Multidimensional Diffusion Processes (Classics in Mathematics)

Classics in Mathematics Daniel W. Stroock S.R.SrinivasaVaradhan

Multidimensional Diffusion Processes

Daniel W.Stroock S.R.SrinivasaVaradhan

Multidimensional Diffusion Processes Reprintofthe 1997 Edition

Springer

Daniel W. Stroock Massachusetts Institute of Technology Department of Mathematics 11 Massachusetts Ave Cambridge, MA 02139-4307 USA S. R. Srinivasa Varadhan New York University Courant Institute of Mathematical Sciences 251 Mercer Street New York, NY 10012 USA

Originally published as Vol. 233 in the series Grundlehren der mathematischen Wissenschaften

Mathematics Subject Classification (2000): 60J60,28A65

Library of Congress Control Number: 2005934787

ISSN 1431-0821 ISBN-10 3-540-28998-4 Springer Berlin Heidelberg New York ISBN-13 978-3-540-28998-2 Springer 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. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springeronline.com © Springer-Verlag Berlin Heidelberg 2006 Printed in Germany 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. Production: LE-TEX Jelonek, Schmidt & Vockler GbR, Leipzig Cover design: design & production GmbH, Heidelberg Printed on acid-free paper

41/3142/YL - 5 4 3 2 1 0

Grundlehren der mathematischen Wissenschaften 233 A Series of Comprehensive Studies in Mathematics

Series editors S.S. Chern J.L. Doob J. Douglas, jr. A. Grothendieck E. Heinz R Hirzebruch E. Hopf S. Mac Lane W. Magnus M.M. Postnikov W. Schmidt D.S. Scott K. Stein J. Tits B.L. van der Waerden

Editor-in-Chief B. Eckmann

J.K. Moser

springer Berlin Heidelberg New York Hong Kong London Milan Paris Tokyo

Daniel W. Stroock S.R. Srinivasa Varadhan

Multidimensional Diffusion Processes

^K Springer

Daniel W. Stroock Massachusetts Institute of Technology Department of Mathematics -j-j Massachusetts Ave Cambridge, MA 02139-4307 USA e-mail: [email protected] S.R. Srinivasa Varadhan New York University Courant Institute of Mathematical Sciences 251 Mercer Street New York, NY 10012 USA e-mail: [email protected] 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 Mathematics Subject Classification (2000): 60J60,28A65

ISSN 0072-7830 ISBN 3-540-90353-4 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 imder 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 © Springer-Verlag Berlin Heidelberg 1979,1997 Printed in Germany 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: design & production GmbH, Heidelberg Printed on acid-free paper

41/3142/db - 5 4 3 2 1 0

To our parents: Katherine W. Stroock Alan M. Stroock S.R. Janaki S.V. Ranga Ayyangar

Contents

Frequently Used Notation

xi

Chapter 0. Introduction

1

Chapter 1. Prehminary Material: Extension Theorems, Martingales, and Compactness 1.0 Introduction 1.1 Weak Convergence, Conditional ProbabiHty Distributions and Extension Theorems 1.2 Martingales 1.3 The Space C([0, oo); R^) 1.4 Martingales and Compactness 1.5 Exercises

7 19 30 36 42

Chapter 2. Markov Processes, Regularity of Their Sample Paths, and the Wiener Measure 2.1 Regularity of Paths 2.2 Markov Processes and Transition Probabilities 2.3 Wiener Measure 2.4 Exercises

46 46 51 56 60

Chapter 3. Parabolic Partial Differential Equations 3.1 The Maximum Principle 3.2 Existence Theorems 3.3 Exercises

65 65 71 79

7 7

Chapter 4. The Stochastic Calculus of Diffusion Theory 4.1 Brownian Motion 4.2 Equivalence of Certain Martingales 4.3 Ito Processes and Stochastic Integration 4.4 Ito's Formula 4.5 Ito Processes as Stochastic Integrals 4.6 Exercises

82 82 85 92 104 107 HI

Chapter 5. Stochastic Differential Equations 5.0 Introduction 5.1 Existence and Uniqueness

122 122 124

Vlll

Contents

5.2 On the Lipschitz Condition 131 5.3 Equivalence of Different Choices of the Square Root 132 5.4 Exercises 134 Chapter 6. The Martingale Formulation 136 6.0 Introduction 136 6.1 Existence 139 6.2 Uniqueness: Markov Property 145 6.3 Uniqueness: Some Examples 149 6.4 Cameron-Martin-Girsanov Formula 152 6.5 Uniqueness: Random Time Change . 157 6.6 Uniqueness: Localization 161 6.7 Exercises 165 Chapter 7. Uniqueness -. 171 7.0 Introduction 171 7.1 Uniqueness: Local Case 174 7.2 Uniqueness: Global Case 187 7.3 Exercises 190 Chapter 8. Ito's Uniqueness and Uniqueness to the Martingale Problem . . 195 8.0 Introduction 195 8.1 Results of Yamadaand Watanabe 195 8.2 More on Ito Uniqueness 204 8.3 Exercises 207 Chapter 9. Some Estimates on the Transition Probability Functions 208 9.0 Introduction 208 9.1 The Inhomogeneous Case 209 9.2 The Homogeneous Case 233 Chapter 10. Explosion 248 10.0 Introduction 248 10.1 Locally Bounded Coefficients 249 10.2 Conditions for Explosion and Non-Explosion 254 10.3 Exercises 259 Chapter 11. Limit Theorems 261 11.0 Introduction 261 11.1 Convergence of Diffusion Process 262 11.2 Convergence of Markov Chains to Diffusions 266 11.3 Convergence of Diffusion Processes: Elliptic Case 272 11.4 Convergence of Transition Probability Densities 279 11.5 Exercises 283 285 Chapter 12. The Non-Unique Case 12.0 Introduction 285 12.1 Existence of Measurable Choices 286 12.2 Markov Selections 290

Contents

IX

12.3 Reconstruction of All Solutions 12.4 Exercises Appendix A.O Introduction A.l Lp Estimates for Some Singular Integral Operators A.2 Proof of the Main Estimate A.3 Exercises Bibliographical Remarks

296 302 304 304 306 315 323 328

Bibliography Index

331 337

Frequently Used Notation

I. Topological Notation. Let (X, p) be a separable metric space. 1) 2) 3) 4) 5) 6) 1) 8) 9)

5° is the interior of B^X. B is the closure of B^X. dB is the boundary of B ^ X. ^x is the Borel field of subsets of X. Ci,(X) is the set of bounded continuous functions/:X -^ R. B(X) is the set of bounded ^^-measurable/:X -• R. Up{X) is the set of bounded p-uniformly continuous/:X -• R. M(X) is the set of probability measures on (X, ^^). |i/||=sup|/(x)| for/6 B(X). xeX

II. Special Notation for Euclidean Spaces 1) K'' is ii-dimensional Euclidean space. 2) 3)

\x\^(^x]Yi''fovxeR'. B(x,r)={yeR':\x-y\ = Y^^yj for X, >; G K 5) S'-' = {xeR':\x\=\}. 6) C(RO = {/eC,(KO: limy(x) = 0}. ix|-oo

7) Co(^) is the set offe CJ^^) having compact support. 8) CJ'('^) is the set off:^-^R possessing bounded continuous derivatives of order up to and including m.

9) cr(^) = n Ct(n m=0

10) C*(^) is the set of/: ^ -• K possessing continuous derivatives of all orders. 11) ci(^) = C'^(^)nCo(^). 12) Cr^C^) for '^ ^ [0, oo) X R'^ is the set off: ^-^ R such that/has m bounded continuous time derivatives and bounded continuous spacial derivatives of order less than or equal to n. 13) L^(^), 1 < p < 00, is the usual L^-space defined in terms of Lebesgue measure on ^.

xii

Frequently Used Notation

14) Lfoc(^) is the set off.^^R

(or C) such that fe n(K) for all compact

III. Path Spaces Notation 1) 2) 3) 4)

C(/, R'^) for / ^ [0, oo) is the set of R'^-valued functions on / into K^. Q,(Q) (see p. 30). Jf,(Jf) (see p. 30). x(t, (jo) (see p. 30).

IV. Miscellaneous Notation 1) 2) 3) 4) 5) 6)

a A 6 is the smaller of the numbers a, b e R. a V b is the larger of the numbers a, b e R. Sd is the set of symmetric non-negative definite d x d real matrices. Sa is the set of nondegenerate elements of S^. ||/4||, where A is a. square matrix, and is the operator norm of A. (7('^), where ^ is a collection of subsets of X, and is the smallest cr-algebra over X containing ^ . 7) (7(^), where J*^ is a set of functions on X into a measurable space, and is the smallest cr-algebra over X with respect to which every element of ^ is measurable. 8) [A], A G K, is the integral part of L 9) i ^ n(a, b) (see p. 92).

Chapter 0

Introduction

The main purpose of this book is to elucidate the martingale approach to the theory of Markov processes. Needless to say, we believe that the approach has many advantages over previous ones, and it is our hope that the present book will convince some other people that this is indeed so. When we began this project we were uncertain whether to proselytize by intimidating the reader with myriad examples demonstrating the full scope of the techniques or by persuading him with a careful treatment of just one problem to which they apply. We have decided on the latter plan in the belief that it is preferable to bore than to batter. The result is that we have devoted what may seem like an inordinate number of pages to a rather special topic. On the other hand, we have endeavoured to present our proofs in such a way that the techniques involved should lend themselves to easy adaptation in other contexts. Only time will tell if we have succeeded. The topic which we have chosen is that of diffusion theory in R'^. In order to understand how this subject fits into the general theory of Markov processes, it is best to return to Levy's ideas about "stochastic differentials." Let x(') be a Markov process with values in R*^ and suppose that for r > 0 and test functions (p e C^(R') (O.I)

E[(p(x(t + h)) - (p(x(t)) Ix(5), 5 < r] = hLMx{t)) + o(h),

h>0,

where, for each r > 0, L, is a linear operator on Co(R'^) into Cj,(K''). It is obvious that Lf must satisfy the weak maximum principle, since if cp achieves its maximum at X® then E[(p(x(t + h)) - (p(x{t))\x(t) = x^] 0, looking like the Gaussian independent increment process with drift b(u ^(0) ^^^ covariance a(r, x(t)). In differential form, this intuitive picture means that (0.5)

dx(t) = G(U x(t)) dp(t) + b(u x(t)) dt

where j?( •) is a ^/-dimensional Brownian motion and a is a square root of a. Indeed, G(U x{t))(P(t -\- h) - P(t)) + b(t, x(t)) will be just such a Gaussian process;

4

0. Introduction

and if {x(s), 0 < s < r} is {P(s): 0 < s < r}-measurable, then cr(r, x{t)) x (p(t + h)- P(t)) + b(t, x(t)) will be conditionally independent of {x(5): 0 < s < r} given x(t). There are two problems of considerable technical magnitude raised by (0,5). First and foremost is the question of interpretation. Since a Brownian path is nowhere differentiable it is by no means obvious that sense can be made out of a differential equation like (0.5). Secondly, even if one knows what (0.5) means, one still has to learn how to solve such an equation before it can be considered to be useful. Both these problems were masterfully handled by Ito, a measure of the success of his solution is the extent to which it is still used. We develop Ito's theory of stochastic integration in Chapter 4 and apply it to equations like (0.5) in Chapter 5. With Chapter 6 we begin the study of diffusion theory along the lines initiated by us in Stroock and Varadhan [1969]. In order to understand this approach, we return once again to (0.1). From (0.1), it is easy to deduce that: E[(p(x(t2))\x(sls0

|^|

\x\ I. Given fi e M(X), the following are equivalent:

8

1. Preliminary Material: Extension Theorems, Martingales, and Compactness

(i) lim„^^ ^ fdii„ = ^ fdfifor every fe Q(A') (a) lim„^^ I fd^„ = ] fdfifor every fe Up(X) where Up(X) is the set of bounded uniformly continuous functions on (X, p) and p is any equivalent metric on X. (Hi) lim sup„_^oo Mn(^) ^ f^(C)for any closed set C in X (iv) lim inf„^oo/i„(G) > fi(G)for any open set G in X (v) lim„^ 1

where p(x, C) = 'mf{p(x, y): ye C}. Then /^ e Up(X) for each k > 1 and fk(x) I xc(x) for each xe X. Therefore p(C) = lim j fkdfi= lim lim j /^ dp„ > lim sup p„(C). fc-^CC «-»QO

k-*co

Finally, to see that (v) implies (i), take/e Q(X) and, given e > 0, choose N and {ai}^~ ^ so that -l = ao 0 and A; > 1, there is an n^ such that inf/z(GJ^)>l-^. S e t K = f]^^, GJ*. Then inf/i(X)> l - £ . Moreover, K is closed in X and is therefore complete. Finally, for any /c > 1

and so K is totally bounded. Thus K is compact and the proof is finished. Q 1.1.4 Theorem. Let T ^ M(X) be given and assume that for each e> 0 there is a compact set K '^ X such that inf ^(K)

>l-e.

Then F is precompact in M(X) {i.e., T is compact). Proof We first recall that if X itself is compact, then by Riesz's theorem and standard results in elementary functional analysis, M(X) is compact. In general we proceed as follows. Choose a metric p on X equivalent to the original one such that (X, p) is totally bounded and denote by X its completion. Then X is compact and we can think of M(A') as being a subset of M(X). Thus it remains to show that if {/i„}f ^ r and n„-* p in M(X\ then p. can be restricted to A" as a probabihty measure //, and /i„ -• /i in M(X). But, by our assumption on F, there is a sequence of compact sets {X,}, / > 1 in X such that p„(Ki) > 1 — 1// for « > 1. Since K, is compact and therefore closed in X, it follows from Theorem 1.1.1 that p(Ki) > \im„^^p„(Ki)>\l//.Thus

4'JK.) =

1. Weak Convergence, Conditional Probability Distributions, and Extension Theorems

11

and so we can restrict fi to (X, ^x) ^s a probability measure //. Finally since /i„ -^ /i in

M(X) lim ^(pd^i„ =

^q)d^

for all (p e Up{X\ and by Theorem 1.1.1 this implies that ^„ -• /i in M(X). Remark. Note that whereas Theorem 1.1.3 relies very heavily on the completeness of X, Theorem 1.1.4 does not use it at all. 1.1.5 Corollary. Let F ^ Cj,(X) be a uniformly bounded set offunctions which are equicontinuous at each point ofX. Given a sequence {//„}f ^ M(X) such that /!„-•// in M(X\ one has lim sup U (pdn„-

j (p dy,

0.

n-*ao 0 e F

Proof Suppose there exists £ > 0 such that lim sup sup \ (P dn„- ^ (p dfi > e. By choosing a subsequence if necessary, we can assume that for each n there is a (p, in F such that I (Pn dfi„ - j (p„dn > £ > 0 . Let M = sup^gf \\(p\\ and choose a compact set K in X such that sup„>i fi„(X\K)<s/SM. It follows from this that fi(X\K) <E/SM. From the Ascoli-Arzela theorem and the Tietze extension theorem, we can find snj/ in Ci,(X) such that I I/A I < M, and a subsequence (p^. = ij/j of{(p„} converging to ij/ uniformly on K. But then J ijjj d^„. - I il/j ^/i| < |J (ij/j - il/) dfij + J (il/j - iP) d^ J ^ dfi„. - ^ip dfi = \\(iljj-il/)d^J-h\\

(il/j-il/)d^„. X\K

i^j - ^) dfi ''X\K

+

J ^ dfi„. - J ij/ dfil

12

1. Preliminary Material: Extension Theorems, Martingales, and Compactness

Therefore I 0 < £ < lim I lAj d^nj - j il/j dfi\

-E=

-£/i

14

1. Preliminary Material: Extension Theorems, Martingales, and Compactness

and Irj — tj\ < e for i <j < n. Therefore (ri+a)/i + - - + r „ / „ > 0 . From the definition of the set F, it follows that (r,+s)g,(x)-^''-^r„g„(x)>0. By letting e -• 0 over the rationals, we conclude that ti9i(x) +

--+t„g„(x)>0,

or equivalently Lj,(/) > 0. Since L;,(/i)= 1 and W is dense in Up(X), Lj^f) defined on W extends uniquely as a non-negative linear functional on U^{X). We continue to call this extension Lj^f). We can view the space Vp{X) as the space C(X\ where X is the completion of the totally bounded space (X, p). Note that X is compact. By the Riesz Representation Theorem, there is a probabihty measure Q^ on (X, edx) such that

Uf)=\f{y)QMy) for a l l / e C(X). Thus we have shown that for all x e X\F, there is a probability measure Q^ on (X, % ) such that (1.3)

gi(x) = f fi(y)QMy)

for all i. (Here we use the notation/to denote the extension of a n / e Up(X) to X.) This shows that the mapping

f(y)QMy) on X\F is Z[X\F]-measurable for a l l / e W, and therefore for a l l / e (7p(X). Moreover, it is easy to see that 1.4)

j ny)QMy), A n {X\F) = £l/(-M]

for all/e U^{X) and ^ G I . Given a compact set K in X, choose {0} a non-decreasing sequence of sub o-algebras whose union generates ^ . Let PQ be a probability measure on (£,#0), and, for each n > 1, let n'^(q,dq') be a transition function from {E.^n-i) to (£,#'„). Define Pn on (E,J^ri)for n>l so that Pn{B) = Jn^q,B)Pn-i(dq)

for

B G ^nl

and assume that, for each n>0, there is a Pn-null set Nn G J%, such that q^Nn.

Be #-„+!, and 5 n An(q) = 0 => n"+\q,B) = 0.

Then there is a unique probability measure P on (E,^) P I ^Q= PQ and, for each n>l, PiB) = fn"{q,B)P(dq')

for

with the properties that

B G J^.

For each n, F„ is probability measure on ^„ and P„+i agrees with P„ on J^„. Hence there is a unique finitely additive P on Q„ ^„ such that P agrees with P„ on ^ „ . We will show that P is countably additive on |J„ J^„ and therefore extends uniquely as a probabihty measure to J^. We have to show that if B„ e ij„ J^„ and J5„i0 then P(B„)iO. Assume that P(B„) > e > 0 for all n. We will produce^a point qef]„B„. We can assume, without loss of generality, that B „ G J ^ „ . For 0 < M < m and B G J^„ we define TT'"- "(q, B) to be XB(Q)- For n > m and B e J^„ we define TT'"' "(q, B) inductively by n"''"{q,B)=

\n"{q',B)n'"'"-'(q,dq).

18

1. Preliminary Material: Extension Theorems, Martingales, and Compactness

Clearly P(B)==jn'''"(q,B)Po(dq)

for

B e ^„.

We also have for n> m 7r'"'% B) = j 71^"^^'V. B)7f"^'(q, dq') for Be ^ „ . Define F« = L : 7 r « " ' t e 5 „ ) > |

n>0.

Then F^+ j ^ F^ and a e/2^+i for 1 < /c < m and « > 0. Let F r ^ = {^:7r'"^^'"teB„)>e/2-^2}. Then

F;;'+^/

^

F;;'^

^ and for n > m TT < ""• %\ i m „ ,' B„) < ; ^ s ^ + rr*'(9», - ^ r ' ) —n/ — y

om+1 —

Hence n-'^'l^qm,

nFrO>£/2'"^^

We can therefore conclude that A„(q„) n (~]Q F ^ ^ ^ =/= 0 ; and so there is a qn+i € An(qn)\Nn with the property that 7r'"+^'"(^;„+i,5„) > £/2'"+2 for all n > 0. By induction on m, we have now shown that a sequence {qmjo' exists with the properties that qm+i e ^m(^m)\^m and inf„7c'"'"(g^,B„) > 0 for all n > 0. In particular XBjqm) = n"'''^(qm,Bm) > 0 and therefore Am(qm) 0. It is easy to check that {J^„, n>0} satisfy (1.9). Next, let {Q"} be a regular conditional probability distribution of Pp^ given (ap"^ _ i) ~ ^ (^F„ _ i) ^^^ define

for all qe Xj and J3 e J^„. It is easily checked that TT" is a transition function from {Xi, ^n-i) to (Xi, J^„) and that it satisfies the condition of Theorem 1.1.9. Thus by that theorem there is a unique probability measure P on {Xj, (T(IJS ^n)) such that P equals Pp^ (jp^ on ^Q and P{B) =

\n%q,B)P{dq)

for all B e J^„. By induction, we see that P equals P on (Jj* J^„. In particular P(A„) = P(A„\ the countable additivity of P implies that P(A„)[0, and the theorem is proved. Q

1.2. Martingales Throughout this section, E will denote a non-empty set of points g. J^ is a c-algebra of subsets of £, and {J«^,: r > 0} is a non-decreasing family of sub aalgebras of ^. Given s > 0 and a map ^ on [s, oo) x £ into some separable metric space (X, D), we will say that 9 is (right-) continuous if ^( •, ^) is (right-) continuous for all ^ e £. If P is a probability measure on (£, ^) and 0: [s, oo) x £ -• X, we will say that 0 is P-almost surely [right-) continuous if there is a P-null set N e ^ such that 9{', q) is (right-) continuous for q 4 N. Given s > 0 and 0 on [s, oo) x £ into a measureable space, 6 is said to be progressively measurable with respect to {J^,: f > 0} a/rer ^ime 5 if for each t > s the restriction of 9 to [s, t] x E is ^[j, f] ^ ^^rn^^asurable. Usually there is no need to

20

1. Preliminary Material: Extension Theorems, Martingales, and Compactness

mention 5 or {^^i t > 0}, and one simply says that 6 is progressively measurable. Note that 9 is progressively measurable with respect to {J^,: t > 0} after time s if and only if 6^, defined by 6^(1, q) = 6{s -I- r, q), is progressively measurable with respect to {^t+s'- ^ ^ ^] ^^er time 0. Thus any statement about a progressively measurable function after time s can be reduced to one about a progressively measurable function after time 0. This remark makes it possible to provide proofs of statements under the assumption that s = 0, even though s need not be zero in the statement itself. Exercises 1.5.11-1.5.13 deal with progressive measurabihty and the reader should work them out. The following lemma is often useful. 1.2.1 Lemma. IfO is a right-continuous function from [s, 00) x £ into (X, D) and if 6(ty ') is ^t-measurable for all t > s, then 6 is progressively measurable. Proof Assume that 5 = 0. Let r > 0 be given and for w > 1 define e„(u,q) = e\^^

At.qj.

Clearly e„ is ^[0, t] ^ ^f-measurable for all n. Moreover, as « -^ 00, 6„ tends to 6 on [0, t] X E. Hence 6 restricted to [0, r] x £ is ^[o, t] ^ ^r^QSiSurable. n Given a probability measure P on (£, ^), 5 > 0, and a function ^ on [s, 00) x £ into C, we will say that (9(t), ^,, P) is a martingale after time 5 if 0 is a progressively measurable, P-almost surely right-continuous function such that 6{t) = 0(t, ') is P-integrable for all t > s and (2.1)

£1^(^2)1^, J = ^(^1)

(a.s., F),

sg(E[e(t^)\^„])

a.s.

Thus, if 0 < f 1 < t2, then E[g(e(h)) I J^.J > g(E[e(t,) IJ^,J) > # ( t O )

a.s.

This completes the proof of the first assertion. The second assertion is immediate in the case when {9(t\ ^ t , P ) is a non-negative submartingale; simply take g(x) = x' on [0, oo). Thus the proof will be complete once we show that (|^(01» ^t, P) is a submartingale if (0(t\ J^,, P) is a martingale. But if (9(t\ J^,, P) is a martingale, then

E[ I ^(^2) I I i ^ J > I E[e(t,) I ^ J I = I d(t,) I and so (10(t) |, i^,, P) is a submartingale.

(a.s., P),

D

1.2.3 Theorem. If(6(t\ 0 and allT>s: (2.3)

p( sup

e(t)>Ae(0

a.s.

(2.11)

E[e(T)\^,„]>e(x„)

a.s.

(2.12)

£[^(r)|i^J>e((T„)

a.s.

From (2.11) and (2.12), respectively, we know that

E[d(x„%e(x„)>x]<E[e(ne(x„)>k] and £[0(ff„), 0(c„) > A] < £[^(T), 0(ff„) > X]. Since 0(r) > 0, these imply: E[e(x„), 0ix„) >X]< E[e{n

sup 0{t) > X]

0 /I 0 1} are uniformly P-integrable families and so 0(T„) -^ ^(T) and e(a„) -^ 0((T) in n(P). Since, by (2.6), E[e(T„)\^J

= e(a„)

a.s.

for all « > 1, the rest of the argument is exactly like the one given at the end of the submartingale case. D 1.2.6 Corollary. Let T: [0, oo) x £ -• [5, 00) be a right-continuous function such that T(t, ')isa bounded stopping time for allt >0 and T(% q) is a non-decreasing function for each q e E. If(0(t), J^,, P) is a (non-negative sub-) martingale after time s, then (6{T(t)), s is a stopping time and (6(t), #",, P) is a (non-negative sub-) martingale after time s, then (^(r AT), J ^ , , P) is a (non-negative sub-) martingale after time s. Proof By Corollary 1.2.6 (6(t A T), J^,^^ , P) is a (sub-)martingale. Thus if ^ e J*^,^, then for ^2 > ti: (>) £[^(^2 AT), A n {Z> r j ] = E[e(ti AT), ^ n {T > tj}], since A n {r > t^} e ^ti^x{t < fj}, and so:

On the other hand, 9(t2 AT) = ^(T) = 0(ti AT) on

(>) £[^(^2 AT), A n{x < r j ] = E[e(ti AT),An{x< Combining these, we get our result,

t^}].

n

The next theorem is extremely elementary but amazingly useful. It should be viewed as the " integration by parts " formula for martingale theory. 1.2.8 Theorem. Let (6(t), J^,, P) be a martingale after time s andrj: [s, 00) x £ ->• C a continuous, progressively measurable function with the property that the variation \rj\(t, q) ofrj(', q) on [s, t] is finite for all t > s and q e E. If for all t > s (2.13)

s u p | e ( u ) | ( | „ | ( t ) + \„(s)\)

< 00,

then {0{t)ri{t) — j ^ 0{s)rj{ds), 9',, P) is a martingale after time s.

27

1.2. Martingales

Proof. Assume s = 0. Using Exercise 1.5.5, one can easily see that jo 0(u)r](du) can be defined as a progressively measurable function. Moreover, (2.13) certainly implies that 0{t)rj{t) — Jo 6(u)rj(du) is F-integrable. Now suppose that 0 < f j < ^2 and that A e J^^j. Then ^(^2)^(^2) - ^(^1)^(^1) - fo(u)rj(dul

A \ = E\ \ (e(t2) - 0(uMdul

A

Since

Emt,)-eit,Mt,iA]

= o.

But if A = r2 - ti, then

(\e(t2) - e(u))r,(du) = lim X ieit^) - ^ i + ^ ^ ) )

and, by (2.13) and the Lebesgue dominated convergence theorem, the convergence is in L^(P). Finally,

^[(^(t2)-^(r.+^A))(,(t, + ^ A ) - , ( r , + ^ for all n > 0 and 1 e(t.)

a.s.

for all ti,t2 € D such that t^ < t2, then (0{t), J^,, P) is a non-negative submartingale after time s. If (2.15)

E[e(t2)\^t,]

= e(t,)

a.s.

for all ti, tje D such that t^ < tj, then {6(t), J^,, P) is a martingale after time s.

28

1- Preliminary Material: Extension Theorems, Martingales, and Compactness

Proof. Assume that s = 0. Clearly the proof boils down to showing that in either case the family {| e(t) \: te [0, T] n D} is uniformly P-integrable for all T e D. Since (2.15) implies (2.14) with |^(*)| replacing ^(•), we need only show that non-negativity plus (2.14) implies that {9(t): t e [0, T] n D} is uniformly Pintegrable. To this end, we mimic the proof of (2.4), and thereby conclude that pi

sup \te[0,T]

e(t)>A0.

^

Combining this with

E[e(t), e(t) >x\< E[e(T% e(t) > x\ <E

e(Ti

sup e(t) > X

t e [0, T] n D,

f 6 [ 0 , 7 ] r^D

we conclude that {9(t): t e [0, T] n D} is uniformly P-integrable.

D

1.2.10 Theorem. Assume that for all t >0 the o-algehra ^^ is countably generated. Let X >s he a stopping time and assume that there exists a conditional probability distribution [Q^] ofP given ^ , . Let 6: [s, co) x E -^ R^ be a progressively measurable, P-almost surely right-continuous function such that 9(t) is P-integrable for all t > s. Then_(6(t), ^ , , P) is a non-negative submartingale after time s if and only if (9(t A T), ^ , , P) is a non-negative submartingale after time s and there exists a P-null set N e ^^ such that for all q' ^ AT, (9(t)X[s, t]('^), ^t^ Gg) ^^ ^ non-negative submartingale after time s. Next suppose that 9: [s, oo) x E^Cisa progressively measurable, P-almost surely right-continuous function such that 9{t) is P-integrable for all t > s. Then {9(t), ^^, P) is a martingale after time s if and only if(9(t A T), ^ , , P) is and there is a P-null set N such that (9(t) - 9{t AT), J^,, Q^) is a martingale after time s for all q' i N.

Proof Assume that s = 0. We suppose that {9(t), ^ , , P) is a martingale. Then by Corollary 1.2.7 so is (9{t AT), ^t^P)- Let 0 < tj < f2» ^ e ^^ and A e ^^^ be given. Then E^[E'^[9{t2\ A\ B n{x
) is a martingale for r > 0. Since (2.16)

P k : e j ^ : T ( ^ ) = T(^')] = l ] = l

we are done. Now suppose {6(t), ^^, P) is a non-negative sub-martingale. Then by Corollary 1.2.7 so is {^{t/\x\ #',,P). By replacing equalities by the obvious inequalities we can conclude that there is a null set N in ^^ such that (0(f), J^,, Q^) is a non-negative submartingale for t > x(q') provided q' ^ N. We note that this is equivalent to X[o, t]('^(Q'))H^) being a non-negative submartingale for t > 0. Again by (2.16) we are done. We now turn to the converse proposition. If 0 tj}]

= E''[e{ti), An{T
t,}]

= E-m,); A]. The submartingale case is proved in the same manner by replacing the equalities by inequalities at the relevant steps. D 1.2.11 Remark. It is hardly necessary to mention, but for the sake of completeness we point out that everything we have said about almost surely right-continuous martingales and submartingales is trivially true for discrete parameter martingales and submartingales. That is, if (£, J^, P) is a probability space, {^„: n >0} a non-decreasing family of sub 0} a. sequence of P-integrable complex valued random variables, such that 6„ is i^„-measurable, then (9„, J^„, P) is a martingale (submartingale) if (9„ is real-valued) and E[e(n-\-l)\^„]^e(n)

a.s.

30

1. Preliminary Material: Extension Theorems, Martingales, and Compactness

for all n>0. The obvious analogues of (1.2.4) through (1.2.10) now hold in this context. Indeed, if one wishes, it is obviously possible to think of this set-up as a special case of the continuous parameter situation in which everything has been made constant on intervals of the form [n, n -\- 1).

1.3. The Space C([0, oo); R"^) In this section we want to see what the theorems in Section 1.1 say when the Polish space is C([0, oo); R^). The notation used in this section will be used throughout the rest of this book. Let Q = Qj = C([0, 00); R'^) be the space of continuous trajectories from [0, oo) into jR'^. Given r > 0 and co e Q let x(u co) denote the position of co in R'^ at time t. Define Dico co')= y ^ supo 0 and T < co (3.6)

sup

lim infF

\x(t) — x(s)\ < p = 1

0<s': x(s, co') = x(s, co) for 0 < s < n].

Therefore if {a)„}o ^ Q has the property

for all iV > 0 then the co determined by x(t, co) = x(r, co„),

0 0 such that (f(x(t)) + Aft, Ml, P) is a non-negative submartingale. 1.4.3 Hypothesis. Given a non-negative f e Co{R% the choice of Aj- in (1.4.2) can be made so that it works for all translates off Under these hypotheses, we are going to develop an estimate for the quantity in (4.2) which depends only on the constants Af. Let e > 0 be given and choose /, € C^iR"^) so that /,(0) = 1, f(x) = 0 for |x| > e, and 0 0 ^(Tn+1 - T„ < ^ I Jt,^) < SApi4.

(a.s., P) on {T„ < oo}.

Proof Let e — p/4 in the preceding discussion and let {Qc^} be a r.c.p.d. of P given M^^. Then we can choose a P-null set F G M^^ SO that

((/r(x(f)) + '4.f)Z[o,,i(T„(a)')), A , e„.) is a non-negative submartingale for all co' ^ f, where/f (x) =/j(x - X(T„(CO'), CO')) if T„(co') < 00 and/f (•) = 1 otherwise. In particular, by Theorem 1.2.5, £«"[/r(x(T„,, A (T„(CO') + ,5)) + A,S\ > 1

38

1. Preliminary Material: Extension Theorems, Martingales, and Compactness

for cd ^ F. In other words, £^-[1

But 0 < 1 -ff

-/f

(X(T„^I A ( T > ' ) + (5))] < A,d, CD' ^ F.

< 1, and T„+ i < T„(a>') + 3 implies that + S))=l

l-ff(x(x„^,A(x„(co') if T„(a)') < 00. Thus

QA^n^i 1} be a non-decreasing sequence of random variables on (£, #") ta/cmgf values in [0, oo) u {oo}, and assume that ^„ is ^„-measurable. Define ^o = ^ ^nd suppose that for some A < 1 and all n>0:

^[exp[-(^„^i-U]|«^J 0 one defines N(q) =

mf{n>0:i„^,(q)>n

then N < oo a.s. and in fact: P(N >k)< e''X\

k>0.

Proof First note that: E[e-^^^^\^„] = e-^"E[cxp[-(Ui

-

U]\^n]

0

then {P„: n > 0} is precompact. Proof The proof here goes by analogy with the proof of Theorem 1.4.6. We must show that (4.7)

limlirnTPJ d^O n-*ao

for all r > 0 and p > 0.

sup

I 0 <s X\ dX which is valid for any non-negative/and any r > 1. See Theorem 3.4 in Chapter 7 of Doob [1952] for details. 1.5.5. Let (£i, J^i), (£2, ^2) and (£3, ^^3) be three measurable spaces. Let F: £1 X £3 -• R be a measurable map. Let ^l{q2, dq^) be a signed measure on (£3, ^2) for each ^2 ^ ^2 and a measurable function of ^2 for each set in ^ 3 . Show that the set AT c: £1 x £2 defined by N= [(quqi)' \ 1/^(^1,^3)1 1/^1(^2, ^^3) < 00 is a measurable subset of (£1 x £2, J^i x ^2) and the function J £(^1, ^3)/i(^2» ^^3) is a measurable function of (^1, ^2) on the set N. Deduce from this that if 6(t, q) is a progressively measurable function and rj(ty q) is a progressively measurable continuous function which is of bounded variation in t over any finite interval [0, T], then Z(r, q) is again a progressively measurable function where Z(r, ^) = f'^(5, q)tj(ds, q) if •'0

= 0 otherwise.

f' I ^(5, ^) I I ;y I (ds, q) < co •'0

44

1. Preliminary Material: Extension Theorems, Martingales, and Compactness

1.5.6. Let £ be a non-empty set and rj a map of [0, oo) x £ into a Polish space (X, d). Define on E the a-field J^ = (T[t]{s): s > 0]. Show that if/is an ^ measurable map of E into a Polish space M, then there exists a ^^z+ -measurable map F of X"^^ into (M,^) and a sequence {t„}f £ [0,oo) such that /(^) = F(^(fi,^), ...,^/(r„,^)•••)»

qeE.

Next, assume that ^/(^ (?) is right continuous for each qeE and define ^t = o-[^(5): 0 < 5 < r]. Given a measurable function 6: [0, oo) x E^M such that for each t, 6(u •) is J^,-measurable, show that there exists a ^[o,x) X ^j^2+-measurable map F of [0,oo) x X^^ and a sequence {t„}5° ^ [0,oo) such that for all t > s and q £ E 0(s At,q) = F(s A r, rj(t^ A r, ^),..., ^(r„ A r, ^),...). In particular conclude that for each fixed r, 6(s A t, ^) is measurable in (5, q) with respect to ^[o, n x -^r» ^i^d hence ^( •, •) is progressively measurable. 1.5.7. Let (£, ^, P) be a probability space and I c J^^ be a sub tr-field. Let (7, ^ ) be a measurable space and F: E x Y-^ R a measurable function (relative to J^ X ^ ) such that sup^^^y £^[|F(*, y)\] < 00. Show that a version G(q, y) of £^[F(', y)|I] can be chosen so that G(*, •) is L x ^ measurable. Suppose now that we have a map /: £ -• 7 which is I-measurable. Assuming that £''[|^(%/(;))|]< 00, show that £ V ( % / ( - ) ) | 2 ] = G(.,/(.))

a.e.

1.5.8. Suppose (9(t), i^,, P) is a martingale on (£, i^, P). Let J^,+o = C]s>t ^s where .^^ is the completion of #"5 in (£, .^, P). (That is J e .^^ if and only if there is an .4 in ^, with A AA a B where Be^ and P(B) = 0.) Show that {6(t), ^t+o, P) is a. martingale. 1.5.9. Use Theorem 1.2.8 to show that if (9(t), #",, P) is a continuous real valued martingale which is almost surely of bounded variation, then for almost all q, 6(t) is a constant in t. Note that this conclusion is definitely false if one drops the assumption of continuity. 1.5.10. Suppose (9{t), ^,, P) is a martingale on (£, J^, P) such that sup,^o £(^(0)^ ^ ^' Show that E{9{t))^ is an increasing function of ? with a finite limit as r -• 00. Use this to show that d{n) tends in mean square to a limit ^(00) as n-* CO. Next, use Doob's inequality to prove

sup 1^(0-^(5)1 >e

0 a non-decreasing family of (T-fields such that ^ = o-(lJ, J^,). Given /I £ [0, oo) x £, we say that A is progressively measurable if XA{'^ •) is a progressively measurable map from [0, oo) x £ into R. Show that the class of progressively measurable sets constitute a (T-field and that a function/: [0, oo) x £ -• (X, ^) is progressively measurable if and only if it is a measurable map relative to the (T-field of progressive measurable sets. 1.5.12. With the same notation as above, let i : £ -> [0,oo] be an extended non-negative real valued function such that for each r > 0, {^: T(^) 0, the map O, restricted to [0, r] x £ is a measurable map of ([0, t] x £, ^[o. t] ^ ^t) into itself. Since the second component of O^ is the identity map it is enough to check that T(^) As: [0,0 x £ ^ [0,r] is ^[o,f] x # i measurable. Since T(^) A S cannot exceed r, we need only check that for each a < r the set {(s, q): T{q) A s < a} is in ^[0, f] X 0 be a non-increasing family of subsets of £ such that A(t) G ^t for each t > 0. Consider the set A=

[j(t,A(t))

=

{(t,q):qEA(t)}.

f>0

Define B(t) = A(t - 0) = P|, 0 and B(0) = A(0). Show that if A is progressively measurable then B = [j(t,B(t))

=

{(t^q):qeB(t)}

t>0

and B\A are progressively measurable too. (Hint: Consider the function/(r, q) = ;f^(r, q). From the fact that/(r, q) is progressively measurable show that/defined by /(t,q)=/(t-0,9)=lini/(i(l-^|,<j| is again progressively measurable. Identify/as XB)

Chapter 2

Markov Processes, Regularity of Their Sample Paths, and the Wiener Measure

2.1. Regularity of Paths Suppose that for each n > 1 and 0 < tj < • • • < r„ we are given a probability distribution F„,...,,„ on the Borel subsets of (R^f. Assume that the family {Ptu...,tJ is consistent in the sense that if {sj,..., 5„_ j} is obtained from {t^, ..., t„} by deleting the kth element t^, then P^^^ ..,s„-i coincides with the marginal distribution ofPf^^ j^ obtained by removing the kth coordinate. Then it is obvious that the Kolmogorov extension theorem (cf. Theorem 1.1.10) applies and proves the existence of a unique probability measure P on (R'^f^- "^^ such that the distribution of (il/(ti), ..., lA(^n)) under P is Pti,...,t„- (Here, and throughout this chapter, i/^ stands for an element of (R'^f^' "^^ and V(0 is the random variable on (R'^f^^ "^^ giving the position of ij/ at time t.) As easy and elegant as the preceding construction is, it does not accomplish very much. Although it establishes an isomorphism between consistent families of finite dimensional distributions and measures on a function space, the function space is the wrong one because it is too large and the class of measurable subsets is too small. To be precise, no subset of (R^f^' "^^ whose description involves an uncountable number of fs (e.g. {ij/: supo 1} ^ [0, oo) such that

Z^(^) = F(^(rO,...,^(0,...),

i^e(RT"^'-

Thus if 5^ = r n [0, iV], f)N=i {^'- ^ |s^ is uniformly continuous} £ A. But, by (1.3), P({^'. ij/ \s^ is uniformly continuous}) = 1 for all N; and so, by the monotone convergence theorem, P(A) = 1. This shows that Q has P-outer measure one. D We are now going to develop a criterion on {P,^ ^J for testing whether the associated P satisfies (1.1). The basic method which we will use goes back to Kolmogorov, although the elegant approach that we employ here is due to Garsia, Rademich, and Rumsey [1970]. It must be admitted that as a tool for studying Markov processes, Kolmogorov's criterion is rather crude by comparison to the machinery which we developed in Section 1.4. On the other hand, it has the important feature that it depends only on the two dimensional marginals P^ ^, and not on any higher order structural properties of the process. This fact makes it more ubiquitous than more refined results. 2.1.3 Theorem. Let p and ^ be continuous, strictly increasing functions on [0, oo) such that p(0) = ^(0) = 0 ami lim, , ^ ^(r) = oo. Given T>0 and (pe C([0, 7], R'l if

then for 0 <s 1} ^ [0,to] as follows. Given f„_ 1, define d„^ i by p(d„_ i) = ip(r„_ J. Choose t„ e (0, ^„_ i) so that (1.4)

I(t„) 0 and S is a countable subset of [0, r ] , then P({il/: yff \s is uniformly continuous}) = 1. Given 5, choose for each N >\ points 0 = to, ^ < ' *' < ^N, N = 7" so that if S^ = {r.^: 0 < i < N}, then S^ c 5^+1 and S c Q * 5^. Then' (1.13)

P({^'' ^ \s is uniformly continuous}) ^lin,.imp(Lsupl^f:if)i^-%(4AH), A/QoN^Qo\! s,teSN r~'^l y —-^ !/

where 7 and )9 are chosen as in Corollary 2.1.4. But if ^^^\') is defined by Nu\ (^f+l,Ar - t)\p{ti,f^) + (t - ti,N)w{U+UN) \p [t) — , ti+l,N — ti,N

^ ^ , ^ , ti^N < t < ti+iN ,

2.2. Markov Processes and Transition Probabilities

51

and \l/^^\t) = il/(T) for t > T, then ij/^^^') is continuous and an easy calculation shows that (1.12) implies E[ I iA(0 - il/^''\s) n < 3TI r - s P ^^

0 < 5 < f < 7.

Thus, Corollary 2.1.4 applied to ij/^^^ yields PIL.

sup

lMz:^^A(4Ap|) 0 there exist numbers a = a r > 0, r = r^ > 1 + OCT, and CT < oo such that (1.14)

j ^ \y - xfP,^,(dx

X dy) < C^^l? - s\^^\

0<s 0 and T e ^^i, it is clear that P(iA(0) G A, ^(t) e r ) = [ P(0, y; r, r)Fo(^>') •'A

= £^[F(0,tp(0);^,r),tp(0)GA], and therefore: P(iA(r) e r I (7[iA(0)]) = P(0, ,A(0); ^ r )

(a.s., P).

Next let 0 < 5 < r and r e ^^a be given and suppose that 0 < MJ < • • • < M„ = 5 and T i , . . . , r„ e ^^d are chosen. Then

p(^A(t/i)eri,...,^A(«„)er„,^(r)6r) = ^«,....,„„,r(ri x " x r „ x r ) \\

Pis;y„;t,r)P

rix...xr„ 'PI = £''[P(s, ^s); t, n

il,(u,) € r „ ..., ^(u„) 6 r j .

Thus an easy application Exercise 1.5.1 implies that (2.3) holds. Finally, assume that P on ((R^f^' *\ ^^Rd)io,oo)) satisfies (2.2) and (2.3). We want to check that its finite dimensional distributions {Q,j^ ^,J are {P^^^ J. Clearly (2.2) implies Qo = PQ- TO complete the identification we use induction on n. If n= 1 and t^ = 0, we have aheady checked Q^^ = P^ j. If n = 1 and ^i > 0, then for T i e ^Rd:

Q,,(r,) = P(iA(ri) e r j = £^P(0, iA(0); r^, T,)] = J P(0, >;; r„ r,)Qo(^y) = | m

yi h. r,)Po(dy)

= ^r.(r,). Now assume that Q,, / : (Ry -• K, we have

t = Pti

t - Then for any bounded measurable

54

2. Markov Processes, Regularity of their Sample Paths, and the Wiener Measure

In particular, if r^, ..., r„+ ^ e ^^d and

/(yi. •••. yn) = xu(yi) ••• XrSyn)P(tn^ 3^n; t„+u r„+i), then, by (2.3),

\\

P(tn^ yn; ^«+i, ^yn+i)^ri

J^yi x ••• x ^y„)

rix...xr„+i

= Pu,...,t„.A^i x . . . x r „ ^ i ) . Thus the proof can be completed by another application of Exercise 1.5.1.

D

Of course, there is no reason why a Markov process should always have to be realized on ({R^f^- °°^ ^(R4)io.ao)). In fact, we want the following definition. 2.2.3 Definition. Let (£, J^, P) be a probability space and {J^^: r > 0 } a nondecreasing family of sub R*^, the triple (^(t), J^,, P) is called a Markov process on (E, ^) with transition probability function P(s, x; t, •) and initial distribution ^ if i(t) is ^,-measurable for all r > 0 and (2.5)

P ( ^ ( 0 ) G r ) = /i(r),

Te^Rd.

and (2.6)

F((^(r) G r IJ^,) = F(s, ^(5); r, F)

(a.s., P)

for all 0 < 5 < r and T e ^^d. Notice that if £ = (R'^y^- °">, J^^ = (^R^T' '^\ and J^, = (T[II/(U): 0 < M < r], then the preceding definition is consistent with the one given in 2.2. L The case in which we will be most interested is when E = Q^^ = Ji,^^ = ^^, and ^(t) = x(t). In fact, if (x(r). Jit, P) is a Markov process on (Q, Jt\ we will call it a continuous Markov process. 2.2.4 Theorem. Let P(s,x; t, •) be a transition probability function such that for each T > 0 there exist a = a j > 0,r = r^ > 1 + a r , an^ C = CT for which (2.7)

sup [ \y - yil'Pit,,

yu t2. dy) 0 and X E R^, there is a unique probability measure P^ ^ on (Q, Jt) such that (2.8)

P,, ^(x{t) = X for all

0 0 and x e P** are given. Define

I

Sy^

if0 0} a nondecreasing family of sub tr-algebras of ^. Given 5 > 0 and a function j8: [0, 00) x E -• R^, we will say that (P{t), ^^, P) is a d-dimensional s-Brownian motion (alternatively, when there is no need to emphasize d or {J^,: r > 0}, ^(•) is an s-Brownian motion under P) if

2.3. Wiener Measure

(/) (a) (in) (iv)

57

P is right-continuous and progressively measurable after time s, p is P-almost surely continuous, P(P(t) = 0 for 0 < t < s) = 1, for all 5 < ^1 < ^2 and F e ^^d P(p(t2) 6 r I J ^ , J = f g,(t2 - t , , y - Pit,)) dy (a.s., P), •'r where g^ is given in equation (2.11).

If s = 0, we will call {P{t\ ^ , , P) a Brownian motion, Clearly (x(r), J^t, i^%) is an s-Brownian motion. In fact, (x(t\ M^, 1^1%) is the canonical s-Brownian motion in that if (P(t\ ^t, P) is any s-Brownian motion and P o j?~ Ms the distribution o(p(') under P on Q (note that by (n), q^ p(', q)isa. map of a set having full P-measure into Q, and therefore P o p~^ {$ well-defined on (Q, e/#)), then P ^ jS" ^ = T^^f\). The next lemma gives a partial answer to the question of why one likes to consider other versions of Brownian motion besides the canonical one. 2.3.1 Lemma. Let (£, ^, P) be a probability space and (p(t), ^^, P) an s-Brownian motion. Denote by #'(J^,) the completion of^(^t) under P and use P to denote its own extention to ^ . For t > 0, set J^,+o = no ^t+d- Then (P(t), ^ , + 0 , P) is again an s-Brownian motion. Proof. Obviously, all we have to check is that P{P(t2) e F | ^^j+o) = jr 9d(h — h^ y - Pih)) {P{t2 + a)), A]- =T7PE \ 9d(t2 -tuy-

P(ti -f e))(P(y) dy, A

Since j5(-) is right-continuous, we can now let e \ 0 and thereby get (3.1).

D

We now want to prove one of the basic properties of Brownian motion, namely: "it starts afresh after a stopping time." The first step is the following lemma.

58

2. Markov Processes, Regularity of their Sample Paths, and the Wiener Measure

2.3.2 Lemma. Let (P(t), ^^, P) he an s-Brownian motion. Given to > s, define PtoiU Q) = P(t + ^0» ^) ~ P(h, q\t >0 and q e E. If is a bounded Jt-measurable function on Q, then (3.2)

£"[4) o ^,„ I ^ , J = E^"\]

(a.s., P).

Proof. We need only prove (3.2) for l, and (^i, ...,„ e Q(R''), then for 0 < fi < • • • < r„: (3.3)

£l(/>i(Ao(^i)) • • • UPto(tn)\ A] = E^'\ct>Mh)) • • • „(x(t„))]F(^).

To this end, let w > s and <j) e Q(/?'') be given; Then for D > M and A € ^ „ : E-'WM'

A] = f Elmv + u)- fS(u,

q))\^MP(dq)

= f j f 4>(y - P{u, q))gAv, y - P(u, q)) dyy(dq) = ( j {y)gAv, y) dyj^i/i). Thus (3.4)

E'{cj>{pM \^u] = \ {y)g,(v, y) dy

(a.s., P).

In particular, (3.4) proves (3.3) when n= \. Next suppose that (3.3) holds for n. Let 0 < fi < ••• < r„+i and (/>i, ..., (/>„+i e Cfc(R'') be given. Applying (3.4) to M = r„ + ro, u = r„+1 - r„, and 0(>') = 0„+ i(y + z), we have: £:''[^„+i(A„+J^n+i-0 + ^)l^r„+J

= I 0n+ i{y)9d(tn^ ^-t^,y-z)dy

(a.s., P).

59

2.3. Wiener Measure

Thus, since (3.3) holds for n: £l0i(/J,„(ti))-n+i(y)9d(tn+i -t„.y-

x ( 0 ) \p(A)

= £ni(^(^i)) •• 0„(x(r„))(/>„,iWr„,O)]P(A) Thus the induction is complete.

D

2.3.3 Theorem. / / {P(t\ i^,, P) is an s-Brownian motion and T is a stopping time satisfying T > s define p^(') by: \P(t^ oo. 2.4.3. Prove Theorem 2.2.6. 2.4.4 Prove that if P(') is any Brownian motion, then

E[\m-p(s)n==c,\t-s\\

o<s 1. Deduce from this that for any 0 < a < 1/2, p(') is almost surely Holder continuous with exponent a on any finite time interval. 2.4.5. Given A > 0, define 5^: Q -^ Q by: x(r, 5^co) = A" ^'^x(h, (o). Show that ir^"^^ is invariant under S^ (i.e., iT^"^^ = iT^'^^ « S^^). Using this fact, prove that a Brownian motion is almost surely not Holder continuous with exponent 1/2 even at one time point. With a little more effort one can show that Brownian motion has an exact modulus of continuity: (231 log ^ | Y'^ (cf. McKean [1969]). 2.4.6. Let a > 0 be given. Using Theorem 2.3.3, derive the following equality. ir^^\x(t) >a) = ii^^^^\x(t) >a,T 0: x(t) > a}. Conclude that ds. ir^'Hr 0,

then the distribution of (^(•) under P is again iT^'^K That is, ((^(r), J^;, P) is a Brownian motion, where ^[ = -g(sl

T) x R'^) n C„([0, T] x R**)

0 < s < T,

then for 0 <s