Z]�2 1 Lv < jp(M,A) and the resultD Corollary. If Z is a local martingale ( Zo = 0), then I Z I Hv = I [ Z, Z];b 2 J I £ v .
where the equality above holds because norms yields martingale. Taking
��-
246 V Stochastic Differential Equations Proof. Since Z is a local martingale, we have that Z Z+0 is a decomposition of Z. Therefore I Z I HP < jp(Z, 0) I [Z, Z]�,:?lb· By Theorem 1 we have I [Z, Z] ZxS2 I L v < I Z I w , hence we have equality. =
=
0
Theorem 2 is analogous to Theorem 5 of Chap. IV. For most of the proofs which follow we need only the case p = 2. Since this case does not need Burkholder's inequalities, we distinguish it from the other cases in the proof. Theorem 2. For 1 < p < oo there exists a constant such that for any = 0, semimartingale - < Proof A semimartingale Z is in lDl, so makes sense. Let = +A = be a decomposition with = 0. Then
Cp
I ZI I sp cp i ZI I Hv . M Z I ZI I sv Mo A0 00 00 I Z I �v E{(Z�)P} < E{(M� + 1 l dAs i )P} < epE{(M�)P + ifo 1 dAsi ) P}, using (a + b)P < 2P - 1 ( aP + bP ) . In the case 2 we have by Doob's maximal quadratic inequality that E{(M�) 2 } < 4E{ M�J 4E{[M, M] 00 }. Z, Zo
=
p
=
=
For general p ,
1 < p < oo, we need Burkholder's inequalities, which state
Cp
for a universal constant which depends only on p and not on the local martingale For continuous local martingales and p > 2 we proved this using Ito's formula in Chap. IV ( Theorem 73 ). For general local martingales page 28 7 . and for all finite p > 1 see, for example, Dellacherie-Meyer Continuing, letting the constant vary from line to line we have
M.
Cp 00 < I Z I �v CpE{(M� )P + ( fo l dAs i )P} 00 < CpE{[M, MJ:2 + ( 1 l dAsi )P} < Cp[)p(M,A)JP,
[46,
and taking p-th roots yields the result.
HP
]
0
Corollary. On the space of semimartingales, the - norm is stronger than the norm, 1 < p < oo.
SP
Theorem 3 (Emery's Inequality ) .
Let Z be a semimartingale, and ! + ! = � (1 < p < oo, 1 < q < oo) . Then
00 1 1 HsdZsl l w < I HI I sv i Z I Hq ·
H E lL,
2 The HP Norms for Semimartingales
247
>O· Recall that we always assume Z0 = 0 H Z MdenoteA be(J�aH8dZ8)t decomposition of Z with Mo Ao 0 a.s. Then Z + H · M + H · A is a decomposition of H · Z. Hence II H · Z l w < jr (H · M , H · A) . Next recall that [H · M, H M] = J H;d[M, M]8, by Theorem 29 of Chap. II. Therefore oo Jr (H · M,H · A) = 1 (100 H;d[M,M]s) 1 12 + 1 I Hs l ldAsl l u < II H�([M , M] �2 + 1 00 l dAs l ) l u < I H� I LP I ([M, M] � 2 + 1 00 l dAsi) I Lq = I H I sPJq (M, A), ·
Proof.
Let a.s., and let
=
=
=
·
where the last inequality above follows from Holder's inequality. The foregoing implies that
I H · Z l w < I H I sPjq (M, A) for any such decomposition Z = M + A. Taking infimums over all such de compositions yields the result. -
-
0
For a process X E lJ]l and a stopping time T, recall that
+ +
xT = Xtl[o,T) Xrl [T, oo) 1 Xtl[o,T) Xr _ l [T,oo ) ·
xT- =
A property holding locally was defined in Chap. I, and a property holding prelocally was defined in Chap. IV. Recall that a property 1r is said to hold has property for each n, where rn locally for a process X if is a sequence of stopping times tending to oo a.s. If the process X is zero at zero (i.e., Xo = 0 a.s. ) then the property 1r is said to hold prelocally if xr _ has property 7r for each n.
xrn 1{Tn>O }
7r
n
SP rn HP ) x 1 {Tn> O}
( resp. Definition. A process X is locally in if there exist stopping is in ( resp. times Tn n l increasing to 00 a.s. such that for each n, < p < oo If Xo 0 then X is said to be prelocally in (resp. if xr n _ is in ( resp. for each n.
( ) 1>
SP HP) = SP SP HP) HP) While there are many semimartingales which are not locally in HP, all semimartingales are prelocally in HP . The proof of Theorem 4 belowclosely parallels the proof of Theorem 13 of Chap. IV. Theorem 4. Let Z be a semimartingale ( Z0 0). Then Z is locally in HP, 1 < < .
=
p
00
.
pre
248
V Stochastic Differential Equations
25 43 Z = M +A, M0 = A0 =
Proof.. By the
Fundamental Theorem of Local Martingales (Theorem of Chap. III) and the Bichteler-Dellacherie Theorem (Theorem of Chap. III) we know that for given E: > 0, has a decomposition 0 a.s., such that the jumps of the local martingale M are bounded by E:. Define inductively
Z
To = 0, t Tk+ l = inf{t > Tk : [M,MJ: 12 + lo i dAs l > k + 1 } . (Tk ) k> l are stopping times increasing to oo a.s. Moreover zTk - = (MTk ) + (Ark - - �Mrk l [rk ,oo) ) = N + C
The sequence
zrk - . Also, since [M, M] rk = [M,M] rk - + (�Mrk )2, 00 Joo (N, C) = I [N,N] � 2 + 1 l dCsl l u"' Tk = I ([M, M]rk - + (�Mrk ) 2) 112 + J{o l dCsii i Loo < ll (k2 + c2) 1/2 + (k + c)I I Loo < Therefore zrk- E H 00 and hence it is in HP as well, 1 < p < Definition. Let Z be a semimartingale in H00 and let a > 0. A finite sequence < Tk is said to a-slice Z if Z = zrk of stopping times 0 = To < T1 < and I ( Z - zT; ) T; + t - u H oo < a, 0 < i < k - 1 . If such a sequence of stopping times exists, we say Z is a-sliceable, and we write Z E S(a). Theorem 5. Let Z be a semimartingale with Z0 = 0 a.s. (i) For a > 0, if Z S(a) then for every stopping time T, zr E S(a) and zT- E S(2a) . (ii) For every a > 0, there exists an arbitrarily large stopping time T such that zT- E S(a) . Proof. Since zr- = MT +(AT- -�Mrl [T, oo) ), and since I ZT I Hoo < I Z I -H 00 always, one concludes I ZT - I Hoo < 2 I ZI I-Hoo, so that (i) follows� Next consider (ii) . If semimartingales Z and Y are a-sliceable, let T[ and TJ be two sequences of stopping times respectively a-slicing Z and Y. By reordering the points T[ and TJ and using (i) , we easily conclude that Z + Y is 8a-sliceable. Next let Z = M + A, Mo = Ao = 0 a.s. , with the local martingale M having jumps bounded by the constant (3 = a/24. By the preceding observation it suffices to consider M and A separately. is a decomposition of we conclude
oo
.
00 .
·· ·
E
D
3 Existence and Uniqueness of Solutions
J;k
249
J�
ldAs l > a/8 or ldAs l > k}. For A, let To = 0, Tk+ l = inf{t > Tk : E S(a/8) for each k, and the stopping times (Tk) increase to oo Then a.s. > For M, let Ro = 0, Rk+l = inf{ t > Rk E each k, and moreover or t > k}. Then
ATk -
MRk - H00 ,
(32 [M, M]
:
[M,M]t - [M,M] Rk
Hence
I (M-MRk ) Rk+ t - I Hoo < I ([M, M] Rk + t - [M, M] Rk ) 112 + � �MRk +lii i L00 = I ((�MRk + 1 )2 + [M, M] Rk+ l - - [M,M] Rk ) 1 12 + � � MRk+tii i L00 < 1 ((32 + (32 ) 1 /2 + fJ I Loo = (1 + v'2)(3. Thus for each k, M R k - E S( (1 + v0,)(3), and since (3 a/24, the result follows.
D
3 Existence and Uniqueness of Solutions
In presenting theorems on the existence and uniqueness of solutions of stochas tic differential equations, there are many choices to be made. First, we do not present the most general conditions known to be allowed; in exchange we are able to give simpler proofs. Moreover the conditions we do give are extremely general and are adequate for the vast majority of applications. For more gen eral results the interested reader can consult Jacod [103, page 45 1 . Second, we consider only Lipschitz-type hypotheses and thus obtain strong solutions. There is a vast literature on weak solutions cf., e.g., Stroock-Varadhan [220]) . However, weak solutions are more natural (and simpler) when the differen tials are the Wiener process and Lebesgue measure, rather than general semi martingales. A happy consequence of our approach to stochastic differential equations is that it is just as easy to prove theorems for coefficients that depend not only on the state Xt of the solution at time t (the traditional framework) , but on the past history of the process X before t as well. We begin by stating a theorem whose main virtue is its simplicity. It is a trivial corollary of Theorem 7 which follows it . Recall that a process is in L if it has caglad paths and is adapted.
]
(
H
Theorem 6. Let Z lR be such that
be a semimartingale with Z0
(i) for fixed x, (t, w)
f---+
j(t, w, x) is in L; and
=
0
and let f : JR+ x f2 x lR
---->
250 V Stochastic Differential Equations (ii) for each (t, w), lf(t, w, x)-f(t, w, y)l < K(w)lx-yl for some finite random variable K. Let X0 be finite and Fo measurable. Then the equation admits a solution. The solution is unique and it is a semimartingale. Of course one could state such a theorem for a finite number of differentials dzi, 1 < j < d, and for a finite system of equations. In the theory of (non-random) ordinary differential equations, coefficients are typically Lipschitz continuous, which ensures the existence and the unique ness of a solution. In stochastic differential equations we are led to consider more general coefficients that arise, for example, in control theory. There are enough different definitions to cause some confusion, so we present all the def initions here in ascending order of generality. Note that we add, for technical reasons, the non-customary condition (ii) below to the definition of Lipschitz which follows. Definition. A function f JR+ x JRn lR is Lipschitz if there exists a (finite) constant k such that (i) lf(t, x) - f(t, y)l < klx - yi, each t E JR+ , and (ii) t f(t, x) is right continuous with left limits, each x E JRn . ---+
:
1---t
f is said to be autonomous if f(t, x)
=
f(x), all t > 0.
Definition. A function f JR + X n X ]Rn lR is random Lipschitz if f satisfies conditions (i) and (ii) of Theorem 6. Let IIlln denote the space of processes X = (X . . . , x n ) where each X i E IIll :
-
1,
(l < i < n).
Definition. An operator F from IJ]ln into 1Ill 1 = lJ]l is said to be process Lipschitz if for any X, Y in IJ]ln the following two conditions are satisfied: (i) for any stopping time T, xr- = yr- implies F(Xf (ii) there exists an adapted process K E L such that
=
F(Yf- , and
Definition. An operator F mapping IJ]ln to = lJ]l is functional Lipschitz if for any X, Y in IJ]ln the following two conditions are satisfied:
1Ill1
(i) for any stopping time T, xr- = yr- implies F(X)T- = F(Yf- , and (ii) there exists an increasing (finite) process K = (Kt)t>O such that IF(X)t F(Y)t l < Kt iiX - Y ll; a.s., each t > 0.
3
Existence and Uniqueness of Solutions
251
Note that if g(t, x) is a Lipschitz function, then f(t, x) = g(t-, x) is ran dom Lipschitz. A Lipschitz, or a random Lipschitz, function induces a process Lipschitz operator, and if an operator is process Lipschitz, then it is also functional Lipschitz. An autonomous function with a bounded derivative is Lipschitz by the Mean Value Theorem. If a function f has a continuous but not bounded derivative, f will be locally Lipschitz; such functions are defined and con sidered in Sect. 7 of this chapter. Let A = (At)t>O be continuous and adapted. Then a linear coeffi cient such as f(t, w, x) = At(w)x is an example of a process Lipschitz co efficient. A functional Lipschitz operator F will typically be of the form F(X) = j (t, w; Xs , s < t), where f is defined on [0, t] X n X D[O, t] for each t > 0; here D[O, t] denotes the space of cadlag functions defined on [0, t]. An other example is a generalization of the coefficients introduced by Ito and Nisio [101], namely
F(X)t =
t
lo g(u, w, Xu )JL(w, du)
for a random signed measure J.l and a bounded Lipschitz function g with constant C(w) . In this case, the Lipschitz process for F is given by Kt (w) = C(w)II JL (w)t ll , where II JL (w)tll denotes the total mass of the measure p(w, du) on [0, t]. Lemmas 1 and 2 which follow are used to prove Theorem 7. We state and prove them in the one dimensional case, their generalizations to n dimensions being simple. Lemma 1. Let 1 < p < oo, let J E let F be functional Lipschitz with F(O) = 0, and suppose supt I Kt(w) l < k a.s. Let Z be a semimartingale in Then the equation H= such that IIZ I IHoo
0. Thus the equations ( *n) and ( *) become respectively t
1 X�_ dZ� , t Xt = 1 + 1 X8 _ dZ8, X� = 1 +
which are elementary continuous exponential equations and have solutions
X� = exp{Zn = exp{ fn (w) ¢( t) } , Xt = exp{ Zt } = exp{ f( w ) ¢ (t) } . We can choose fn such that limn __,00 E{f� } = 0 but lim n-+oo E{f�} =/= 0 for but xn does not converge to X = 1 p > 2. Then the zn converge to 0 in for any p > 1. Indeed, limn __,00 E {!� } =/= 0 for p > 2 (since f = 0) in implies limn -+ oo E{ e tfn } =/= 1 for any t > 0. The next result does not require that the coefficients be bounded, because there is only one, fixed, semimartingale differential. Theorem 10, 11, and 13 all have as well versions in Theorem 9, but we state and prove
H2
8P,
82 H2 only the 82 versions. as
as
Let J, Jn E lDl; Z be a semimartingale; F, pn be functional Lipschitz with constants K, Kn , respectively; and let xn , X be the unique solutions of equations ( *n) and ( *), respectively. Assume that (i} Jn , J are in 82 and limn -+oo Jn = J in 82; Fn (X) = F(X) in -82, where X is the solution of (*); and (ii} lim oo -+ n (iii} max(supn Kn , K) < a < oo a.s. for a non-random constant a, and Z E S ( 2)sa ). Then limn -+oo xn = X in -82 where xn is the solution of ( *n) and X is the solution of ( *). Proof. Let xn and X be the solutions of equations ( *n) and ( *), respectively. Theorem 10.
•
Then
We let yn = (F(X) - Fn (X))- Z, and we define a new functional Lipschitz operator en by ·
Then
cn (O) = 0. If we set U = X - xn, we obtain the equation = J - r + yn + cn ( U) - . z.
u
260
V Stochastic Differential Equations
Since Z E H00 , by Emery's inequality (Theorem 3) we have yn ----> 0 in H2 and hence also in 82 (Theorem 2 ). In particular II Yn ll s2 < oo, and therefore by Lemma 2 in Sect. 3 we have ,
11 U IIs 2 < C( a , Z)IIJ - r + yn l l s2 , -
-
where C( a, Z) is independent of n, and where the right side tends to zero as n ----> oo. Since U = X - x n , we are done. o
We now wish to localize the results of Theorems 9 and 10 so that they hold for general semimartingales and exogenous processes Jn , J. We first need a definition, which is consistent with our previous definitions of properties holding locally and prelocally (defined in Chap. IV, Sect. 2 ) . Definition. Processes Mn are said to converge locally (resp. prelocally) in 8P (resp. HP) to M if Mn, M are in 8P (resp. HP) and if there exists a sequence of stopping times Tk increasing tooo a.s. such that limn_, 00 II ( Mn M ) Tk l{ Tk >O} lisp = 0 (resp. limn _,oo II( Mn - Mfk - l l sp = 0) for each k > 1 (resp. 8P replaced by HP).
Let J, Jn E lDl; Z be a semimartingale ( Z0 = 0); and F, pn be functional Lipschitz with Lipschitz processes K, Kn , respectively. Let xn, X be solutions respectively of Theorem 11.
X;' = J;' +
1t Fn (Xn )8_dZ"'
Assume that {i) Jn converge to J prelocally in 82 ; (ii} Fn(X) converges to F(X) prelocally in 82 where X is the solution of (*); and (iii) max (supn Kn , K) < a < oo a.s. (a not random). Then limn _, oo xn = X prelocally in 82 where xn is the solution of ( *n) and X is the solution of ( *). Proof By stopping at T- for an arbitrarily large stopping time T we can assume without loss of generality that Z E S( Jsa ) by Theorem 5, and that 2 2 Jn converges to J in 8 and F (X n ) converges to F( X) in 82 , by hypothesis. Next we need only to apply Theorem 10. 0 We can recast Theorem 11 in terms of convergence in ucp (uniform conver
gence on compacts, in probability) , which we introduced in Sect. 4 of Chap. II in order to develop the stochastic integral.
4 Stability of Stochastic Differential Equations Let J n , J E lDl; Z be a semimartingale (Zo = 0); and F,
261
Corollary. pn be functional Lipschitz with Lipschitz processes K, Kn , respectively. Let X , x n be as in Theorem 11. Assume that (i) Jn converges to J in ucp, (ii) Fn (X) converges to F(X) in ucp, and (iii) max(supn Kn , K) < a < oo a.s. (a not random) . n = X in ucp. Then lim x oo n--+ Proof Recall that convergence in ucp is metrizable; let d denote a distance compatible with it. If xn does ,not converge to 0 in ucp, we can find a subsequence n' such that infn, d ( Xn , 0) > 0. Therefore no sub-subsequence (X n ) can converge to 0 in ucp, and hence x n" cannot converge to 0 prelocally in 82 as well. Therefore to establish the result we need to show only that for any subsequence n ', there exists a further subsequence n" such that x n" con verges prelocally to 0 in 82 . This is the content of Theorem 12 which follows, 0 so the proof is complete. "
Let Hn , H E lDl. For Hn to converge to H in ucp it is necessary and sufficient that there exist a subsequence n ' such that limn' _,00 Hn, = H, prelocally in 82 . Proof We first show the necessity. Without loss of generality, we assume that by iteration a decreasing sequence of subsets (N k ) of HN =={1,0. 2,We3, .construct . . }, such that lim sup IH� I = 0 a.s. --+oo n Theorem 12.
n ENk O < s Noo' O <s 0. By replacing N with N' we can assume without loss of generality that tends to 0 uniformly on compacts, almost surely. We next define
I H;' I
Hn
Tn 8
n
=
inf{t > 0 = inf
T > m n m.
: I Hf l > 1},
Then Tn and 8n are stopping times and the 8n increase to oo a.s. Indeed, for each k there exists N (w) such that for n > N (w), SUPo-<s k and 8N(w) (w) > k. Next, define -
Lm =
(Hm ) ( Sn!l.n) - .
I H;'(w) l
262
V Stochastic Differential Equations
Then which tends to
(L m )* < o<s< sup [H;'[,
n
0 a.s. Moreover, when > n, (Lm )* < sup [H;'[ < sup [H;' [ < 1. m
O <s< Sn
O<s< Tm
Hence by the Dominated Convergence Theorem we have that (Lm )* tends to 0 in L2, which implies the result. For the sufficiency, suppose that Hn does not converge to H in ucp. Then there exist t0 > 0, 8 > 0, c: > 0, and a subsequence such that P{(Hn' H);0 > 8 } > c: for each n'. Then II (Hn ' - H);0 [[ £2 > 8JE for all n'. Let Tk tend to oo a.s. such that (Hn' ) tends to 0 in 82 , each k. Then there exists K > 0 such that P(T k < t 0 ) < 8f for all k > K. Hence -
We conclude
u�.
H Tk -
n'
8JE < 8f , a contradiction. Whence Hn converges to H
m
0
Recall that we have stated and proven our theorems for the simplest case of one equation ( rather than finite systems ) and one semimartingale driving term (rather than a finite number ) . The extensions to systems and several driving terms is simple and essentially only an exercise in notation. We leave this to the reader. An interesting consequence of the preceding results is that prelocal 8P and prelocal HP convergence are not topological in the usual sense. If they were, then one would have that a sequence converged to zero if and only if every subsequence had a sub-subsequence that converged to zero. To see that this is not the case for 82 for instance, consider the example given in the comment following Theorem 9. In this situation, the solutions x n converge to X in u�. By Theorem 12, this implies the existence of a subsequence such that limn' --> oo x n' = X, prelocally in 82 . However we saw in the comment that x n does not converge to X in 82 • It is still a priori possible that x n converges to X prelocally in 8 2 , however. In the framework of the example a stopping time is simply a non-negative random variable. Thus our counter example is complete with the following real analysis result (see Protter [199, page 344] for a proof ) . There exist non-negative functions fn on [0, 1 ] such that limn -->oo J01 fn (x) 2 dx = 0 and lim supn--+ oo JA (fn (x))Pdx = +oo for all p > 2 and all Lebesgue sets A with strictly positive Lebesgue measure. In conclusion, this counterexample gives a sequence of semimartingales x n such that every subsequence has a sub-subsequence converging prelocally in 8P, but the sequence itself does not converge prelocally in 8P , (1 < p < oo) .
n'
4
Stability of Stochastic Differential Equations
263
Finally, we observe that such non-topological convergence is not as un usual as one might think at first. Indeed, let Xn be random variables which converge to zero in probability but not a.s. Then every subsequence has a sub subsequence which converges to zero a.s., and thus almost sure convergence is also not topological in the usual sense. As an example of Theorem 10, let us consider the equations
t
1 Fn (xn )sdWs + 1 t Gn (Xn )8ds , t t Xt = Jt + lo F(X) s dWs + 1 G(X)8ds
Xf = Jf +
where W is a standard Wiener process. If ( Jn - J)t converges to 0 in L 2 , each t > 0, and if pn, en , F, G are all functional Lipschitz with constant K < oo and are such that (Fn( X ) - F(X))t and (Gn (X) - G(X))t converge to 0 in L 2 , each t > 0, then (X n - X)t converges to 0 in L2 as well, each t > 0. Note that we require only that F n (X ) and cn (X) converge respectively to F( X) and G(X) for the one X that is the solution, and not for all processes in ]])) . One can weaken the hypothesis of Theorem 9 and still let the differentials vary, provided the coefficients stay bounded, as the next theorem shows. Theorem 13. Let Jn , J E ]])) ; Z, zn be semimartingales (Z0 = Zo = 0 a.s.);
and F, pn be functional Lipschitz with Lipschitz processes K, Kn , respectively. Let x n , X be solutions of ( *n) and ( *), respectively. Assume that (i) Jn converges to J pre locally in 82 ; (ii} Fn (X) converges to F(X) prelocally in 82 , and the coefficients Fn , F are all bounded by c < oo; (iii) zn converges to Z prelocally in H2 ; and {iv) max(sup n Kn , K) < a < oo a.s. (a not random). Then lim xn = X prelocally in -82 • -HX> n Proof By stopping at T- for an arbitrarily large stopping time T we can assume without loss that Z E S ( }s) by Theorem 5, and that Jn converges 2 n 2 to J in 8 , p (X) converges in 82 to F( X), and zn converges to Z in H2 , all by hypothesis. We then invokeTheorem 9, and the proof is complete. -0
The assumptions of prelocal convergence are a bit awkward. This type of convergence, however, leads to a topology on the space of semimartingales which is the natural topology for convergence of semimartingale differentials, just as ucp is the natural topology for processes related to stochastic integra tion. This is exhibited in Theorem 15. Before defining a topology on the space of semimartingales, let us recall that we can define a "distance" on ]])) by setting, for Y, Z E ]]))
V
264
Stochastic Differential Equations
r(Y ) =
2- n E{ 1 /\ sup l}, IYt L t O < O
and d(Y, Z) = r(Y- Z). This distance is compatible with uniform convergence on compacts in probability, and it was previously defined in Sect. 4 of Chap. II. Using stochastic integration we can define, for a semimartingale X,
f(X) = Hsup r(H · X) I I 0) by Tx(Y) = min( , sup( Y)) . E
oo.
x
x
-
x
,
Then Tx is functional Lipschitz with Lipschitz constant Consider the equations
t �n = J[' 1 (Ta+c+l Fn )(Yn) s- dZ�. +
1, for each x > 0.
266
V Stochastic Differential Equations
Then, by Theorem 13, y n converges to X prelocally in 82 • By passing to yet another subsequence, if necessary, we may assume that Fn (X) tends to F(X) and yn tends to X uniformly on compacts almost surely. Next we define The stopping times 8k increase a.s. to oo. By stopping at 8k -, we have for n > k that (Yn - X) * and ( Fn(X) - F (X)) * are a.s. bounded by 1. (Note that stopping at 8k - changes Z to being in S ( }sa ) instead of S ( 4)sa ) by 2 Theorem 5.) Observe that
+
I Fn (Y n ) l < j Fn ( Yn ) - Fn ( X) j + I Fn ( X) - F ( X) j jF ( X) j < a( Yn - X) * + (Fn ( X) - F(X) ) * + F(X) * < a + 1 + c, whence (Ta+ c + l pn )(Yn) = Fn(Yn). We conclude that, for an arbitrarily large stopping time R, with Jn and zn stopped at R, y n is a solution of t Yfn = J? + Fn ( Yn )s _dZ� ,
1
which is equation ( *n ) . By the uniqueness of solutions we deduce yn = xn on [0, R). Since yn converges to X prelocally in 82 , we thus conclude xn converges to X prelocally in 82 • It remains only to remove the hypothesis that supt Kt(w) < a < oo. Since we are dealing with local convergence, it suffices to consider sups 0 be such that P( Kt < a) > 0, and define
ilm = {w : Kt (w) < a + m}.
Then ilm increase to n a.s. and as in the proof of Theorem 7 we define Pm by Pm (A) = P ( Aiilm), and define :F[' = .Ft l n , , the trace of Ft on ilm . Then Pm � P, so that by Lemma 2 preceding Theorem 25 in Chap. IV, if zn converges to Z prelocally in H2 (P), then zn converges to Z prelocally in H2 ( Pm) as well. Therefore by the first part of this proof, xn converges to X in ucp under Pm , each m > 1. Choose c > 0 and m so large that P(il�) < c . Then
n < lim Pm ((Xn - X); > 8) + P(il�) 8 P(( X); > X ) n ---+ oo n --+ oo < lim Pm(( Xn - X); > 8) + c n-HX> lim
and since c > 0 was arbitrary, we conclude that xn converges to X in ucp on [0, t] . Finally since t was arbitrary, we conclude xn converges to X in ucp. 0
4
Stability of Stochastic Differential Equations
267
Another important topic is how to approximate solutions by difference solutions. Our preceding convergence results yield two consequences (Theo rem 16 and its corollary) . The next lemma is a type of Dominated Convergence Theorem for stochas tic integrals and it is used in the proof of Theorem Lemma (Dominated Convergence Theorem) . Let p, q , r be given such and let that 1. 1. = 1.r , where < r < oo . Let Z be a semimartingale in Hn EP SP such that 0, < E SP, all n > Suppose lim n_, oo all Then
16.
+q (t,w).
1n IH l Y
Hq , Hf_ (w) =
1.
=
Hq , there exists a decomposition of Z, Z N + A, such that ]q (N,A) I [N, N] �2 + 1 00 l dA .II I Lq < 00 . Let en be the random variable given by 00 00 en = ( 1 (H;-_ ) 2d[N, N] s ) 1 / 2 + 1 IH-: II dAs l· The hypothesis that I Hn l < Y implies en < Y* ([N, N]�2 + 1 00 i dAs I ) a.s. However 00 00 I Y* ( [N,N]�2 + 1 j dA si ) I Lr < I Y* I Lvi i [N, N]�2 + 1 jdA sll i Lq = I Y I sv]q (N,A) < Proof.
Since
ZE
=
=
oo.
Thus en is dominated by a random variable in Lr and hence by the Dominated Convergence Theorem en tends to 0 in Lr . D We let an denote a sequence of random partitions tending to the identity. 3 Recall that for a process Y and a random partition a = {0 < ) , then ( Yt )s > O is left continuous with right limits (and adapted, of course) . It is convenient to have version ya
of ya
E ill>, occasionally, so we define ya + L Yrk [Tk ,Tk+I) · =
k
a
1
3 Random partitions tending to the identity are defined in Chap. II, preceding
Theorem 21.
268
V
Stochastic Differential Equations
Let J E S2 , let F be process Lipschitz with Lipschitz process K < a < oo a.s. and F(O) E S2 . Let Z be a semimartingale in S( 2)s) , and let X( a) be the solution of Theorem 16.
t Xt = Jt + l F(X k}, we have that sk is a stopping time, and limk _,oo sk = oo a.s. By now stopping at Sk - we have that J is bounded, hence also in S2 , and Z E S( }sa ) . An analogous argument gives us that F(O) can be assumed 4 bounded ( and hence in S2 ) as well; hence Z E S( Js)· We now can apply 2 Theorem 16 to obtain the result. To remove the assumption that K < a < oo a.s. , we need only apply an argument like the one used at the end of the proofs of Theorems 7, 8 and 15. D :
Theorem 1 6 and its corollary give us a way to approximate the solution of a general stochastic differential equation with finite differences. Indeed, let X be the solution of
where Z is a semimartingale and F is process Lipschitz. For each random < TJ:n } , we see that the random variables partition an = { 0 = T0 < T{' < X(an ) r;: verify the relations (writing a for an , X for X( an ), Tk for TJ:) ·
·
·
Xro = Jo, Xrk + ' = Xrk Jrk+ l - Jrk + F(xa + ) rk (Zrk + l - Zrk ).
+
Then the solution of the finite difference equation above converges to the solution of ( *), under the appropriate hypotheses. As an example we give a means to approximate the stochastic exponential.
270
V
Stochastic Differential Equations
Let Z be a semimartingale and let X = £(Z ) , the stochastic exponential of Z. That is, X is the solution of t Xt = 1 + lo Xs _dZ8•
Theorem 17.
Let an be a sequence of random partitions tending to the identity. Let kn - 1 X n 11 (1 + (ZTi+ 1 - zrt )) . =
i= l
Then lim xn = X in ucp. n-->oo Proof. Let yn be the solution of yt
=
t
1 + lo ysan dZs,
equation (*a ) of Theorem 16. By the corollary of Theorem 16 we know that yn converges to X = £ ( Z ) in ucp. Thus it suffices to show yn = x n . Let an = {0 T0 < T{' < · · · < T,rJ . On (Tin , TZ+ 1 J we have =
�n
= Y,Pn + Y,Pn (zr,+. l - zrt ) = Y,Pn (1 + (Z Ti+1 - zr,n )) . '
'
'
Inducting on i down to 0 we have
�n = 11 (1 + ( ZT7+1
for Tt < t < we have that
j
-
zT? )),
T[+ 1 . Since zT7+ 1 - zrjn = 0 for all j > i when Tt < t < T[+ 1 , y n x n ' and the theorem is proved. 0 =
5 Fisk-Stratonovich Integrals and Differential Equations
In this section we extend the notion of the F isk-Stratonovich integral given in Chap. II, Sect. 7, and we develop a theory of stochastic differential equa tions with Fisk-Stratonovich differentials. We begin with some results on the quadratic variation of stochastic processes. Definition. Let H, J be adapted, cadlag processes. The quadratic covari ation process of H , J denoted [H, J] = ( [H, J] t ) t > o , if it exists, is defined to be the adapted, cadlag process of finite variation on compacts, such that for any sequence an of random partitions tending to the identity,
5 Fisk-Stratonovich Integ rals and Differential Equations
271
n T T , + Z ' � H (H lim J H )(Jr;+ , JTt ) San (H, ) = 0J0 + i = [H, J] < Tf: . is the sequence 0 = T0 < T{' < with convergence in ucp, where n A process H in ill> is said to have finite quadratic variation if [H, H] t exists and is finite a.s. , each t > 0. If H , J, and H +J in ill> have finite quadratic variation, then the polarization identity holds: 1 [H, J] = 2 ([H + J, H + J] - [H, H] - [J,J] ) . For X a semimartingale, in Chap. II we defined the quadratic variation of X using the stochastic integral. However Theorem 22 of Chap. II shows every semimartingale is of finite quadratic variation and that the two definitions are lim n�oo
L....t
n--+oo
_
an
· ··
consistent. Notation. For of finite quadratic variation we let denote the con tinuous part of the ( non-decreasing paths of Thus,
H
[H, H]c [H, H]. ) [H, H]t = [H, H] f + O<s , be a semimartingale, and assume exists. denoted J; The Fisk-Stratonovich integral of with respect to is defined to be
H
dX8,
X
H
X,
[H, X]
H8_
o
rJo t Hs- odXs - Jor t H8_dX8 + 21 [H,X]�.
To consider properly general Fisk-Stratonovich differential equations, we need a generalization of Ito's formulas ( Theorems 32 and 33 of Chap. II ) . Since Ito's formula is proved in detail there, we only sketch the proof of this generalization. be an Theorem 18 (Generalized Ito's Formula) . Let X = n-tuple of semimartingales, and let f JR + x 0 x lRn --> lR be such that
(X l , . . . , Xn)
:
(i} there exists an adapted FV process A and a function g such that
t
f(t,w,x) = lo g(s,w,x)dA8, (s, w) g(s, w, x) is an adapted, jointly measurable process for each and J; supxE K j g ( s , w , x) l l dA s l < oo a.s. for compact sets K. t---+
x,
V Stochastic Differential Equations (ii} the function g of (i} is C2 in x uniformly in s on compacts. That is,
272
n
xi ) s,w,x))( ( sup{ Y x i i s 0, t > 0, let A(c, t) be a set of jumps of X that has a.s. a finite number of times and let B = B(c,t ) be such that L s EB (L\X8)2 < c2 , where A U B exhaust the jumps of X on (0, t]. Then ) ,Xtk f(tk, W l:J(tk,w,Xtk+1) k=O t (t ) f(tk,W, Xt ) ,Xtk k, (t k, ,XtJ + L J ) f( k,W,Xtk W + W LJ + l k l k, B k, A Next consider the second term on the right side of
* ,
m- 1
s,
m-1
=
By Taylor's formula, and letting
L\kX denote Xtk+l - Xtk ,
274
V
Stochastic Differential Equations
k,B
2 X (t tk,W,X Lf + Lf k,W,Xtk)(!:l.kX) tk)!:l.k ( � xx x k k 1 "' 2 t X X X tk, - L.. w, + f k, w, Xt ( ( tk )!:l. k ) J k) (!:l.k xx x 2 k R t ,Xt k,w,Xt + l: ( 1) k · + k k,B By Theorems 21 and 30 of Chap. II, the first sums on the right above con to J� fx (s- , w, Xs - )dXs and � J� fxx (s -, w, Xs - )d[X , X]8, re verge in spectively. The third sum converges a.s. to - :2: Ux ( W , Xs - )!:l.Xs + � fxx (s - , w, Xs - )(!:l.X8)2}. By condition (ii) on the function g, we have =
,
A
ucp
s EA
where
S-,
rt(w, c:+) = lim sup rt(w,o). Next we let tend to 0; then rt(w,c:+)[X,X]t tends to 0 a.s., and finally combining the two series indexed by A we see that :2: {f(s-,w,Xs) - f(s- ,w,Xs-) - fx (s-,w,Xs - )!:l.Xs - 21 fxx (s-,w,X8 - )(!:l.X8) 2 } E
o.J.c:
sEA(c:, t )
tends to the series
L {f(s - ,w,Xs) - f(s-,w,Xs - ) - fx (s-,w,Xs - )D.Xs ( I:o< s
w, x
w x
x
s
w x
w,
First assume that for fixed (t, w ) the function f and all its first partials are bounded functions of x. Then by optional stopping at times T- we can assume without loss of generality that f and all its first partials (in x) are bounded functions. Let fk be a sequence of functions on �+ x n x �d which are C2 on �d such that fk and its first partials converge uniformly on �d respec tively to f and its corresponding first partials. Moreover we can take all the fk bounded and Lipschitz continuous with Lipschitz constant c, independent of k. For simplicity take d = 1. Let an be a sequence of random partitions tending to the identity. We write, for a process Z,
Proof.
where
an = {0 = T0 < T{' < . . · < lkn }. Since fk and their first partials converge to f and its first partials uni formly, we have fk - !£ is Lipschitz with constant Eke , which tends to 0 as k, £ tend to oo . Therefore, letting denote an for a given n and writing fk (X) for fk (t, Xt ) , we have w,
T
276
V Stochastic Differential Equations
IST ( !k (X)) - ST ( h(X)) \ < ST (( /k - h)(X)) + 2{ST ( !k (X))ST (( !k - /£ )(X ))} 1 1 2 (*) < (c�" + 2cck£)Sr (X) by the Lipschitz properties. Since fk (X ) and f£(X) are semimartingales, we know ( by Theorem 22 of Chap. II ) that San ( /k (X)) and Sa n ( !£ (X )) converge in ucp respectively to [fk (X), fk (X)] and [h(X), /£ (X)] . By restricting our attention to an interval ( s, t], we then have from ( *) that \ { [fk (X), /k (X)]t - [fk (X), fk (X)]s} - { [f£(X ), !£ (X ) ]t - [ !£ (X ), fl' (X)]s}l < (ck£ + 2cc k£){ [X, X] t - [X, X]s }· Since this is true for all 0 < s < t < oo , we deduce that Therefore d [fk(X), fk (X)] is a Cauchy sequence of random measures, con verging in total variation norm on (0, oo ) , a.s. In addition, by Ito's formula (Theorem 18) and Theorem 29 of Chap. II we have that
from Theorem
23 of Chap. II we also know that ( D.fk (X)) 2 = A[/k (X), fk (X)] .
Therefore
[fk (X), fk (X)]t =
t
1 c::: (s-, w,Xs- )2d [X, X]� + O = Defining :f. Then increase to oo a.s. Also, since the quadratic variation is a path = property, for all < Therefore is of finite quadratic variation, and since by the preceding for = 1) -+
whence the result.
0
Y
Y
Note that the subspace of processes that have a representation = 1, . . . is a finite-dimensional semimartin where X = (X 1 gale is not fixed) and E 1 is a vector subspace of the space of processes of finite quadratic variation. As an immediate consequence of Theorem 19 we have the following extension of Theorem 34 of Chap. II.
f (X (d , Xd),
. . . Xd) , , f C,
Let X be a semimartingale and let f be C2. Then t f(Xt) -J(Xo) = 10+ f'(Xs_)odXs + O<s< L t {f(X.) -f(Xs_) -J'(Xs_) f1Xs}. Proof Note that f' is C 1 , so that f' (X) is in the domain of definition of the F-S integral by Theorem 19. Also by definition we have t{ f (Xs ) odXs = {t J'(Xs )dXs + � [ X]� . !'(X), ' 2 Theorem 20.
h+
By Theorem
h+
19 and polarization we know that
t [J' (X),X]� = 1 f'' (X8 _ )d[X,X] � ,
and thus the result follows by Ito's formula (Theorem
32, Chap. II ) .
0
Theorem 20 has an obvious generalization to the multi-dimensional case. We omit the proof which is an analogous consequence of Theorem 33 of Chap. II.
V
278
Stochastic Differential Equations
Let X = (X l , . . . ,X n ) be an n-tuple of semimartingales, and let f JR_n JR. have second order continuous partial derivatives. Then f (X) is a semimartingale and the following formula holds:
Theorem 21. ---+
:
f(Xt ) - f(Xo) t of ) . n n o = L 1 OX (Xs - o dX� + L {f(Xs) - J(Xs - ) - L oxfi (Xs - )�X�. } . O<s 0). Note that f is C2• We have Then Since
�----'
.
that for t < T,
df(Yt) = 2Y� o dY = 2Y� · a(Yt) dZt = 0. Therefore I Yt l = llxo ll for t < T and thus by continuity T = oo a.s. Corollary. Let B = (B\ . , En )' be n-dimensional Brownian motion, o
a (x)
=
.
I - ��2 , and let X be the solution of .
.
'
Then X is a Brownian motion on the sphere of radius
I x0 II·
D
let
282
V
Stochastic Differential Equations
In Theorem 36 of Sect. 6 we show that the solution X is a diffusion. By Proof Theorem 24 we know that X always stays on the sphere of center 0 and radius ll xo II · It thus remains to show only that X is a rotation invariant diffusion,
since this characterizes Brownian motion on a sphere. Let U be an orthogonal matrix. Then UB is again a Brownian motion, and thus it suffices to show
d( UX)
=
a(UX)
o
d( UB) .
The above equation shows that UX is statistically the same diffusion as is X, and hence X is rotation invariant. The coefficient a satisfies a(Ux)
= Ua(x) U' ,
and therefore
d( UX) = U dX = Ua( X) dB = Ua(X)U'U dB = a( UX) d( U B), o
o
o
o
and we are done.
D
The F-S integral can also be used to derive explicit formulas for solutions of stochastic differential equations in terms of solutions of (non-random) or dinary differential equations. As an example, consider the equation t t + Xo +
Xt =
lo f(X8) dZs lo g(X8)ds, o
Z is a continuous semimartingale, Z0 = 0. (Note that J; g(Xs) ds = so we have not included Ito's circle in this term.) Assume that f J; Cg(Xs)ds, is 2 and that j,g, and ff ' are all Lipschitz continuous. Let u = u(x, z) be the unique solution of au (x , z) = f(u(x, z)) oz u(x, 0) = x. Then tz �� = f' (u(x, z)) ��, and �� (x, 0 ) = 1 , from which we conclude that z ��(x,z) = exp{fo f'(u(x,v))dv }. Let Y = (Yt) t >o be the solution of {Zs Yt = xo + Jto exp{- Jo f'(u(Y8,v))dv}g(u(Y8, Z8))ds g (x, Zs)) is Lipschitz, this would which we assume exists. For example if �� a (x, Zs) where
suffice.
o
x
5 Fisk-Stratonovich Integrals and Differential Equations
283
With the notation and hypotheses given above, the solution X of (*4) is given by
Theorem 25 .
Proof.
Using the F-S calculus we have
t 8u
t 8u
u(yt, Zt) u( xo , 0) + Jo ox (Ys, Zs) o dYs + Jro oz (Ys, Zs) o dZs Zs t t { Xo + J exp { j f'(u(Ys, v))dv} o dYs + J f(u(Y8, Z8)) o dZ8 • o o =
=
Since
0
dYs ds
= exp{ -
{ Zs f (u(Ys, v))dv}g( u(Y8 , Z8)), I
Jo
we deduce
By the uniqueness of the solution, we conclude that
Xt = u(Yt, Zt) ·
0
We consider the special case of a simple Stratonovich equation driven by a (one dimensional) Brownian motion B. As a corollary of Theorem 25 we obtain that the simple Stratonovich equation Xt has a solution
=
Xo +
t
1 f(Xs ) o dEs
Xt = h - 1 (Bt + h(Xo)), where h(x )
=
1 r Jo f(s) ds + C.
The corresponding Ito stochastic differential equation is
By explicitly solving the analogous ordinary differential equation (without using any probability theory) and composing it, we can obtain examples of stochastic differential equations with explicit solutions. This can be useful when testing simulations and numerical solution procedures. We give a few examples. Example. The equation
dXt
=
-
1 a Xtdt + a yI 2 2
has solution Xt = sin (aBt + arcsin(Xo)).
1
-
Xt dBt 2
284
V
Stochastic Differential Equations
Example. The following equation has only locally Lipschitz coefficients and thus can have explosions: for m :f. 1 ,
has solution Xt = (XJ - m - a(m - 1)Bt) ! Example. Our last example can also have explosions: ,
= .
dXt = Xt (1 + X'f)dt + (1 + Xt2 )dBt has the solution Xt = tan(t + Bt + arctan(Xo )). The Fisk-Stratonovich integrals also have an interpretation as limits of sums, as Theorems 26 through 29 illustrate. These theorems are then useful in turn for approximating solutions of stochastic differential equations. Theorem 26. Let H be cadlag, adapted, and let X be a semimartingale. Assume [H, X] exists. Let an = {0 = T0 < Tf < · · · < Tk:, } be a sequence of random partitions tending to the identity. If H and X have no jumps in flH8ilX8 = 0, all t > 0), then common (i. e.,
Lo<s + Hrr> ) (XTn hm i
'
>+I
i+t
Tn X - ;)
equals the F-S integral J� H8_ dX8 in ucp. Proof It follows easily from the definition of [H, X] at the beginning of this o
section that tl[H, X]t = flHtflXt and limn---. oo I.:(Hrz+ , - Hrt ) (XT;''t , X Tn ) = [H, X] - H0X0 . Thus if H and X have no jumps in common we conclude [H, X] = [H, X]c + H0X0 . Observing that ;
the result follows from Theorems 21 and 22 of Chap. II. Corollary 1. If either H or X in Theorem 2 1 is continuous, then
where H0_ = 0, in ucp.
D
5 Fisk-Stratonovich Integrals and Differential Equations
285
X
Corollary 2. Let and Y be continuous semimartingales, and let an = {0 = T0 < TJ' < · · · < TJ:;, } be a sequence of random partitions tending to the identity. Then hm .
n-HXl
1 -(Y L 2 r:> + Yr:•>+ 1 )(XT•:'+ 1 - XT:' ) = 1 " Ys - o dX8, •
'
.
'
0
with convergence in ucp.
Proof
By Theorem 22 of Chap. II, any semimartingale Y has finite quadratic variation. Thus Corollary 2 is a special case of Theorem 26 ( and of Corol lary 1). D
Let H be cadlag, adapted, of finite quadratic variation, and suppose :L:o <s< t I�Hsl < oo a. s., each t > 0. Let X be a semimartingale, and let an = [Tr }o< i 0 we define
Jt =
L �Hs l { l.:lHs l >.s} ·
O< s < t
Then J"' is also a cadlag, adapted process, and it has paths of finite variation on compacts. Theorem 28. Let H be cadlag, adapted, of finite quadratic variation, and let
X be a semimartingale. Assume [H, X] exists. Let an = {�n } be a sequence of random partitions tending to the identity. Let ]"' be as defined above. Then
with convergence in ucp. Proof. Let H"' = H - J"'. Then the jumps of H"' are bounded by
c.
Moreover,
Therefore the first limit at t > 0 is
0 gives the result. That is, since [H"' H "'] t1 1 2 [X ' X] t1 / 2 � l ll Hs"' !lX I "'"'
and letting c tend to
$
and since
0.
. s(Yr,+ 1 - Yr,n )) - f'(Yr,n ) I } . •
Since f' and Y are continuous, Also, on [0, t] ,
F.( ti+ l t i ), where the t i are understood to be in an . Suppose a is continuous on [0, t]. Theorem 30.
L •
.
-
Then
which tends to 0. Therefore
t ' l�� 2:: 1 a(s)ds = ).. 1 a(s)ds. n ' tA
.
Moreover since continuous functions are dense in result ( * ) holds for all a in L 1 ( [0, t] , ds ) .
L 1 ([0, t], ds ) ,
the limiting
5 Fisk-Stratonovich Integrals and Differential Equations
Next suppose
289
H is a continuous, adapted process, and set H>. ,n = L Ht , 1 ( t , , t7 J · '
Then taking limits in ucp we have
and using integration by parts, this equals lim { H>.,n
n
· (XY) - ( H>. ,n X_ ) · Y - (H>.,n y ) · X }.
By Theorem 2 1 of Chap. II, this equals
lim{LHt, (xt 7 yt 7 -xt, yt, ) -LHt,Xt, (Yt7 -yt, ) -LHt,Yt, (xt7 -Xt ' ) } n i i i [X, Y] t = J; J.ds, by the result ( * ) we conclude lim L Ht , (x t7 - x t ' )(Y t7 - yt' ) = lim l: Ht, { [X, Yf7 - [X, Y ] t' } ( ** ) n i n i t>. li� J.ds Ht, t = .A lo H8d [X, Y] 8•
However since
=
2:: 1' '
'
We now turn our attention to the statement of the theorem. Using the Mean Value Theorem we have
1 2:: 1 f (�7 )J-t(d.A)(Xt'+ l - xt' ) '
= L f( Yt J(Xt'+ l t
- x t' ) + L t
(* * *)
fol J-t(d.A)f'(YtJ(�; - YtJ(Xt'+ l - xt, )
{ � (d..\) f � s{ f' (Yt, +s (�>. - yt. ) ) - f' (Yt, )}(�>. -Yt. )(X t '+ l_x t , ). + ""' ' � lo lo '
'
The first sum tends in ucp to J f (Y8 )dX8 by Theorem 21 of Chap. II. The second sum on the right side of ( * * * ) can be written as
V
290
Stochastic Differential Equations
:L •
1 M(d>.)f'(YtJ(yt� - YtJ (Xt� - xt• ) 1 + L fo M (d>.) J' (YtJ (�; - YtJ (xti+l - xt; ) . 1
•
The first sum above converges to
by
(
** ) ,
and the second sum can be written as
K>.,n . X
where
'
K;,n Li f'(YtJ(yt� - YtJ l(t;,t,+ 1J (s ) . =
K>., n
Then limn · X converges to 0 locally in ucp by the Dominated Conver gence Theorem (Theorem 32 of Chap. IV) . Finally consider the third sum on the right side of * * * ) . Let
Fn (w)
Then
(
=
1{ (d>.)F (w)iyt.\ Xt'+ l - ti X l M n ' - Yt, I I
lim '"""' � Jo
n
•
+ s (yt.\' - YtJ ) - f'(YtJ I . IJ'(Yt, t;E.) Fn {L(yt� - yt, ) 2 } 1/2 {L(Xti+l - Xt, )2 } 1/2 1
•
•
= 0 a.s. and the summations stay bounded in probability. This since lim completes the proof. Since Y is continuous, [Y, X ] = [Y, X] c , whence if a = 1/2 we obtain the Fisk-Stratonovich integral. 0
Fn
Corollary. Let X be a semimartingale, Y a continuous semimartingale, and f be C1. Let [X, Y] be absolutely continuous and let = { tf} be a sequence of non-random partitions tending to the identity. Then
an
with convergence in
ucp.
6 The Markov Nature of Solutions
Proof
Let J.L ( d>. ) = c { 1; 2} (d>.), point mass at apply Theorem 30.
291
1/2. Then J AJ.L(d>.) = 1/2, and 0
Note that if Y is a continuous semimartingale and B is standard Brownian motion, then [Y, B] is absolutely continuous as a consequence of the Kunita Watanabe inequality. Therefore, if f is C 1 and an are partitions of [0, t], then
with convergence in probability. 6 The Markov Nature of Solutions
One of the original motivations for the development of the stochastic integral was to study continuous strong Markov processes (that is, diffusions) , as solutions of stochastic differential equations. Let B = (Bt ) t >o be a standard Brownian motion in JR.n . K. Ito studied systems of differential equations of the form t t Xt = Xo + f (s, Xs)dBs + g(s, Xs)ds,
1
1
and under appropriate hypotheses on the coefficients f, g he showed that a unique continuous solution exists and that it is strong Markov. Today we have semimartingale differentials, and it is therefore natural to replace dB and ds with general semimartingales and to study any resulting Markovian nature of the solution. If we insist that the solution itself be Markov then the semimartingale differentials should have independent increments (see Theorem 32); but if we need only to relate the solution to a Markov process, then more general results are available. For convenience we recall here the Markov property of a stochastic pro cess which we have already treated in Chap. I. Assume as given a filtered probability space (!1, :F, (:Ft ) t >O, P) satisfying the usual hypotheses.6
Definition. A process Z with values in JR.d and adapted to IF = (:Ft )t > O is a simple Markov process with respect to IF if for each t > 0 the a-fields :Ft and a{Zu ; u > t} are conditionally independent given Zt . Thus one can think of the Markov property as a weakening of the property of independent increments. It is easy to see that the simple Markov property is equivalent to the following. For u > t and for every f bounded, Borel measurable,
E{ f (Zu) [ :Ft } = E{ f (Zu ) [ a{ Zt} } .
One thinks of this a.S "the best prediction of the future given the past and the present is the same as the best prediction of the future given the present." 6
See Chap. I, Sect. 1 for a definition of the "usual hypotheses" (page 3).
292
V Stochastic Differential Equations
Using the equivalent relation ( * ) , one can define a transition function for a Markov process as follows, for s < t and f bounded, Borel measurable, let
Ps,t (Z. , J ) = E {f (Zt ) IF. } .
Note that if f ( ) = 1 A (x ) , the indicator function of a set A, then the preceding equality reduces to
x
P(Zt E A I Fs) = Ps,t( Zs, l A) · with A, we often write Ps, t(Z8, A) on the right side above.
Identifying 1A When we speak of a Markov process without specifying the filtration of a algebras (.Ft)t >o, we mean implicitly that J1 = a {Z8; s < t } , the natural filtration generated by the process. It often happens that the transition function satisfies the relationship
Ps,t = Pt-s for t > s. In this case we say the Markov process is time homogeneous, and the transition functions are a semigroup of operators, known as the transition semigroup (Pt)t > O · In the time homogeneous case, the Markov property becomes
P(Zt + s E AI.Ft) = Ps(Zt , A).
A stronger requirement that is often satisfied is that the Markov property hold for stopping times. Definition. A time homogeneous simple Markov process is strong Markov if for any stopping time T with P(T < oo ) = 1 , s > 0,
P( Zr+ s E AI.Fr) = P. ( Zr , A)
or equivalently
E{f(Zr+s)IFr} = P, (Zr , f), for any bounded, Borel measurable function f .
The fact that we defined the strong Markov property only for time ho mogeneous processes is not much of a restriction, since if X is an 1R.d-valued simple Markov process, then it is easy to see that the process Zt = (Xt , t) is an JR_d+ 1 -valued time homogeneous simple Markov process. Examples of strong Markov processes (with respect to their natural fil trations of a-algebras) are Brownian motion, the Poisson process, and indeed any Levy process by Theorem 32 of Chap. I. The results of this section will give many more examples as the solutions of stochastic differential equations. Since we have defined strong Markov processes for time homogeneous pro cesses only, it is convenient to take the coefficients of our equations to be autonomous. We could let them be non-autonomous, however, and then with an extra argument we can conclude that if X is the solution then the process Yt = (Xt , t) is strong Markov. We recall a definition from Sect. 3 of this chapter.
6 The Markov Nature of Solutions
Definition. A function f JR.+ x JR.n exists a finite constant k such that :
293
JR. is said to be Lipschitz if there
---->
i f(t, x) - f (t, y)i < kix - yj, each t E JR.+, and t �-----' f(t, x ) is right continuous with left limits, each x E JR.n . f is said to be autonomous if f(t, x) = f(x). (i) (ii)
In order to allow arbitrary initial conditions, we need (in general) a larger probability space than the one on which Z is defined. We therefore define
f! = JR.n
-=t :F�. A random variable Z defined on f! is considered to be extended automatically to n by the rule Z(w) = Z(w), when w = (y, w). We begin with a measurability result which is an easy consequence of S ect 3 of Chap. IV. Theorem 31. Let zi be semimartingales ( 1 < j < d), HX a vector of adapted processes in ][)) for each x E JR.n , and suppose ( x, t, w) �-----' Hf (w) is /3 ® /3+ ® :F .
measurable. 7 Let FJ be functional Lipschitz and for each x E JR.n , xx is the unique solution of
There exists a version of xx such that (x, t, w) �-----' Xf (w) is /3 ® /3+ ® :F measurable, and for each x, Xf is a cadlag solution of the equation. Proof Let X0 (x, t, w) = Hf (w) and define inductively
1
t
x n +l (x, t,w) i = Htx + L FJ (xn (x, · , ·))s-dZ1. j=l 0 The integrands above are in L, hence by Theorem 63 in Chap. IV there exists d
measurable, cadlag versions of the stochastic integrals. By Theorem 8 the processes xn converge ucp to the solution X for each x. Then an application 0 of Theorem 62 of Chap. IV yields the result. 7 B denotes the Borel sets on JR.n ; B+ the Borel sets on JR.+ .
294
V Stochastic Differential Equations
We state and prove the next theorem for one equation. An analogous result (with a perfectly analogous proof) holds for finite systems of equations. Theorem 32. cesses, Z0 = 0, Xo be as in ( * )
Let Z = ( Z 1 , . . . , zd ) be a vector of independent Levy pro and let (fJ ) 1 < j < d, 1 < i < n, be Lipschitz functions. Let and let X be the solution of d
l
x; = Xo + L fj (s�, Xs - )dZ{ j= l 0 Then X is a Markov process, under each pY and X is strong Markov if the fj are autonomous. Proof We treat only the case n = 1. Let T be an lF stopping time, T < oo a.s. Define gr = a{ Z?+u � Z?; u > 0, 1 < j < d } . Then gr is independent of :Fr under P , since the ZJ are Levy processes, as a consequence of Theorem 32 of Chap. I. Choose a stopping time T < oo a.s. and let it be fixed. For u > 0 -y
t
0
define inductively
Y0 ( x, T, u ) = x, d T+u +l yn (x, T, u) = x + L fJ (v�, yn ( x, T, v �))dZt. j=l T Also, let X (x, T, u) denote the unique solution of d rT+u X( x, T, u) = x + 2:. }7 fJ (v� , X (x, T, v -))dZ� , j=l T taking the jointly measurable version (cf., Theorem 31). By Theorem 8 we know that X (x, T, u) is g r measurable. By approximating the stochastic integral as a limit of sums,- we see by induction that yn (x, T, u) is gT measurable as well. Under Px we have X(Xo , T, u ) = X(x, T, u) a.s. , and yn (Xo , T, u ) = yn (x, T, u) Px- a . s . , also. By uniqueness of solutions and us ing Theorem 31, for all u > 0 a.s. X(X0, 0, T + u) = X(X(Xo, 0, T) , T, u).
l
There is no problem with sets of probability zero, due to (for example) the continuity of the flows. (See Theorem 37.) Writing Ex to denote expectation on n with respect to Px , and using the independence of :Fr and gr (as well as of :Fr and g r ), we have for any bounded, Borel function h
Ex {h(X(Xo, 0, T + u)) i:Fr } = E{h(X(x, 0, T + u))i:Fr } 1 IR = E{h(X(X (x, O, T), T, u))}1 IR = j (X(x, 0, T)) 1 IR ,
6 The Markov Nature of Solutions
295
where j(y) = E{h(X(y, T, u)} . The last equality follows from the elementary fact that E{F(H, ·)IH} = f (H) , where f(h) = E{F(h, ·)}, if F is independent of 1i and H is 1i measurable. This completes the proof, since the fact that Ex {h ( X ( Xo , 0, T + u))I.Fr } is a function only of X(x, 0, T) implies that Ex {h (X ( Xo , 0, T + u))I.Fr}
= Ex {h ( X ( Xo, 0, T + u))IX(Xo, 0, T)}.
0
It is interesting to note that Theorem 32 remains true with Fisk-Straton ovich differentials. To see this we need a preliminary result. Theorem 33. Let Z = (Z 1 , . . . , zd ) be a vector of independent Levy pro cesses, Zo = 0. Then [Zi , ZJ ] c = 0 if i 1:- j, and [Zi , Zi ]r = at, where a = E{ [Zi , zi ]i} . First assume that the jumps of each zi are bounded. Then the mo ments of zi of all orders exist (Theorem 34 of Chap. I) , and in particular Mf Zf - E{Zf} is an martingale for each i, with E{Zf} = tE{zn. By independence Mi MJ is also a martingale and hence [Mi, M1 ] = 0 by Corol lary 2 of Theorem 27 of Chap. II. Therefore [Zi , z1]r = [M i , M1 Jr = 0 as well. Next consider A� [Zi , zi ]t = [M i , Mi ]t. It is an immediate consequence of approximation by sums (Theorem 22 of Chap. II) that Ai also has inde pendent increments. Since
Proof =
L2
A� = [M i , M i lr + :L (�M;) 2 , O<s 0
-
-
Y 0 (x, T, u) = x rT+u yn+1 (x, T, u) = x + Jr f(v-, yn (x, T, v-))dZv +
rT+u
g(v - , Yn (x, T, v-))dv,
Jr and let X(x, T, u) denote the unique solution of rT+u rT+u X(x, T, u) = x + J f(v -, X(x, T, v-))dZv + J g(v- , X(x , T, v-))d v. r r
6 The Markov Nature of Solutions
297
Next define gT = a{ZT+ u - ZT ; u > 0} fo r the same stopping time T. As in the proof of Theorem 32 we see that yn (x, T, u) is g T measurable for each n > 1 and that X (x, T, u) is gT measurable as well. Next let h be Borel and bounded. measurable and bounded, and let G E Observe that
gT
E{h ( XT)GIFT} = h(XT)E{G I Fr } = h (XT)E{GIZT} = j (Xr , Zr ) , lR is Borel. Since XT +u = X(XT , T, u) by the uniqueness
where j JR2 of the solution, the theorem now follows from the Monotone Class Theorem (Theorem 8 of Chap. I) . 0 :
---t
Theorem 35 can also be shown to be true with Stratonovich differentials, but the proof is more complicated since the quadratic variation process is an additive functional of Z, rather than a deterministic process (as is the case when Z is a Levy process). For this type of result we refer the reader to Qinlar-Jacod-Protter-Sharpe [34]. Each of Theorems 32, 34 and 35 can be interpreted with X having an arbitrary initial distribution 1-L· Indeed for 1-L a probability measure on (JR, B), define -J-L P on f2 by -J-L P (A) = fn�. P ( A) /-L (dx ) , for A E B 0 F. Then the conclusions of the three theorems are trivially still valid for p�-' , and the distribution of X0 is 1-L · Traditionally the most important Markovian solutions of stochastic dif ferential equations are diffusions. Suppose as given a space (f2, F, ( Ft ) t > o , P) satisfying the usual hypotheses. Definition. An adapted process X with values in JRn is a diffusion9 if it has continuous sample paths and if it satisfies the strong Markov property. A restatement of Theorems 32 and 34 yields the following. Theorem 36. Let B (B l , . . . , Bd ) be a standard Brownian motion on JRd , B 0 = 0, and let (fj) 1 < j < d, 1 < i < n, gi be autonomous Lipschitz x
-
=
functions. Let Xb be as in Theorem 32 and let X be the solution of
Then X is a diffusion. If the coefficients U)h<j o is a true velocity process for X. Using integration by parts we can verify that
6 The Markov Nature of Solutions is an explicit solution for t
e-"'t vt = Vo + 1 =
V. Indeed t 8 Vs ( - o:e "' )ds + 1 e - "' s dv;, t
t
299
t
Vo - o: 1 v;, e-"'8 ds + 1 e - "'8 o:V, ds + 1 e - "' 8 adBs
Vo + lo e- "' 8 adB8 , t
=
and we are done. Since V0 and the Brownian motion B are independent (by construction) , we see that when V0 has a Gaussian distribution then V is a Gaussian process. If o: is negative and we take a 2 = -1/(2o:), then V is a stationary Gaussian process. Example. Consider next the equation t Xs - ds + Bt, (0 < 1) t < Xt = 1 s 0 -
lo
B a standard Brownian motion. For each to, 0 < to < 1, there is a solution which is a diffusion on [0, to]. By the uniqueness of solutions if t 1 < t0 then the solution for [0, t0] agrees with the solution for [0, h] on the interval [0, h]. Thus we have a solution on [0, 1). If we can show that limt__, 1 Xt = 0, then the solution extends by continuity to [0, 1] and we will have constructed a diffusion X on [0, 1] with X0 = X1 = 0, 0 for
known as the Brownian bridge. 1 An application of integration by parts shows that the solution of the Brow nian bridge equation is given by t 1 f < 1). (0 t dEs Xt = (1 - t) < }0 1 - s
Write Xt = f(t)Mt , where f(t) = (1 -t) and Mt = J�(1 -s) - 1 dB8• To see that limt __, 1 Xt = 0, we study Mt, making the change of variables t = u(1 + u) 1 . Then 0 < t < 1 corresponds to 0 < u < oo , and define -
N is clearly a continuous 9u martingale, and moreover [N, N]u = u; hence N
is a standard Brownian motion by Levy's Theorem (Theorem 39 of Chap. II) . It is then easy to show, as a consequence of the Strong Law of Large Numbers, that 11 Nt = 0 a.s. lim t -->oo
t
-
1 0 The Brownian bridge is also known as tied down Brownian motion, and alterna tively as pinned Brownian motion. 1 1 See, for example, Breiman [23, page 265] .
300
V Stochastic Differential Equations
Let g(t,w) = Nt(w)jt so that limt_, o g(C 1 , w) a.s. We then have, replacing t with (1 - t)
=
0 a.s. Thus, limt_, tN1; t = 0 o
lim ( l - t)M1 / ( 2 -t) = 0 a.s., tlim ---> 1 ( l t)N1 ; ( 1 - t) = t---> 1 and therefore limu _, 1 (1 - u)Mu = 0 a.s., and hence limt_,1 Xt = 0 a.s. We now know that X is a continuous diffusion on [0, 1], and that X0 = X1 = 0 a.s. Also X is clearly a semimartingale on [0, 1), but it is not obvious that X is a semimartingale on [0, 1]. One needs to show that the integral f01 11X sJ ds < oo a.s. To see this calculate E { xn, 0 < t < 1, -
By the Cauchy-Schwarz inequality
Therefore
1 l Xs l 1 E { I Xs l } { { E{ }0 1 - s ds} = }0 1 - s ds 1 Js( 1 - s) ds < oo < { ' - }0 1 - s
whence J01 11Xs� ds < oo a.s. Therefore the solution X is a semimartingale on [0, 1]. Finally we remark that one can construct a similar Brownian bridge from any value a to any value b, on an interval of arbitrary finite length T . Using the interval [0, r] one has the equation
Xt = a + with solution
1t b - Xs ds + Bt, 0
T - S
1
0 t}, where ( is the explosion time (cf., Theorem 38}. If (!�) are globally Lipschitz then ( = oo. Let Dk (t, w, x) 8�k X(t, w, x). Then for each process (X(·,w, x),D(·, w, x)) is identically cadlag, and it is the solution Xoftheequations ( D) on [0, ((x, · )) . Proof We will give the proof in several steps. In Step 1 we will reduce the -
problem to one where the coefficients are globally Lipschitz. We then resolve the first system (for X) of (D), and in Step 2 we will show that, given X, there exists a "nice" solution D of the second system of equations, which depends
306
V
Stochastic Differential Equations
continuously on x. In Step 3 we will show that Dt is the partial derivative in of X i in the distributional sense. 14 Then since it is continuous (in x), we can conclude that it is the true partial derivative. Step 1. Choose a constant N. Then the open set {x : ( ( x, w) > N} is a countable union of closed balls. Therefore it suffices to show that if B is one of these balls, then on the set r = {w : \fx E B, ((x, w) > N} the function x X ( t, w, x) is continuously differentiable on B. However by Theorem 38 we know that for each w E r, the image of X as x runs through B is compact in vn with 0 < t < N, hence it is contained in a ball of radius R in JRn, for R sufficiently large. We fix the radius R and we denote by K the ball of radius R of JRn centered at 0. Let
Xk
�--t
E B and 0 < t < N, X(t, w, x) E K}. We then condition on A. That is, we replace P by PA , where PA (A) P(A I A) = P�1��) . Then PA « P, so Z is still a semimartingale with re spect to PA (Theorem 2 of Chap. II) . This allows us to make, without loss of generality, the following simplifying assumption: if x E B then ((x, w) > N, and X(t, w,x) E K, 0 < t < N. Next let h JRn lR be c= with compact support and such that h(x) = 1 if x E K and replace f with fh. Let Z be implicitly stopped at the (constant stopping) time N. (That is, zN replaces Z.) With these assumptions and letting PA replace P, we can therefore assume-without loss of generality that the coefficients in the first equation in (D) are globally Lipschitz and bounded. Step 2. In this step we assume that the simplifying assumptions of Step 1 hold. We may also assume by Theorem 5 that Z E S(f3) for a {3 which will be A = {w
:
:
for x
--+
specified later . If we were to proceed to calculate formally the derivative with respect to of Xi , we would get
Xk
Therefore our best candidate for the partial derivative with respect to the solution of the system
Xk is
1 4 These derivatives are also known as derivatives in the generalized function sense.
7 Flows of SDE: Continuity and Differentiability
D
307
and we can make (Di).byNaturally X(s,w,x), Xs x rewriting the above equation
and let be the matrix explicit this dependence on
=
as
where the summations over now show that =
a
and j are implicit (Einstein convention) . We Fix E !Rn and let is continuous Then
. (D�, . . . , DJ:) i n Dk x, x y Vs(w) Dk(s, w, x) - Dk(s,w, y). V: i 1t { of� x _ of� (Xsy ) JY.Y } dZsa. jx D X ) ( s ks ;::. - ks ;::. - - UXj t UXj 1t { ofUXj ;::. � (XsY- )} Djxks - dzsa. ;::. � (Xsx- ) - ofUXj 1t of� ;::.UXj (XsY- ) { Djxks - - d,ksY- } dZsa. Hl(x, y , k) lot �f�xi (Xf_ )Vj_ dZ� t Hl 1 Vj_dYji, where Yji I::' J; ��: (X�_)dZ::. Note that by Step 1 we know that ��: (X�_ ) is bounded; therefore since za. E S((3), the Yji are in S(c(3) for a constant c. If (3 is small enough, by Lemma 2 (preceding Theorem 7) we have that for each p > 2 there exists a constant Cp ( Z) such that =
=
0
=
0
+
0
+
=
+
=
1
However we can also estimate IIHII sp . If we let as
Ji
=
� � j= 1
{ Oof� (XsX- ) - of� Y ) } jx OX ( Xs - D ks - ' XJ.
.
J
I::' J� J�8dZ';, and therefore by Emery's inequality (Theorem 3) i i Z I I H= , II H I I sp < cpiiJIIsp which in turn implies then Hf = we have that
1
-
IIVIIsp -
-
< Cp (Z) IIJIIsp - .
We turn our attention to estimating II Jll SP . Consider first the terms of� ( X x ) _ of�
;::.ux · s - uX;::. · (XsY- ). J
J
308
V
Stochastic Differential Equations
1, the functions �;:. are Lipschitz in By the simplifying assumptions of takes its values in Therefore Theorem 37 applies, and as we and saw in its proof (inequality * * * )) we have that
xx
K,
Step
K.
J
( of� (Xsx ) _ of� (XsY ) x Y < < K(p, Z) l x - Y l · X X K I I s 2p uX · ux · s2p Next consider the terms D�� . We have seen that these terms are solutions of the system of equations V,ktx = k 1t Dtksx- of� (Xsx- ) dZsa. uxg �
�
J
J
61 +
�
0
and therefore they can be written as solutions of the exponential system
t DJktx = oki 1 Dksex- d J x with YJsx = J; �;� (X�_ )dZ�. As before, by Lemma 2, +
0
Y.
ls '
J and using the Cauchy-Schwarz inequality gives I J I sp < l ��: (x:_) - ��: (Xf_) l s2p i D�x l s2p < C2p(Z) I x - Yl l ,
Recalling the definition of
A
which in turn combined with previous estimates yields
II V IIsp < C(p, Z) l x - Y l ·
Since V was defined to be that (with >
Vs(w) = Dk(s,w,x) - Dk(s,w, y ), we have shown
p n) E { sup I Dk(s, w, x) - Dk(s, w, y ) I P } < C(p, Z)PI I x - Y I P , s t
where B2n denotes the Borel sets of IR2n . Let .>. be normalized Lebesgue mea sure of Finally define P = >. x P. We can as in the proof of Step 2 assume that Z E S((3) for (3 small enough and then, S2 , --+oo (X(ur), D(ur) = (X, D) in rlim by Theorem 16. Therefore there exists a subsequence rq such that
K.
M = L: sup jj (X(rq ) , D(rq ) ) - (X, D) I I q=l t 00
E L 1 ( dP).
310
V Stochastic Differential Equations
The function M = M(w, x) is in L 1 (>. x P) , and therefore for P-almost all w the function x f---t M(w, x) E L 1 (d>.). For w not in the exceptional set, and t fixed it follows that
= (X, D) (X(rq), D(rq)) qlim oo ---+
>. a. e. Further, it is bounded by the function M(w, · )+I I (X (t, w, · ) , D( t, w, · )) I
which is integrable by hypothesis. This gives convergence in the distributional sense, and the proof is complete. D
39 as a theorem. Theorem 40. Let Z be as in {H1) and let the functions (!�) in {H2) have locally Lipschitz derivatives up to order N, for some N, 0 < N < oo . Then there exists a solution X (t, w, x) to We state the following corollary to Theorem
Xf = Xi +
rt
f�(Xs - )dZ';, 1 < i < n, L o a = 1 J° m
which is N times continuously differentiable in the open set {x : ((x, w) > t}, where ( is the explosion time of the solution. If the coefficients (f�) are globally Lipschitz, then ( = oo . Proof. If N = 0, then Theorem 40 is exactly Theorem 38. If N = 1 , then Theorem 40 is Theorem 39. If N > 1 , then the coefficients of equations (D) have locally Lipschitz derivatives of order N - 1 at least. Induction yields (X, D) E c N - 1 , whence X E CN. D
Note that the coefficients (f�) in Theorem 40 are locally Lipschitz of order N if, for example, they have N 1 continuous partial derivatives; that is, if f� E cN + l (JRn), for each i and a , then (!�) are locally Lipschitz of order N.
+
8 Flows as Diffeomorphisms: The Continuous Case
In this section we will study a system of differential equations of the form
xj
= Xi + L= 1 lot° F� (X)s- dZ';, m
1 < i < n,
a semimartingales Z"' are assumed to have continuous paths
with where the Zo = 0. The continuity assumption leads to pleasing results. In Sect. 10 we consider the general case where the semimartingale differentials can have jumps. The flow of an equation such as ( *) is considered to be an 1Rn-valued function
oo l x ll s< N Proof By Theorem 43 the equality Theorem 45 . x m
m,
is valid for all r E JR. 0 let yx = For Theorem 41.) Then
I xx - X0 1 - 1 . (Note that yx is well-defined by ( ) I Yx l = l x l - l £( A - I(x ,O) ) I Yx - YYI < IIX x - XYIIIIXx - X0 I -1 I X Y - X0 l -1 = l x - Yl l l x i -1 I Y I - 1 £( A 1 (x, y) )£( A - I ( x, 0) )£( A - I (y, 0) ) . Define y oo = 0. The mapping x � x l x l -2 inspires a distance d on JRn \ { 0} b d(x, Y ) = lllxi-i i iyllI I " I ndee d ' x Y ( )2 x X Yl l l Y l l x l 2 - I I Y I 2 II 2 = l x i i Y I x i=
***
Y
By Holder ' s inequality we have that
I £(A 1 (x , y) )£(A - l (x , O)£(A-l (y, 0)) llsr < 1 £ ( A l (x, y)) I -s3r 1 £ ( A-l (x, o)) ll -s3r 1 £ (A - l (y, o)) I -s3r
and therefore by the corollary to Theorem 43 we can find a sequence of stop ping times (Te)e>I increasing to oo a.s. such that there exist constants Ce with ( using * * *))
(
Next set
318
V
Stochastic Differential Equations A
yx =
{ yyxoollx ll-02
'
= '
0 < llxll < oo , llxll = 0.
Then II (Y x - YY ) Tt llsr < Cf llx - Y llr on JR.n , and by Kolmogorov's Lemma (Theorem 72 of Chap. IV) , there exists aA jointly continuous version of (t, x) � �x , on JR.n . Therefore lim l x ii ->O yx exists and equals 0. Since 0 (Yx ) -1 = II X x ll x ll -2 - X0 ll, we have the result.
Let za be continuous semimartingales, 1 < a < m, and F be an n x m matrix of process Lipschitz operators. Let X be the solution of ( * ) Let cp : JR.n ....-. ]Rn be the flow cp(x) = X(t, w, x). Then for almost all w one has that for all t the function cp is surjective and moreover it is a homeomorphism from lR. n to lR.n . Proof. As noted preceding Theorem 41, the flow cp is continuous from JR.n to vn , topologized by uniform convergence on compacts; hence for a.a. w it is continuous from lR.n to lR.n for all t. The flow cp is injective a.s. for all t by Theorem 44. Next observe that the image of lR.n under cp, denoted cp(lR.n ) , is closed. Indeed, let cp(JR n ) denote its closure and let y E cp (JRn ) . Let (x k ) denote a sequence such that limk -> oo cp (xk ) = y . By Theorem 45, lim supk ->oo ll x k I < oo, and hence the sequence (x k ) has a limit point x E JR. n . Continuity implies cp( x) = y , and we conclude that cp(lR.n ) = cp(JRn ) ; that is, cp(lR.n ) is closed. Theorem 46.
.
Then, as we have seen, the set {x k } is bounded. If Xk does not converge to x, there must exist a limit point z =I x. But then cp(z) = y = cp (x), and this violates the injectivity, already established. Therefore cp- 1 is continuous. Since cp is a homeomorphism from lR.n to cp (lR.n ), the subspace cp(JR. n ) of lR.n is homeomorphic to a manifold of dimension n in lR.n ; therefore by the theorem of the invariance of the domain (see, e.g., Greenberg [84, page 82]), the space cp(lR.n ) is open in lR.n . But cp (lR.n ) is also closed and non-empty. There is only one such set in lR. n that is open and closed and non-empty and it is 0 the entire space JR.n . We conclude that cp ( lR.n ) = JR.n . Comment. The proof of Theorem 46 can be simplified as follows: extend to the Alexandrov compactification JR� = lR. n U { oo } of lR.n to cp by cp (x) =
{
cp( x ) , oo ,
cp
X E JRn ' X = 00 .
Then cp is continuous on JR� by Theorem 45, and obviously it is still injective. Since JR.� is compact, cp is a homeomorphism of JR.� onto cp(JR.� ). However JR.� is topologically the sphere sn ' and thus it is not homeomorphic to any proper subset (this is a consequence of the Jordan-Brouwer Separation Theorem (e.g. , Greenberg [84, page 79]). Hence cp(JR�) = JR.� .
8 Flows as Diffeomorphisms: The Continuous Case
319
We next t urn our attention to determining when the flow is a diffeomor phism of JR.n. Recall that a diffeomorphism of JR.n is a bijection (one to one and onto) which is coo and which has an inverse that is also coo. Clearly the hypotheses on the coefficients need to be the intersection of those of Sect. 7 and process Lipschitz. First we introduce a useful concept, that of right stochastic exponentials, which arises naturally in this context. For given n, let Z be an n x n matrix of given semimartingales. If X is a solution of
n x n matrix of semimartingales and I is the identity matrix, then X = £( Z ), the (matrix-valued) exponential of Z. Since the space of n n matrices is not commutative, it is also possible to consider right stochastic integrals, denoted where X is an
x
where Z is an n x n matrix of semimartingales and H is an n x n matrix of (integrable) predictable processes. If denotes matrix transpose, then 1
(Z : H) = (H' · Z')', and therefore right stochastic integrals can be defined in terms of stochastic integrals. Elementary results concerning right stochastic integrals are collected in the next theorem. Note that J Y_dZ and [Y, Z] denote n x n matrix-valued processes here.
Let Y, Z be given n x n matrices of semimartingales, H an n x n matrix of locally bounded predictable processes. Then, (i) Y"tZt - YoZo = f� Ys _ dZs + f� (dYs)Zs - + [Y, Z]t i (ii) [H · Y, ZJ = H · [Y, ZJ; and (iii) [Y, Z : HJ = [Y, ZJ : H. Moreover ifF is an n n matrix of functional Lipschitz operators, then there exists a unique n n matrix of ]])) -valued processes which is the solution of
Theorem 47.
x
x
Proof
The first three identities are easily proved by calculating the entries of the matrices and using the results of Chap. II. Similarly the existence and uniqueness result for the stochastic integral equation is a simple consequence D of Theorem 7. Theorem 4 7 allows the definition of the right stochastic exponential.
V
320
Stochastic Differential Equations
Definition. The right stochastic exponential of an n x n matrix of semi is the ( unique ) matrix-valued solution of the denoted martingales equation
t: R (Z), t Xt = I + lo (dZs)Xs - ·
Z,
We illustrate the relation between left and right stochastic exponentials in the continuous case. The general case is considered in Sect. 10 ( see Theo = rem 63 ) . Note that Theorem 48. Let be an n x n matrix of continuous semimartingales with
t:R (Z) E(Z ') ' . Z R and t: ( -Z + [Z, Z]) are inverses; that is, E(Z)t: R (-Z + Zo 0.=Then E(Z) [Z, Z]) I. Proof Let U = E(Z) and V = t:R (- Z + [Z,Z]). Since U0V0 = I, it suffices to show that d (Ut vt) = 0, all t > 0. Note that dV = ( -dZ + d[Z, Z])V, (dU)V (UdZ)V, and d[U, V] Ud[Z, V] = -Ud[Z,Z]V. =
=
=
Using Theorem 47 and the preceding,
d(UV) UdV + (dU)V + d[U, V] U( -dZ + d[Z, Z]) V + UdZV - Ud[Z, Z]V =
=
=
0,
0
and we are done.
The next theorem is a special case of Theorem 40 (of Sect. 7 ) , but we state it here as a separate theorem for ease of reference. Theorem 49. Let be continuous semimartingales with = 0,
1 , , zm ) zg (Z 1 < i < m, and let (!�), 1 < i < n, 1 < a < m, be functions mapping to with locally Lipschitz partial derivatives up to order N, 1 < N < and bounded first derivatives. Then there exists a solution X ( t, w, x) to •
•
•
JRn
oo,
JR,
such that its flow r.p : x ---> X(x, t, w ) is N times continuously differentiable on JRn . Moreover the first partial derivatives satisfy the linear equation ( l < i < n). where 8f. is Kronecker's delta.
9 General Stochast ic Exponentials and Linear Equations
321
Observe that since the first partial derivatives are bounded, the coefficients are globally Lipschitz and it is not necessary to introduce an explosion time. Also, the value N = oo is included in the statement. The explicit equation for the partial derivatives comes from Theorem 39. Let denote the n x n matrix-valued process
D
( 0)
The process by
D is the right stochastic exponential [R (Y), where Y is defined f � j � i dzsa . dYs = (X 8 ) a= l a
L..J ax J
,
.
Combining Theorems 48 and 49 and the above observation we have the im portant following result . Theorem 50. With the hypotheses and notation of Theorem 49, the matrix a.s. is non-singular for all t > 0 and x E Theorem 5 1 . Let be continuous semimartingales and let (!�),
Dt
(Z 1 , . . . , zm )
JRn ,
be functions mapping JR n to JR, with partial derivatives of all orders, and bounded first partials. Then the flow of the solution of
1 < i < n , 1 < a < m,
t m r x: = Xi + L Jo f�(Xs)dZ';, 1 < i < a= 1 0 is a diffeomorphism from JRn to JRn . Proof. Let r.p denote the flow of X. Since (f�h< i< n, l 1. Therefore all of the "large" jumps of Let xt denote the solution of *) driven by the semimartingales Outside of the interval the solution is
T0 T1 T2 z a ,J = ( "' 1 ,j a z za ,j ( (Tj - l, T1 )
(Z"'h Ti .
Next define the linkage operators
using vector and matrix notation. We have the following obvious result. Theorem 57. The solution of (*) is equal to, for 1 be a sequence of points in !R n such that lim i-+oo g(xi ) = y E !Rn . We will show that y E g(!Rn ). Let Xi = tivi , where t i > 0 and I vi I = 1. By choosing a subsequence if necessary we may assume that Vi converges to v E sn , the unit sphere, as i tends to oo. Next observe that the sequence (t;)i> l must be bounded by condition (i) in the theorem: for if not, then t i = I xi I tends to oo
along a subsequence and then llg(xi k ) ll tends to oo by (i), which contradicts that limi-+oo g(xi ) = y. Since (ti)i> l is bounded we may assume limi-+oo ti = t o E !Rn again by taking a subsequence if necessary. Then limi-+oo xi = tov, and by the continuity of g we have y = limi-+oo g(xi ) = g(tov). To show g is injective (i.e., one-to-one) , we first note that g is a local homeomorphism, and moreover g is finite-to-one. Indeed, if there exists an infinite sequence (xn ) n > l such that g(xn ) = Yo , all n, for some yo , then by condition (i) the sequence must be bounded in norm and therefore have a cluster point. By taking a subsequence if necessary we can assume that Xn tends to x (the cluster point), where g(xn) = yo all n. By the continuity of g we have g(x ) = y0 as well. This then violates the condition that g is a local homeomorphism, and we conclude that g is finite-to-one. Since g is a finite-to-one surjective homeomorphism, it is a covering map. 16 However since !Rn is simply connected the only covering space of !Rn is !Rn (the fundamental group of !Rn is trivial). Therefore the fibers g-1 (x) for x E !Rn 0 each consist of one point, and g is injective. ,
The next step is to show that the functions x �---' X� (x, w) and x �---' x� + 1 ( x, w) of Theorem 58 satisfy the two conditions of Theorem 59 and are J
16
For the algebraic topology used here, the reader can consult, for example, Munkries [183, Chapter 8].
10 Flows as Diffeomorphisms: The General Case
331
thus diffeomorphisms. This is done in Theorems 62 and 64. First we give a result on weak injectivity which is closely related to Theorem 41.
Let za be semimartingales, 1 < a < m with Z0 = 0, and let F be an n x m matrix of process Lipschitz operators with non-random Lipschitz constant K. Let Hi E IDl, 1 < i < n (cadlag, adapted). If'f:.r;: 1 I Za l -w"' < for > 0 sufficiently small, then the flow of the solution of Theorem 60.
c: ,
c;
is weakly injective. 1 7 Proof Let x, y E !Rn , and let xx , XY denote the solutions of the above equa tion with initial conditions x, y, respectively. Let u = x - y, U = x x - XY, and
V = F(X x) _ - F(XY)_ . Then V E L and \ V I < K I U- 1. Also,
!::.. Us 'f:.a V8a t::..zr; li t::.. Us I < l �a l l t::..zr; I < CI I Us- l c < 21 1 \ Us - 1 if c is small enough. Consequently 1 \ Us l > � I \ Us - 1 · Define T = inf{t > 0 : 0}. Then Ut - =/= 0 on [0, T) and the above implies Ut =/= 0 on [0, T) as Utwell.- =Using Ito's formula for f(x) = log l x l , in the proof of Theorem 41 we have
= and moreover (using the Einstein convention Therefore to leave the summations implicit)
as
For
s fixed, let
1 7 We are using vector and matrix notation, and the Einstein convention on sums.
The Einstein convention is used throughout this section.
332
V Stochastic Differential Equations
so that the last sum on the right side of equation ** ) can be written Js . By Taylor's Theorem
(
l.::o <s 0 sufficiently small, then for r E IR there exist uniformly locally bounded predictable processes H"' (x,y ) and K"'·f3 (x ,y), which depend on r, such that II X x - XY II'" = l x - YI I '"E (A,. (x, y)) where x x is the solution of on
_
1
The semimartingale A,. is given by
H'"'
10 Flows as Diffeomorphisms: The General Case
333
Lo <s 0. Corollary. Let Z be a square matrix of semimartingales. If I Z I w"' < E for E > 0 sufficiently small, then [R (Z)t is invertible for all t > 0. Proof If I Z I H = < c then the jumps of Z are bounded by E > 0 and therefore the process of Theorem 63 is a well-defined semimartingale. Theorem 64. Let (Z"'h< a < m be semimartingales with Z0 = 0, and let f be a matrix of coefficients satisfying Hypotheses (H2) and (H3). Let X be the -
·
=
vt
W
0
0
unique solution of
The Jacobian matrix Dic(t, x ) = 8�k Xi (t, x ) is invertible for each t > 0 provided I Z I H= < E for sufficiently small E > 0. w,
w,
10 Flows as Diffeomorphisms: The General Case Proof By Theorem 39 (in Sect. 7) the Jacobian matrix
stochastic exponential equation
337
D satisfies the right
t {}fi 1 Dt = I + ( , (Xs- )dZ�)Ds- , 0
{)
Xk
and the matrix semimartingale differential ��: (X8_ )dZ'; satisfies the hy potheses of the corollary of Theorem 63, whence the result. D Before stating the principal result of this section, we need to define two subsets of !Rm ; recall that under Hypotheses (Hl ), (H2), and (H3), that Z ( Z, h < n < m is a given m-tuple of semimartingales and that f(x) = (f�(x)) is an n x m matrix of coo functions. Let =
V = {z E !Rm H(x) = X + f(x)z is a diffeomorphism of !Rn } I = {z E !Rm H(x) = X + f(x)z is injective in !Rn }. :
:
Clearly V c I.
Let Z and f be as given in Hypotheses {Hl), (H2), and (H3), and let X be the solution of t Xt = X + f(Xs - )dZ8•
Theorem 65.
1
The flow of X is a.s. a diffeomorphism of !Rn (resp. trajectories of X from different initial points a.s. never meet) for all t if and only if all the jumps of Z belong to V (resp. all the jumps of Z belong to I). Proof Recall the processes X? (x, w) and X;' +1 (x, w) defined in Theorem 57, and the linkage operator Hi (x) = x + f(x)Ll Zr; , define� immediately preced J
ing Theorem 57. By hypothesis the linkage operators flJ (x) are clearly diffeo morphisms of !Rn (resp. injective), and by Theorems 62 and 64, Hadamard's conditions are satisfied (Theorem 59) , and therefore the functions x X? (x, w) and x X;' + l (x, w) are diffeomorphisms of !Rn if c > 0 is taken small enough in the definition of the stopping times (TJ )j > 1 , which it is always possible to do. Therefore by Theorem 58 the flow c.p x Xt(x, w) is a.s. a diffeomorphism of !Rn for each t > 0. The necessity is perhaps the more surprising part of the theorem. First observe that by Hadamard's Theorem (Theorem 59) the set V contains a neighborhood of the origin. Indeed, if z is small enough and x is large enough then llf(x)zll < llxll /2 since f is Lipschitz, which implies that II H(x)ll > llxll/2 and thus condition (i) of Hadamard's Theorem is satisfied. On the other hand H'(x) = I + f'(x)Z is invertible for all x for llzll small enough because f'(x) is bounded; therefore condition (ii) of Hadamard's Theorem is satisfied. Since f(x) is coo (Hypothesis (H3)), we conclude that V contains a neighborhood of the origin. r-t
J
r-t
:
-+
V Stochastic Differential Equations
338
To prove necessity, set
F1 = {w : 3 > 0 with �Z8(w) E vc} , F2 = {w : 3 > 0 with �Z8(w) E Ic} . Suppose P(F1 ) > 0. Since V contains a neighborhood of the origin, there exists an c > 0 such that all the jumps of Z less than c are in V. We also take c so small that all the functions x f---> Xj.i ( x) are diffeomorphisms as soon as the linkage operators Hk are, all k < i. Since the jumps of Z smaller than c are in V, the jumps of Z that are in vc must take place at the times Ti . Let Ai = {w : �Zri E V, all i < j, and �Zri E vc} . Since P(F1 ) > 0, there must exist a j such that P ( Ai ) > 0. Then for w E Ai , w) is a diffeomorphism, but Hi (x, w) is not a diffeomorphism. xLetf--->woX E _A1(x,and to T1 (wo). Then x f---> Xt0 (x , wo) is not a diffeomorphism, s s
Ti
and therefore
=
P {w 3 t such that x ----> Xt (x, w) is not a diffeomorphism} > 0, :
and we are done. The proof of the necessity of the jumps belonging to I to D have injectivity is analogous. Corollary. Let Z and f be as given in Hypotheses (Hl), (H2), and (H3), and let X be the solution of
Xt = X +
t
1 f(Xs - )dZ8•
Then different trajectories of X can meet only at the jumps of Z.
Proof
We saw in the proof of Theorem 65 that two trajectories can intersect only at the times T1 that slice the semimartingales za into pieces of H= norm less than c. If the za do not jump at Tio for some ]o, however, and paths of X intersect there, one need only slightly alter the construction of Tj0 (cf. , the proof of Theorem 5, where the times T1 were constructed) , so that Tjo is not included in another sequence that c-slices (Zn h< a < m , to achieve a contradiction. (Note that if, however, (Zn h< a < m has a large jump at Tj0 , D then it cannot be altered.) 1 1 Eclectic Useful Results on Stochastic Differential
Equations
We begin this collection of mostly technical results concerning stochastic dif ferential equations with some useful moment estimates. And we begin the moment estimates with preliminary estimates for stochastic integrals. The first result is trivial, and the second is almost as simple.
11 Eclectic Useful Results on Stochastic Differential Equations
Lemma. For any predictable (matrix-valued) process and for 0 < t < 1 , we have
339
H and for any p > 1 ,
Since 0 < t < 1 , we have that ds on [0, 1] is a probability measure, and since f(x) = xP is convex for p > 1 , we can use Jensen's inequality. The result then follows from Fubini's Theorem. 0
Proof
Lemma. For any predictable (matrix-valued) process sional Brownian motion B, and for any p > 2, and for 0 a constant Cp depending only on p such that
H and multidimen < t < 1 , there exists
The proof for the case p = 2 is simply Doob's inequality; we give it for the one dimensional case.
Proof
where the last equality is by Fubini's Theorem. For the case Burkholder's inequality (see Theorem 48 of Chap. IV,
p>
2, we use
where the last inequality follows from Jensen's inequality (recall that p > 2 so that f(x) = xP/ 2 is a convex function, and since 0 < t < 1 we have used that ds on [0, 1 J is a probability measure) and Fubini's Theorem. 0 Theorem 66. Let Z be a d-dimensional Levy process with E{ll Zt I I LP } for 0 < t < 1 , H a predictable (matrix-valued) process and p > 2. Then exists a finite constant Kp such that for 0 < t < 1 we have
< oo there
Proof We give the proof for the one dimensional case. 1 8 Since the Levy pro cess Z can be decomposed Zt = bt + aBt + Mt , where M is a purely dis continuous martingale (that is, M is orthogonal in the L 2 sense to the stable subspace generated by continuous L 2 martingales) , it is enough to prove the inequality separately for Zt = bt, and Zt = aBt > and Zt = Mt . The preceding
two lemmas prove the first two cases, so we give the proof of only the third. 18
An alternative (and simpler) proof using random measures can be found in [105] .
V Stochastic Differential Equations
340
Cp
vary Kp(·) < < 2k + 1 .
In the computations below, the constants and the functions from line to line. Choose the rational number k such that 2 k p Applying the Burkholder-Gundy inequalities 1 9 for p > 2 we have
Set
= oo. 2 } vM(dx) Msl aM := E{[M, M] I} = E{L J i x l 2 I � < s< l Since [M, M] is also a Levy process, we have that [M, M] t - a M t is also a martingale. Therefore the inequality above becomes p p t tr /2 r E{ Jo HsdMs } 2, we decompose it as Zt = bt + cBt + Mt where B is a standard Brow nian motion and M is a martingale independent from B and L2 orthogonal to all continuous martingales. We then define
which is finite since by assumption E{IZtiP} < oo and p > 2. Here we consider the multidimensional case: both Z and the solution x x of equation ( * ) are assumed to be d-dimensional, where d > 1 . 2 1 We let x: · ( k ) represent a k-th derivative of the flow of xx . Theorem 70. For k > 1 assume that cr is differentiable k times and that
all partial derivatives of cr of order greater than or equal to one are bounded. Assume also that Z has moments of order kp with p > 2. With the nota tion defined right before this theorem, we have that there exists a constant K(k,p, cr, 'T/p) such that for 0 < s < 1, E { sup 1 x; , ( k) I P} < K. 8
We recall from Theorem 39 that x x together with the unique solution of the system of equations
Proof
xx, ( k)
constitute
(D) as long as k > 2, and with a slightly simpler formulation if k = 1 . ( Note that this is the same equation (D) as on page 305.) We can write the second equation above in a slightly more abstract way
Xtx ,(k)
1 'Vcr(Xx )Xx,(k) dZ t
=
0
8-
s-
8
if k ;:::: 2; and if k = 1 the equation is the simpler 21
One could also allow the dimensions of Z and xx to be different, but we do not address that case here.
344
V Stochastic Differential Equations
t
xx ,( l )
=
I +
d
10t 'Vcr(Xx )Xsx-,( l) dZs . s-
We next observe that the components of Fk , i (x< 1 l, . . . , x< k -i + l ) ) are sums of terms of the form
k -i+ l II j =l
C>j II r=l
where
L jai J
=
( ** )
k,
and where xU ) , l is the l-th component of xU) E 1Rd1 , and an "empty" product equals 1. We thus want now to prove that
t E{sup 1 'Vi cr(Xs'x- )F t
p
· (Xsx-,( l) ' . . . ' Xsx-,( k -i+ l ) )dZs } 0 almost surely. Let T = inf {t > 0 : Xt = 0}. Then P(T < ) = 0. In words, if the coefficient of the equation is of the form xcr (x) with cr ( x) continuous on [0, ) then the solution X stays strictly positive in finite time if it begins with a strictly positive initial condition. We also remark that in oo
oo ,
the applied literature the equation is often written in the form
= x �x) , but Note that if cr(x) is any function, one can then write we then need the new function o-�x) to be continuous, well-defined for all nonnegative x, and we also need for a unique solution to exist that has no explosions. This of course amounts to a restriction on the function Since being Lipschitz continuous is a sufficient condition to have a unique and non-exploding solution, we can require in turn for to be both bounded and Lipschitz, which of course implies that xcr(x) is itself Lipschitz.
cr(x) ( ) ""
xcr(x)
cr
cr.
Proof of Theorem 71. Define Tn = inf{t > 0 : Xt = 1/n or Xt = Xo V n}, and note that P ( Tn > 0) = 1 because P(Xo > 0) = 1 and Z is continuous.
Using Ito's formula up to time Tn we have
cr
and since is assumed continuous, it is bounded on the compact set [1/n, n] , say by a constant c. Since the stopping times Tn j T we see that on the event {T < oo} the left side of the above equation tends to oo while the right side 0 remains finite, a contradiction. Therefore P (T < oo ) = 0.
Z
Theorem 71 has an analogue when is no longer assumed to be continuous. Theorem 72. Let be a semimartingale and let be a bounded, continuous
cr Z function such that there exists a unique non-exploding solution of t Xt = Xo + 1 cr(Xs_)X8_dZ8 with Xo > 0 almost surely. If in addition we have I� Zs I < l �li: almost surely for all s > 0 for some c with 0 c 1, then the solution X stays positive a.s. for all t > 0.
1
En
o
Z
(Z
un (
Eo
t { Vt = 1 +
n t
v;,_d rJ n Ys
for T < < Tn + I · Show that the (unique) solution of the equation
t
1 IXs- l dZs L::n ->o Xf1[Tn,Tn+ d(t), where xr Xrn un (En Z)t, for Tn < Xt = 1 +
= is given by Xt = < Tn + l · Exercise 11. Show that if M is an L2 martingale with 6.M8 > -1, and (M, M)00 is bounded, then £(M)00 > 0 a.s. and it is square integrable. (See [14 7].) Exercise 12. Let M be a martingale which is of integrable variation. Suppose 16.Ms I is bounded. Show that also that the compensator of the process the local martingale E(M) is also of integrable variation. Exercise 13. Let be a semimartingale with decomposition = L + V where L is a local martingale with 6.L8 of. - 1 for all finite s , and V has paths of finite variation on compacts. Let
t
L::s 1 } and also Xt = Zt - Jl. O<s< t Then X is also a Levy process, and since X has bounded jumps it is in LP for all > 1, and in particular E{Xl} < each t > 0. Let lHl(Xl) denote IF expanded by the adjunction of the random variable X1 . Let = lHl(X l ) the filtration generated by IHI (Xl ) and lHl(J[). Then lHl(Z1 ) But lHl(Jf), a semimartingale on lHl(Xl), and since J 1 is independent of X we have Xby isTheorem 2 that X is a (JK, P) semimartingale. Therefore by Stricker's Theorem, X is an (lHl, P) semimartingale; and since Jl is a finite variation process adapted to lHl we have that Z is an (lHl, P) semimartingale as well. D The most important example of the above theorem is that of Brownian motion, which was Ito's original formula. In this case let lHl = lHl(B1 ), and we have as a special case the lHl decomposition of Brownian motion: t B� - Bs t B� - Bs t B� - Bs f f Bt = (Bt - h - s ds) + }0 - s ds = f3t + h - s ds. Note that the martingale f3 in the decomposition is continuous and has [tion. /3, f3]tA=simple t, which by Levy's theorem gives that f3 is also a Brownian mo calculation gives Ito's original formula, for a process H which is IF predictable: t t t 1 1 B1 o HsdBs = B1 1o Hsdf3s + B1 o Hs B11 -- sBs ds 1t BlHsd/38 + 1t B1 Hs B11 -- Bs s ds 0t 0 = lo B1 HsdBs and are all independent, we conclude that M is an martin Since Y, gale. Since M is adapted to lHl, by Stricker's Theorem it is also an !HI martingale, and thus is an semimartingale. Finally we drop all integrability assumptions on We let
p
oo,
=
lK
C lK.
V
2 Initial Expansions
359
where since the random variable B 1 is 1t0 measurable, it can be moved inside the stochastic integral. We can extend this theorem with a simple iteration; we omit the fairly obvious proof. Corollary. Let Z be a given Levy process with respect to a filtration IF, and let 0 = to < t 1 < < tn < oo. Let lHl denote the smallest filtration satis fying the usual hypotheses containing IF and such that the random variables Zt1 , , Ztn are all 1t0 measurable. Then Z is an lHl semimartingale. If we have a countable sequence 0 = to < i t < < tn < , we let T = supn tn , with lHl the corresponding filtration. Then Z is an lHl semimartingale on [0, T ) . ·
•
•
·
·
•
·
·
·
·
·
·
We next give a general criterion (Theorem 5) to have a local martingale remain a semimartingale in an expanded filtration. (Note that a finite varia tion process automatically remains one in the larger filtration, so the whole issue is what happens to the local martingales.) We then combine this theo rem with a lemma due to Jeulin to show how one can expand the Brownian filtration. Before we begin let us recall that a process X is locally integrable if there exist a sequence of stopping times (Tn ) n >l increasing to oo a.s. such that E{IXrn l {Tn >O} I} < oo for each n. Of course, if Xo = 0 this reduces to the condition E{IXrn f} < oo for each n.
Let M be an IF local martingale and suppose M is a semimartin gale in an expanded filtration JHI. Then M is a special semimartingale in JHI. Proof. First recall that any local martingale is a special semimartingale. In particular the process Mt = sups 0, and let J = 1A, Jt = E{ J l:Ft } , the cadlag version of the martingale. Let j = inft Jt . Then j > 0 on { J = 1}. We have a.s.
Proof
Consider next by Jensen's inequality, and this is equal to (E{1A l:Ft} - 11(0, u]) + , where 11 is the law of Rt . Continuing with ( *) we have
where
nP (A n ) > E{1An 1 Rsasds} = E{ 1 1An R8a8ds} 00 00 E{1A n Rsl :Fs }a . ds} > E{ 1
oo ER{ (1 /\ plies
I Hn , x 1 . -ER{(l l\ i Hn, x Xu l)} Ep{(1 1\ I Hn, x · Xui )qu (·, x)}
0 = hm ---+ oo C
·
n
=
qu (-, x) E L 1 (dP). Therefore the limit x (under (dTJu )), and we conclude n · Xu !)} = 0, IH lim E{(1 /\ n ->oo
for a.a. x (under (dTJu )) such that the integrand in (*) is zero for a.a.
of
which is a contradiction. Hence X must be a semimartingale for the filtration lHl0 , where Hf = :Ft V a {L}. Let X = M + A be a decomposition of X under lHl0 . Since lHl0 need not be right continuous, t�e local martingale M need not be right _continuous. However if we d�fine Mt = Mt if t is rational; and limu"-,t u EIQ Mu if t is not rational; then Mt is a right continuous martingale for the -filtration lHl where Ht = n u >t H� . Letting At = Xt - Mt' we have that Xt = Mt +At is an lHl decomposition of X, and thus X is an lHl semimartingale.
,
0
A simple but useful refinement of Jacod's Theorem is the following where we are able to replace the family of measures TJt be a single measure 7]. Theorem 11. Let L be a random variable with values in a standard Borel space (JE, £), and let Qt (w, dx) denote the regular conditional distribution of L
given :Ft, each t > 0. Then there exists for each t a positive a-finite measure 'T]t on (lE, £) such that Qt ( w, dx) « 'T]t ( dx) a. s. if and only if there exists positive a-finite measure ry(dx) such that Qt(w, dx) « ry(dx) for all w, each t > 0. In this case TJ can be taken to be the distribution of L. Proof. It suffices to show that the existence of TJt for each t > 0 implies the existence of TJ with the right properties; we will show that the dis tribution measure of L is such an 7]. As in the proof of Theorem 10 let (x, w) �----* qt(x, w) be £ 0 :Ft measurable such that Qt(w, dx) = qt(x, w)TJt(dx). Let a t(x) = E{qt(x, w)}, and define
one
(rt ) {q�(x(�w)) ' X, W
=
0,
t
if at(x) > O, otherwise.
365 Note that at(x) = 0 implies qt(x, ·) = 0 a.s. Hence, qt(x, w) = rt(x, w)at (x) a.s.; whence rt (x, w)at(X)T/t(dx) is also a version of Qt(W, dx). Let T/ be the law of L. Then for every positive £ measurable function g we 2 Initial Expansions
have
J g(x)ry(dx) = E{g(L)} = E{ k g(x)Qt(·, dx) } = E{ k g(x)qt(x, ·)r]t(dx)} = k g(x)E{qt(x, ·)}'f/t(dx) = 1 g(x)at(x)ryt(dx)
from which we conclude that at(X)T/t(dx) = rt(w, x)ry(dx), and the theorem is proved.
ry(dx).
Hence,
Qt (w, dx)
= 0
We are now able to re-prove some of the previous theorems, which can be seen as corollaries of Theorem 11. Corollary 1 {Independence) . Let L be independent of the filtration IF. Then every IF semimartingale is also an JHI ( L) semimartingale.
Since L is independent of Ft, E{g(L) I Ft} = E{g(L)} for any bounded, Borel function g. Therefore
Proof
E{g( L) IFt} =
1 Qt(w, dx)g(x) = k ry(dx)g(x) = E{g(L)},
from which we deduce Qt(W, dx) = ry(dx), and in particular Qt(w, dx) a.s. , and the result follows from Theorem 11.
«
ry(dx)
0
Corollary 2 ( Countably-valued random variables) . Let L be a random variable taking on only a countable number of values. Then every IF semi martingale is also an JHI ( L ) semimartingale.
Proof Let L take on the values . . . . The distribution of L is given by ry(dx) = 2:::: � P( L = o:i )Ea , (dx), where Ea , (dx) denotes the point mass at o:i . With the notation of Theorem 11, we have that the regular conditional distribution of L given :Ft, denoted Qt(w, dx), has density with respect to TJ 1
o: 1 , 0:2 , 0:3 ,
given by
The result now follows from Theorem
11.
0
366
VI Expansion of Filtrations
Corollary 3 (Jacod's Countable Expansion) . Let A = (A1 , A 2 , . . . ) be a sequence of events such that Ai n Aj = 0, i f. j, all in F, and such that U� 1 Ai = D. Let lHl be the filtration generated by IF and A, and satisfying the usual hypotheses. Then every IF semimartingale is an lHl semimartingale. Proof. Define L = I:� 1 2 - i 1A ; . Then lHl = the preceding corollary.
JHI(L) and we need only to apply D
Next we consider several examples. Example (Ito's example) . We first consider the original example of Ito, where in the standard Brownian case we expand the natural filtration IF with a{B1 }. We let lHl denote JHI(Bl). We have
j
E{g(B 1 )I:Ft} = E{g(B1 - Bt + Bt) I:Ft} = g(x + Bt)"lt(dx) where "lt(dx) is the law of B1 - Bt and where we have used that B 1 - Bt is in dependent of Ft. Note that "lt(dx) is a Gaussian distribution with mean 0 and variance (1 -t) and thus has a density with respect to Lebesgue measure. Since Lebesgue measure is translation invariant this implies that Qt(w, dx) « dx a.s., each t < 1. However at time 1 we have E{g(Bl ) I F1 } = g(Bl ), which yields Q 1 (w, dx) = E{ B1 (w ) } (dx), which is a.s. singular with respect to Lebesgue mea sure. We conclude that any IF semimartingale is also an lHI(B 1 ) semimartingale, for 0 < t < 1 , but not necessarily including 1. This agrees with Theorem 7 which implies that there exist local martingales in IF which are not semimartin gales in lHI(Bl ). Our next example shows how Jacod ' s criterion can be used to show a somewhat general, yet specific result on the expansion of filtrations. Example (Gaussian expansions) . Let IF again be the standard Brownian filtration satisfying the usual hypotheses, with B a standard Brownian motion. Let V = J;' g(s)dB8, where J000 g(s)2ds < oo , g a deterministic function. Let a = inf {t > 0 : ft00 g(s) 2 ds = 0}. If h is bounded Borel, then as in the previous example
l g(s)dBs + 1= g(s)dBs)I:Ft} t J h(l g(s)dBs + x) 'f/t (dx),
E{h(V) I :Ft} = E{h( =
t
where "'t is the law of the Gaussian random variable ft00 g(s)dB8• If a = oo , then "'t is non-degenerate for each t, and "'t of course has a density with respect to Lebesgue measure. Since Lebesgue measure is translation invariant, we conclude that the regular conditional distribution of Qt(W, dx) of V given :Ft also has a density, because
Qt (w, h) = E{h(V) IFt} =
2 Initial Expansions
t
J h(fo g(s)dBs + x)TJt(dx) .
367
Hence by Theorem 10 we conclude that every lF semimartingale is an !Hl(V) semimartingale. Example (expansion via the end of a stochastic differential equa tion) . Let B be a standard Brownian motion and let X be the unique solution of the stochastic differential equation
t
t
Xt = Xo + lo a(Xs)dBs + lo b(Xs)ds
where a and b are Lipschitz. In addition, assume a and b are chosen so that for h Borel and bounded,
J
E{h(X1 )IFt} = h(x)n(1 - t, Xt, x)dx n(1 - t , u , x) is a deterministic function.2 Thus Qt (w, dx) = n(1 t, Xt(w), x)dx, and Qt(w, dx) is a.s. absolutely continuous with respect to Lebesgue measure if t < 1. Hence if we expand the Brownian filtration lF by initially adding Xb we have by Theorem 10 that every lF semimartingale is an !Hl( X1 ) semimartingale, for 0 < t < 1.
where
The mirror of initial expansions is that of filtration shrinkage. This has not been studied to any serious extent. We include one result (Theorem 12 below) , which can be thought of as a strengthening of Stricker's Theorem , from Chap. II. Recall that if X is a semimartingale for a filtration lHI, then it is also a semimartingale for any subfiltration o be a bounded measurable process. It can be shown that there exists a predictable process PH, also bounded, such that for any predictable stopping time T one has The process PH is called the predictable projection of H. It follows that for the optional projection, for each stopping time have 0Hr = E{HriFr} a.s. on {T < oo}
T we
whereas for the predictable projection we have that
PHr = E{HriFr- } a.s. on {T < oo} for any predictable stopping time T. For fixed times t > 0 we have then of course 0Ht = E{Ht iFt} a.s. , and one may wonder why we don't simply use the "process" (E{Ht!Ft})t>o instead of the more complicated object 0H. The reason is that this is defined only almost surely for each t, and we have
an uncountable number of null sets; therefore it does not uniquely define a process. (Related to this observation, note in contrast that in some cases (E{HtiFt})t>o might exist even when °H does not) . We begin our treatment with two simple lemmas. Lemma. Let lF c
