second edition - tsinghua universityfaculty.math.tsinghua.edu.cn/~zliang/paper/ioannis karatzas,...

Ioannis Karatzas Steven E. Shreve

Brownian Motionand Stochastic Calculus

Second Edition

With 10 Illustrations

Springer-VerlagNew York Berlin Heidelberg LondonParis Tokyo Hong Kong Barcelona

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Ioannis KaratzasDepartment of StatisticsColumbia UniversityNew York, NY 10027USA

Editorial BoardJ.H. EwingDepartment of

MathematicsIndiana UniversityBloomington,Indiana 47405USA

Steven E. ShreveDepartment of MathematicsCarnegie Mellon UniversityPittsburgh, PA 15213USA

F.W. GehringDepartment of

MathematicsUniversity of MichiganAnn Arbor,Michigan 48109USA

P.R. HalmosDepartment of

MathematicsUniversity of Santa ClaraSanta Clara,California 95053USA

AMS Classifications: 60G07, 60H05

Library of Congress Cataloging-in-Publication DataKaratzas, Ioannis.

Brownian motion and stochastic calculus/ Ioannis Karatzas, StevenE. Shreve.-2nd ed.

p. cm.-(Graduate texts in mathematics; 113)Includes bibliographical references and index.ISBN 0-387-97655-8 (New York).-ISBN 3-540-97655-8 (Berlin)1. Brownian motion processes. 2. Stochastic analysis.

I. Shreve, Steven E. II. Title. III. Series.QA274.75.K37 1991530.4'75-dc20 91-22775

The present volume is the corrected softcover second edition of the previously published hard-cover first edition (ISBN 0-387-96535-1).

Printed on acid-free paper.

© 1988, 1991 by Springer-Verlag New York, Inc.All rights reserved. This work may not be translated or copied in whole or in part without thewritten permission of the publisher (Springer-Verlag, 175 Fifth Avenue, New York, New York10010, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use inconnection with any form of information storage and retrieval, electronic adaptation, computersoftware, or by similar or dissimilar methodology now known or hereafter developed is forbid-den. The use of general descriptive names, trade names, trademarks, etc. in this publication, evenif the former are not especially identified, is not to be taken as a sign that such names, asunderstood by the Trade Marks and Merchandise Marks Act, may accordingly by used freelyby anyone.

Typeset by Asco Trade Typesetting Ltd., Hong Kong.Printed and bound by R.R. Donnelley and Sons, Harrisonburg, VA.Printed in the United States of America.

9 8 7 6 5 4 3 2 1

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


To Eleni and Dot


Preface

Two of the most fundamental concepts in the theory of stochastic processesare the Markov property and the martingale property.* This book is writtenfor readers who are acquainted with both of these ideas in the discrete-timesetting, and who now wish to explore stochastic processes in their continuous-time context. It has been our goal to write a systematic and thorough exposi-tion of this subject, leading in many instances to the frontiers of knowledge.At the same time, we have endeavored to keep the mathematical prerequisitesas low as possible, namely, knowledge of measure-theoretic probability andsome familiarity with discrete-time processes. The vehicle we have chosen forthis task is Brownian motion, which we present as the canonical example ofboth a Markov process and a martingale. We support this point of view byshowing how, by means of stochastic integration and random time change,all continuous-path martingales and a multitude of continuous-path Markovprocesses can be represented in terms of Brownian motion. This approachforces us to leave aside those processes which do not have continuous paths.Thus, the Poisson process is not a primary object of study, although it isdeveloped in Chapter 1 to be used as a tool when we later study passage timesand local time of Brownian motion.

The text is organized as follows: Chapter 1 presents the basic properties ofmartingales, as they are used throughout the book. In particular, we generalizefrom the discrete to the continuous-time context the martingale convergencetheorem, the optional sampling theorem, and the Doob-Meyer decomposi-tion. The latter gives conditions under which a submartingale can be written

* According to M. Loeve, "martingales, Markov dependence and stationarity are the only threedependence concepts so far isolated which are sufficiently general and sufficiently amenable toinvestigation, yet with a great number of deep properties" (Ann. Probab. 1 (1973), p. 6).


viii Preface

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

Chapter 2 contains three different constructions of Brownian motion,as well as discussions of the Markov and strong Markov properties forcontinuous-time processes. These properties are motivated by d-dimensionalBrownian motion, but are developed in complete generality. This chapter alsocontains a careful discussion of the various filtrations commonly associatedwith Brownian motion. In Section 2.8 the strong Markov property is appliedto a study of one-dimensional Brownian motion on a half-line, and on abounded interval with absorption and reflection at the endpoints. Manydensities involving first passage times, last exit times, absorbed Brownianmotion, and reflected Brownian motion are explicitly computed. Section 2.9is devoted to a study of sample path properties of Brownian motion. Resultsfound in most texts on this subject are included, and in addition to these, acomplete proof of the Levy modulus of continuity is provided.

The theory of stochastic integration with respect to continuous martingalesis developed in Chapter 3. We follow a middle path between the originalconstructions of stochastic integrals with respect to Brownian motion and themore recent theory of stochastic integration with respect to right-continuousmartingales. By avoiding discontinuous martingales, we obviate the need tointroduce the concept of predictability and the associated, highly technical,measure-theoretic machinery. On the other hand, it requires little extra effortto consider integrals with respect to continuous martingales rather thanmerely Brownian motion. The remainder of Chapter 3 is a testimony to thepower of this more general approach; in particular, it leads to strong theoremsconcerning representations of continuous martingales in terms of Brownianmotion (Section 3.4). In Section 3.3 we develop the chain rule for stochasticcalculus, commonly known as It6"s formula. The Girsanov Theorem of Sec-tion 3.5 provides a method of changing probability measures so as to alterthe drift of a stochastic process. It has become an indispensable method forconstructing solutions of stochastic differential equations (Section 5.3) and isalso very important in stochastic control (e.g., Section 5.8) and filtering. Localtime is introduced in Sections 3.6 and 3.7, and it is shown how this conceptleads to a generalization of the Ito formula to convex but not necessarilydifferentiable functions.

Chapter 4 is a digression on the connections between Brownian motion,Laplace's equation, and the heat equation. Sharp existence and uniquenesstheorems for both these equations are provided by probabilistic methods;applications to the computation of boundary crossing probabilities are dis-cussed, and the formulas of Feynman and Kac are established.

Chapter 5 returns to our main theme of stochastic integration and differ-ential equations. In this chapter, stochastic differential equations are driven


Preface ix

by Brownian motion and the notions of strong and weak solutions are pre-sented. The basic Ito theory for strong solutions and some of its ramifications,including comparison and approximation results, are offered in Section 5.2,whereas Section 5.3 studies weak solutions in the spirit of Yamada &Watanabe. Essentially equivalent to the search for a weak solution is thesearch for a solution to the "Martingale Problem" of Stroock & Varadhan.In the context of this martingale problem, a full discussion of existence,uniqueness, and the strong Markov property for solutions of stochastic differ-ential equations is given in Section 5.4. For one-dimensional equations it ispossible to provide a complete characterization of solutions which exist onlyup to an "explosion time," and this is set forth in Section 5.5. This section alsopresents the recent and quite striking results of Engelbert & Schmidt con-cerning existence and uniqueness of solutions to one-dimensional equations.This theory makes substantial use of the local time material of Sections 3.6,3.7 and the martingale representation results of Subsections 3.4.A,B. Byanalogy with Chapter 4, we discuss in Section 5.7 the connections betweensolutions to stochastic differential equations and elliptic and parabolic partialdifferential equations. Applications of many of the ideas in Chapters 3 and 5are contained in Section 5.8, where we discuss questions of option pricingand optimal portfolio/consumption management. In particular, the Girsanovtheorem is used to remove the difference between average rates of returnof different stocks, a martingale representation result provides the optimalportfolio process, and stochastic representations of solutions to partial differ-ential equations allow us to recast the optimal portfolio and consumptionmanagement problem in terms of two linear parabolic partial differentialequations, for which explicit solutions are provided.

Chapter 6 is for the most part derived from Paul Levy's profound study ofBrownian excursions. Levy's intuitive work has now been formalized by suchnotions as filtrations, stopping times, and Poisson random measures, but theremarkable fact remains that he was able, 40 years ago and working withoutthese tools, to penetrate into the fine structure of the Brownian path and toinspire all the subsequent research on these matters until today. In the spiritof Levy's work, we show in Section 6.2 that when one travels along theBrownian path with a clock run by the local time, the number of excursionsaway from the origin that one encounters, whose duration exceeds a specifiednumber, has a Poisson distribution. Levy's heuristic construction of Brownianmotion from its excursions has been made rigorous by other authors. We donot attempt such a construction here, nor do we give a complete specificationof the distribution of Brownian excursions; in the interest of intelligibility, wecontent ourselves with the specification of the distribution for the durationsof the excursions. Sections 6.3 and 6.4 derive distributions for functionalsof Brownian motion involving its local time; we present, in particular, aFeynman-Kac result for the so-called "elastic" Brownian motion, the for-mulas of D. Williams and H. Taylor, and the Ray-Knight description of


x Preface

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

The writing of this book has become for us a monumental undertakinginvolving several-people, whose assistance we gratefully acknowledge. Fore-most among these are the members of our families, Eleni, Dot, Andrea, andMatthew, whose support, encouragement, and patience made the whole en-deavor possible. Parts of the book grew out of notes on lectures given atColumbia University over several years, and we owe much to the audiencesin those courses. The inclusion of several exercises, the approaches taken toa number of theorems, and several citations of relevant literature resultedfrom discussions and correspondence with F. Baldursson, A. Dvoretzky,W. Fleming, 0. Kallenberg, T. Kurtz, S. Lalley, J. Lehoczky, D. Stroock, andM. Yor. We have also taken exercises from Mandl, Lanska & Vrkoe (1978),and Ethier & Kurtz (1986). As the project proceeded, G.-L. Xu, Z.-L. Ying,and Th. Zariphopoulou read large portions of the manuscript and suggestednumerous corrections and improvements. Careful reading by Daniel Oconeand Manfred Schal revealed minor errors in the first printing, and these havebeen corrected. However, our greatest single debt of gratitude goes to MarcYor, who read much of the near-final draft and offered substantial mathemat-ical and editorial comments on it. The typing was done tirelessly, cheerfully,and efficiently by Stella De Vito and Doodmatie Kalicharan; they have ourmost sincere appreciation.

We are grateful to Sanjoy Mitter and Dimitri Bertsekas for extendingto us the invitation to spend the critical initial year of this project at theMassachusetts Institute of Technology. During that time the first four chap-ters were essentially completed, and we were partially supported by the ArmyResearch Office under grant DAAG-299-84-K-0005. Additional financial sup-port was provided by the National Science Foundation under grants DMS-84-16736 and DMS-84-03166 and by the Air Force Office of Scientific Researchunder grants AFOSR 82-0259, AFOSR 85-0360, and AFOSR 86-0203.

Ioannis KaratzasSteven E. Shreve


Contents

Preface vii

Suggestions for the Reader xvii

Interdependence of the Chapters xi x

Frequently Used Notation xxi

CHAPTER 1

Martingales, Stopping Times, and Filtrations 1

1.1. Stochastic Processes and a-Fields 1

1.2. Stopping Times 61.3. Continuous-Time Martingales 11

A. Fundamental inequalities 12B. Convergence results 17C. The optional sampling theorem 19

1.4. The Doob-Meyer Decomposition 211.5. Continuous, Square-Integrable Martingales 301.6. Solutions to Selected Problems 381.7. Notes 45

CHAPTER 2

Brownian Motion 47

2.1. Introduction 472.2. First Construction of Brownian Motion 49

A. The consistency theorem 49B. The Kolmogorov-entsov theorem 53

2.3. Second Construction of Brownian Motion 562.4. The Space C[0, co), Weak Convergence, and Wiener Measure 59

A. Weak convergence 60


xii Contents

B. Tightness 61C. Convergence of finite-dimensional distributions 64D. The invariance principle and the Wiener measure 66

2.5. The Markov Property 71A. Brownian motion in several dimensions 72B. Markov processes and Markov families 74C. Equivalent formulations of the Markov property 75

2.6. The Strong Markov Property and the Reflection Principle 79A. The reflection principle 79B. Strong Markov processes and families 81C. The strong Markov property for Brownian motion 84

2.7. Brownian Filtrations 89A. Right-continuity of the augmented filtration for a

strong Markov process 90B. A "universal" filtration 93C. The Blumenthal zero-one law 94

2.8. Computations Based on Passage Times 94A. Brownian motion and its running maximum 95B. Brownian motion on a half-line 97C. Brownian motion on a finite interval 97D. Distributions involving last exit times 100

2.9. The Brownian Sample Paths 103A. Elementary properties 103B. The zero set and the quadratic variation 104C. Local maxima and points of increase 106D. Nowhere differentiability 109E. Law of the iterated logarithm 111F. Modulus of continuity 114

2.10. Solutions to Selected Problems 1162.11. Notes 126

CHAPTER 3

Stochastic Integration 128

3.1. Introduction 1283.2. Construction of the Stochastic Integral 129

A. Simple processes and approximations 132B. Construction and elementary properties of the integral 137C. A characterization of the integral 141D. Integration with respect to continuous, local martingales 145

3.3. The Change-of-Variable Formula 148A. The Ito rule 149B. Martingale characterization of Brownian motion 156C. Bessel processes, questions of recurrence 158D. Martingale moment inequalities 163E. Supplementary exercises 167

3.4. Representations of Continuous Martingales in Terms ofBrownian Motion 169A. Continuous local martingales as stochastic integrals with

respect to Brownian motion 170


Contents xiii

B. Continuous local martingales as time-changed Brownian motions I73C. A theorem of F. B. Knight 179D. Brownian martingales as stochastic integrals 180E. Brownian functionals as stochastic integrals 185

3.5. The Girsanov Theorem 190A. The basic result 191

B. Proof and ramifications 193C. Brownian motion with drift 196D. The Novikov condition 198

3.6. Local Time and a Generalized It8 Rule for Brownian Motion 20IA. Definition of local time and the Tanaka formula 203B. The Trotter existence theorem 206C. Reflected Brownian motion and the Skorohod equation 210D. A generalized Ito rule for convex functions 212E. The Engelbert-Schmidt zero-one law 215

3.7. Local Time for Continuous Semimartingales 2173.8. Solutions to Selected Problems 2263.9. Notes 236

CHAPTER 4

Brownian Motion and Partial Differential Equations 239

4.1. Introduction 2394.2. Harmonic Functions and the Dirichlet Problem 240

A. The mean-value property 241B. The Dirichlet problem 243C. Conditions for regularity 247D. Integral formulas of Poisson 251E. Supplementary exercises 253

4.3. The One-Dimensional Heat Equation 254A. The Tychonoff uniqueness theorem 255B. Nonnegative solutions of the heat equation 256C. Boundary crossing probabilities for Brownian motion 262D. Mixed initial/boundary value problems 265

4.4. The Formulas of Feynman and Kac 267A. The multidimensional formula 268B. The one-dimensional formula 271

4.5. Solutions to selected problems 2754.6. Notes 278

CHAPTER 5

Stochastic Differential Equations 281

5.1. Introduction 2815.2. Strong Solutions 284

A. Definitions 285B. The Ito theory 286C. Comparison results and other refinements 291D. Approximations of stochastic differential equations 295E. Supplementary exercises 299


xiv Contents

5.3. Weak Solutions 300A. Two notions of uniqueness 301B. Weak solutions by means of the Girsanov theorem 302C. A digression on regular conditional probabilities 306D. Results of Yamada and Watanabe on weak and strong solutions 308

5.4. The Martingale Problem of Stroock and Varadhan 311A. Some fundamental martingales 312B. Weak solutions and martingale problems 314C. Well-posedness and the strong Markov property 319D. Questions of existence 323E. Questions of uniqueness 325F. Supplementary exercises 328

5.5. A Study of the One-Dimensional Case 329A. The method of time change 330B. The method of removal of drift 339C. Feller's test for explosions 342D. Supplementary exercises 35 I

5.6. Linear Equations 354A. Gauss-Markov processes 355B. Brownian bridge 358C. The general, one-dimensional, linear equation 360D. Supplementary exercises 361

5.7. Connections with Partial Differential Equations 363A. The Dirichlet problem 364B. The Cauchy problem and a Feynman-Kac representation 366C. Supplementary exercises 369

5.8. Applications to Economics 371A. Portfolio and consumption processes 371B. Option pricing 376C. Optimal consumption and investment (general theory) 379D. Optimal consumption and investment (constant coefficients) 381


CHAPTER 6P. Levy's Theory of Brownian Local Time 399

6.1. Introduction 3996.2. Alternate Representations of Brownian Local Time 400

A. The process of passage times 400B. Poisson random measures 403C. Subordinators 405D. The process of passage times revisited 411E. The excursion and downcrossing representations of local time 414

6.3. Two Independent Reflected Brownian Motions 418A. The positive and negative parts of a Brownian motion 418B. The first formula of D. Williams 421C. The joint density of (W(t), L(t), f (t)) 423


Contents xv

6.4. Elastic Brownian Motion 425A. The Feynman-Kac formulas for elastic Brownian motion 426B. The Ray-Knight description of local time 430C. The second formula of D. Williams 434

6.5. An Application: Transition Probabilities of Brownian Motion withTwo-Valued Drift 437


Bibliography 447

Index 459


Suggestions for the Reader

We use a hierarchical numbering system for equations and statements. Thek-th equation in Section j of Chapter i is labeled (j.k) at the place where itoccurs and is cited as (j.k) within Chapter i, but as (i j.k) outside Chapter i. Adefinition, theorem, lemma, corollary, remark, problem, exercise, or solutionis a "statement," and the k-th statement in Section j of Chapter i is labeled j.kStatement at the place where it occurs, and is cited as Statement j.k withinChapter i but as Statement i.j.k outside Chapter i.

This book is intended as a text and can be used in either a one-semester ora two-semester course, or as a text for a special topic seminar. The accompany-ing figure shows dependences among sections, and in some cases amongsubsections. In a one-semester course, we recommend inclusion of Chapter 1and Sections 2.1, 2.2, 2.4, 2.5, 2.6, 2.7, §2.9.A, B, E, Sections 3.2, 3.3, 5.1, 5.2,and §5.6.A, C. This material provides the basic theory of stochastic integration,including the Ito calculus and the basic existence and uniqueness results forstrong solutions of stochastic differential equations. It also contains mattersof interest in engineering applications, namely, Fisk-Stratonovich integralsand approximation of stochastic differential equations in §3.3.A and 5.2.D,and Gauss-Markov processes in §5.6.A. Progress through this material canbe accelerated by omitting the proof of the Doob-Meyer DecompositionTheorem 1.4.10 and the proofs in §2.4.D. The statements of Theorem 1.4.10,Theorem 2.4.20, Definition 2.4.21, and Remark 2.4.22 should, however, beretained. If possible in a one-semester course, and certainly in a two-semestercourse, one should include the topic of weak solutions of stochastic differentialequations. This is accomplished by covering §3.4.A, B, and Sections 3.5, 5.3,and 5.4. Section 5.8 serves as an introduction to stochastic control, and so werecommend adding §3.4.C, D,.E, and Sections 5.7, and 5.8 if time permits. Ineither a one- or two-semester course, Section 2.8 and part or all of Chapter 4


xviii Suggestions for the Reader

may be included according to time and interest. The material on local timeand its applications in Sections 3.6, 3.7, 5.5, and in Chapter 6 would normallybe the subject of a special topic course with advanced students.

The text contains about 175 "problems" and over 100 "exercises." Theformer are assignments to the reader to fill in details or generalize a result,and these are often quoted later in the text. We judge approximately two-thirds of these problems to be nontrivial or of fundamental importance, andsolutions for such problems are provided at the end of each chapter. Theexercises are also often significant extensions of results developed in thetext, but these will not be needed later, except perhaps in the solution ofother exercises. Solutions for the exercises are not provided. There are someexercises for which the solution we know violates the dependencies amongsections shown in the figure, but such violations are pointed out in theoffending exercises, usually in the form of a hint citing an earlier result.


Interdependence of the Chapters

Chapter 1

2.1

2.2 2.3

2.4

2.5

2.6

2.7 2.8

§2.9.A, B, E §2.9.C, D, F

3.2 624.2

4.3 3.3 3.6

§3.4.A,4.4 B 3.7

§3.4.C, D, E

5.1 35 55

5.2 5.3

§5.6.A, C 5.4 6.3

§5.6.B, D 5.7 [ 6.4[ 5.8,

L 6.5 1


Frequently Used Notation

I. General Notation

Let a and b be real numbers.(1) A means "is defined to be."(2) anbA min la, bl.(3) a v bA max {a, b}.(4) a+ A max{a, 0}.(5) a- A max{ -a,0}.

II. Sets and Spaces

(1) No A {0,1, 2, ... }.(2) Q is the set of rational numbers.(3) Q+ is the set of nonnegative rational numbers.(4) Rd is the d-dimensional Euclidean space; R1 = R.(5) B, A {x e Rd; Ilx II < (p. 240).(6) (Rd)1°') is the set of functions from [0, co) to Rd (pp. 49, 76).(7) C[0, cc)" is the subspace of (Rd)'0'') consisting of continuous functions;

C[0, oo)1 = C[0, co) (pp. 60, 64).(8) D[0, co) is the subspace of RE°'"') consisting of functions which are right

continuous and have left-limits (p. 409).(9) 0E), CNE), (E): See Remark 4.1, p. 312.

(10) CI 2 ( [0, T) x E), 2 ((3, T) x E): See Remark 4.1, p. 312.(11) .29, Sf(M), .29*, 2) *(M): See pp. 130-131.(12) P1(M), P1 *(M): See pp. 146-147.(13) .1/2(11`2): The space of (continuous) square-integrable martingales (p. 30).(14) ../rf-1.(.11`.1"): The space of (continuous) local martingales (p. 36).

xxii Frequently Used Notation

III. Functions

(1) sgn(x) =1; x > 0,

- 1; x < 0.x e A,

(2) 1A(x) 5.1;10; x A.

(3) p(t; x, y)/2e-(x-Y)212'; t > 0, x, ye R (p. 52).

nt(4) p+ (t; x, y) p(t; x, y) ± p(t; x, - y); t > 0, x, y e R (p. 97).(5) It]] is the largest integer less than or equal to the real number t.

IV. a-Fields

(1) .4(U): The smallest 6 -field containing all open sets of the topologicalspace U (p. 1).

(2) at(c [0, co)), A(C[0, oo)d). See pp. 60, 307.(3) a(W): The smallest o--field containing the collection of sets W.(4) o-(Xs): The smallest a-field with respect to which the random variable X,

is measurable.(5) o-(Xs; 0 < s < t): The smallest a-field with respect to which the random

variable X, is measurable, V s e [0, t].(6) A o-(Xs; 0 s t), ..Foo A o-(U,,o,97,): See p. 3.

(7) nc>0.Fr+c, 0-(us<g-,$): See p. 4.(8) .97T: The a-field of events determined prior to the stopping time T; see

p. 8.

(9) + The a-field of events determined immediately after the optionaltime T; see p. 10.

(10) .97 W A o-(A x B;Ae.F,Bel: The product a-field formed from thea-fields ,F and W.

V. Operations on Functions

d 02(1) A A E ,: The Laplacian (p. 240).

ax;(2) .91, sit: Second order differential operators; see pp. 281, 311.

VI. Operations on Processes

(1) Os, Os: Shift operator at the deterministic time s and the random time S;see pp. 77, 83.

Frequently Used Notation xxiii

(2) /1"(X) A j) XS dMs: The stochastic integral of X with respect to M. Seep. 141 for M e .41(.2, X e ,99* (M); see p. 147 for M e dr.'" X e 1)*(M).

(3) Mt* -A maxs<, I Msl: See p. 163 for M e dif`.1".(4) <X>: The quadratic variation process of X e d12 (p. 31) or X e dr'

(p. 36).(5) <X, Y>: The cross-variation process of X, Y in /2 (p. 31) or in i`.'"

(p. 36).(6) II XL, II XII: See p. 37 for X e .412-

VII. Miscellaneous

(1) mT(X, 6) A sup{ I Xs - X,I; 0 -s<t-T,t-s-.5}; See p. 33.(2) mT(a),6)4 max { I w(s) - w(t)I; 0 -s<t - s 61: See p. 62.(3) D: The closure of the set D Rd.

(4) DC: The complement of the set D.(5) OD: The boundary of the set D Rd.

(6) ID 4' inf{t > 0; W e Dc}: The first time the Brownian motion W exits fromthe set D Rd (p. 240).

(7) Tb A inf{t > 0; W = 6}: The first time the one-dimensional Brownianmotion W reaches the level be (p. 79).

(8) 1",(t) 1(0)(Ws)ds: The occupation time by Brownian motion of thepositive half-line (p. 273).

(9) P. P: Weak convergence of the sequence of probability measures{P};',°=1 to the probability measure P (p. 60).

(10) X,, '4 X: Convergence in distribution of the sequence of random variables{X}.°=, to the random variable X (p. 61).

(11) P": Probability measure corresponding to Brownian motion (p. 72) or aMarkov process (p. 74) with initial position x e Rd.

(12) Pµ: Probability measure corresponding to Brownian motion (p. 72) or aMarkov process (p. 74) with initial distribution tt.

(13) ,P, ,ACP: Collections of PP-negligible sets (p. 89).(14) 1(a), Z(o-): See pp. 331, 332.(15) Id: The (d x d) identity matrix.(16) meas: Lebesgue measure on the real line (p. 105).

CHAPTER 1

Martingales, Stopping Times,and Filtrations

1.1. Stochastic Processes and a-Fields

A stochastic process is a mathematical model for the occurrence, at eachmoment after the initial time, of a random phenomenon. The randomness iscaptured by the introduction of a measurable space called the samplespace, on which probability measures can be placed. Thus, a stochastic processis a collection of random variables X = {X,; 0 < t < co} on (a .97), whichtake values in a second measurable space (S, 99), called the state space. Forour purposes, the state space (S, 9') will be the d-dimensional Euclidean spaceequipped with the a-field of Borel sets, i.e., S = Rd, = R(1 W), where 9B(U)will always be used to denote the smallest a-field containing all open sets ofa topological space U. The index t e [0, co) of the random variables X, admitsa convenient interpretation as time.

For a fixed sample point w a 12, the function X,(w); t > 0 is the samplepath (realization, trajectory) of the process X associated with w. It providesthe mathematical model for a random experiment whose outcome can beobserved continuously in time (e.g., the number of customers in a queueobserved and recorded over a period of time, the trajectory of a moleculesubjected to the random disturbances of its neighbors, the output of a com-munications channel operating in noise).

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

2 1. Martingales, Stopping Times, and Filtrations

1.1 Definition. Y is a modification of X if, for every t 0, we haveP[X, = y] = 1.

1.2 Definition. X and Y have the same finite-dimensional distributions if, forany integer n > 1, real numbers 0 < t, < t2 < < t < cc, and A e .4(Ord),we have:

P[(X,,,... , Xin) e A] = ..., Y,n)EA].

1.3 Definition. X and Y are called indistinguishable if almost all their samplepaths agree:

P[X, = Yt; V 0 < t < cc] = 1.

The third property is the strongest; it implies trivially the first one, whichin turn yields the second. On the other hand, two processes can be modifica-tions of one another and yet have completely different sample paths. Here isa standard example:

1.4 Example. Consider a positive random variable T with a continuous dis-0; t T

tribution, put X, 0, and let Y, = Y is a modification of X, since1; t = T

for every t > 0 we have P[Y, = = PET z t] = 1, but on the other hand:P[Y, Xi; V t 0] = 0.

A positive result in this direction is the following.

1.5 Problem. Let Y be a modification of X, and suppose that both processeshave a.s. right-continuous sample paths. Then X and Y are indistinguishable.

It does not make sense to ask whether Y is a modification of X, or whetherY and X are indistinguishable, unless X and Y are defined on the sameprobability space and have the same state space. However, if X and Y havethe same state space but are defined on different probability spaces, we canask whether they have the same finite-dimensional distributions.

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

, Xt.) e A] = Y,n)EA].

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

1.1. Stochastic Processes and o -Fields 3

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

For technical reasons in the theory of Lebesgue integration, probabilitymeasures are defined on a-fields and random variables are assumed to bemeasurable with respect to these a-fields. Thus, implicit in the statement thata random process X = {Xi; 0 t < co } is a collection of (Rd, .4(Rd))-valuedrandom variables on (fl, F), is the assumption that each X, is /.4(E4d)-

measurable. However, X is really a function of the pair of variables (t, w), andso, for technical reasons, it is often convenient to have some joint measurabilityproperties.

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

(t, Xt(co): ( [0, co) x M( [0, oo)) .F) -÷ (Rd, mold))

is measurable.

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

f,X,' dt < co a.s. P, and ji EX, dt = E X, dt.

There is a very important, nontechnica reason to include a-fields in thestudy of stochastic processes, and that is to keep track of information. Thetemporal feature of a stochastic process suggests a flow of time, in which, atevery moment t > 0, we can talk about a past, present, and future and can askhow much an observer of the process knows about it at present, as com-pared to how much he knew at some point in the past or will know at somepoint in the future. We equip our sample space (f2, ,F) with a filtration,i.e., a nondecreasing family {A; t > 0} of sub-a-fields of .97 for0 s < t < co. We set d.6-co =-- a(Ut>o

Given a stochastic process, the simplest choice of a filtration is that gen-erated by the process itself, i.e.,

A' A o-(Xs; 0 < s < t),

the smallest a-field with respect to which X, is measurable for every s e [0, t].

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

1.7 Exercise. Let X be a process, every sample path of which is RCLL (i.e.,right-continuous on [0, co) with finite left -hand limits on (0, co)). Let A be theevent that X is continuous on [0, to). Show that A e

1.8 Exercise. Let X be a process whose sample paths are RCLL almost surely,and let A be the event that X is continuous on [0, to). Show that A can fail tobe in .??-tox, but if {.";; t > 0} is a filtration satisfying .??7,' t 0, and Fto iscomplete under P, then A Eo.

Let {,56;; t > 0} be a filtration. We define cr(Us 0 and Ft, nE>0,,,E to be the a-field of eventsimmediately after t > 0. We decree .50_ A .yo and say that the filtration {..F,}is right- (left-)continuous if = (resp., 3,7,_) holds for every t > 0.

The concept of measurability for a stochastic process, introduced in Defini-tion 1.6, is a rather weak one. The introduction of a filtration {g} opens upthe possibility of more interesting and useful concepts.

1.9 Definition. The stochastic process X is adapted to the filtration {.F,} if, foreach t > 0, X, is an ..F- measurable random variable.

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

1.10 Exercise. Let X be a process with every sample path LCRL (i.e., left -continuous on (0, co) with finite right-hand limits on [0, co)), and let A be theevent that X is continuous on [0, to]. Let X be adapted to a right-continuousfiltration {.Ft}. Show that A e "o.

1.11 Definition. The stochastic process X is called progressively measurablewith respect to the filtration {,;} if, for each t > 0 and A e gii(Rd), the set{(s, a)); 0 < s < t, w e SZ, Xs(w) e A} belongs to the product a-field .4([0, t])..Ft; in other words, if the mapping (s, co) 1-+ X s(w): ([0, t] x S2, .4([0, t]) 0 g") -+(Rd M(Rd)) is measurable, for each t 0.

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

1.1. Stochastic Processes and a-Fields 5

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

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

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

PROOF. We treat the case of right-continuity. With t > 0, n > 1, k = 0, 1,

..., 2" - 1, and 0 < s < t, we define:

X.1")((0) = X(k+1)02^(W) for -kt <s< k+lt2" - 2"

as well as Xnw) = Xo(w). The so-constructed map (s, w) X 1") (co) from[0, t] x SI into Rd is demonstrably .4([0, t]) ..Ff-measurable. Besides, byright-continuity we have: lim, )(no) = )(Jo)), V (s, w) E [0, t] x a There-fore, the (limit) map (s, XS(w) is also 9([0, t]) .F,- measurable.

1.14 Remark. If the stochastic process X is right- or left-continuous, butnot necessarily adapted to then the same argument shows that X ismeasurable.

A random time T is an "--measurable random variable, with values in[0, 09].

1.15 Definition. If X is a stochastic process and T is a random time, we definethe function XT on the event { T < co} by

XT(w) A XT00(co).

If Xco(co) is defined for all w e SI, then X,- can also be defined on SI, by settingXT(w) A X00(o) on { T = co }.

1.16 Problem. If the process X is measurable and the random time T is finite,then the function XT is a random variable.

1.17 Problem. Let X be a measurable process and T a random time. Showthat the collection of all sets of the form {XT e A }; A e R(R), together with theset { T = oo}, forms a sub-a-field of We call this the o--field generated by XT.

We shall devote our next section to a very special and extremely useful classof random times, called stopping times. These are of fundamental importancein the study of stochastic processes, since they constitute our most effectivetool in the effort to "tame the continuum of time," as Chung (1982) puts it.

1.2. Stopping Times

Let us keep in mind the interpretation of the parameter t as time, and of the(7-field as the accumulated information up to t. Let us also imagine that weare interested in the occurrence of a certain phenomenon: an earthquake withintensity above a certain level, a number of customers exceeding the safetyrequirements of our facility, and so on. We are thus forced to pay particularattention to the instant T(w) at which the phenomenon manifests itself for thefirst time. It is quite intuitive then that the event {o); T(co) < t}, which occursif and only if the phenomenon has appeared prior to (or at) time t, should bepart of the information accumulated by that time.

We can now formulate these heuristic considerations as follows:

2.1 Definition. Let us consider a measurable space (11, S) equipped with afiltration {Fr}. A random time T is a stopping time of the filtration, if the event{T < t} belongs to the (7-field ,97 for every t > 0. A random time T is anoptional time of the filtration, if { T< t} for every t 0.

2.2 Problem. Let X be a stochastic process and T a stopping time of {,Fix}.Suppose that for any w, ol e LI, we have X,(co) = Xt(o)') for all t e [0, T(w)] n[0, co). Show that T(w) = T(w').

2.3 Proposition. Every random time equal to a nonnegative constant is a stoppingtime. Every stopping time is optional, and the two concepts coincide if thefiltration is right-continuous.

PROOF. The first statement is trivial; the second is based on the observation{ T < t} = Unc°---1 t - (1/n)} e ..Ft, because if T is a stopping time, then{T < t - (l/n) } g --(1/n) for n 1. For the third claim, suppose that Tis an optional time of the right-continuous filtration {,F,}. Since {T < =nE,01T< t + el, we have { T < t} E Ft+, for every t 0 and every e > 0;whence T e ..Ft+ =

2.4 Corollary. T is an optional time of the filtration {..F,} if and only if it is astopping time of the (right-continuous!) filtration {g",+}.

25 Example. Consider a stochastic process X with right-continuous paths,which is adapted to a filtration {A}. Consider a subset F .4(Fid) of the state

1.2. Stopping Times 7

space of the process, and define the hitting time

11,-(w) = inf { t >_ 0; X, (co) e F}.

We employ the standard convention that the infimum of the empty set isinfinity.

2.6 Problem. If the set T in Example 2.5 is open, shoWthat Hr is an optionaltime.

2.7 Problem. If the set F in Example 2.5 is closed and the sample paths of theprocess X are continuous, then 11,- is a stopping time.

Let us establish some simple properties of stopping times.

2.8 Lemma. If T is optional and B is a positive constant, then T + B is a stoppingtime.

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

{T + 0 < t} = IT < t - 0} e ,F0_0), g Sr.,. 0

2.9 Lemma. If T, S are stopping times, then so are T A S, T v S, T + S.

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

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

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

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

U ft > T > r, S > t - r},re Q+0<r<t

where Q+ is the set of rational numbers in [0, co). Membership in ,°?7, is nowobvious. CI

2.10 Problem. Let T, S be optional times; then T + S is optional. It is astopping time, if one of the following conditions holds:

(i) T > 0, S > 0;(ii) T > 0, T is a stopping time.

2.11 Lemma. Let {T} n°11 be a sequence of optional times; then the random times

sup Tn, inf T, lim T, lim 'Tn> 1 n> 1 n-,:o n-co

are all optional. Furthermore, if the Tn's are stopping times, then so is supn, 1 Tn.

PROOF. Obvious, from Corollary 2.4 and from the identities

co .{ sup T,, 5 t} =n {7,, 5 t} and { inf T < t} = U I T,, < tl. El

n >1 n=1 n>1 n=1

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

A' n IT t} = IT t} n (An IT t})`,

it is enough to check only that A n IT < t} e Ft, t > 0.

2.12 Definition. Let T be a stopping time of the filtration {A}. The o--field FTof events determined prior to the stopping time T consists of those events A e Ffor which An {T < 4 E, for every t > 0.

2.13 Problem. Verify that FT is actually a 6 -field and T is FT-measurable.Show that if T(w) = t for some constant t > 0 and every w e n, then FT = A.

2.14 Exercise. Let T be a stopping time and S a random time such that S > Ton n. If S is FT-measurable, then it is also a stopping time.

2.15 Lemma. For any two stopping times T and S, and for any A e Fs, we haveA n {S < TI e FT. In particular, if S _-_ Ton Si, we have Fs FT.

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

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

which shows readily that the left-hand side is an event in A.

2.16 Lemma. Let T and S be stopping times. Then .."°-"T A s = FT n Fs, and eachof the events

IT < SI, IS < TI, IT 5 SI, IS 5 TI, IT = SI

belongs to FT n Fs.

PROOF. For the first claim we notice from Lemma 2.15 that FT,,s _c_ FT n Fs.In order to establish the opposite inclusion, let us take A E .9-'s n FT and

1.2. Stopping Times 9

observe that

An{SA Tst}=An[ISstlulTstl]= [A n IS tl] u [A n { T t}] e F

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

other hand, consider the stopping time R = S A T, which, again by virtueof Lemma 2.15, is measurable with respect to Fr. Therefore, {S < T} ={R < T} e Fr. Interchanging the roles of S, T we see that {T > S }, {T < S}belong to Fs, and thus we have shown that both these events belong toFT n Fs. But then the same is true for their complements, and consequentlyalso for {S = T}.

2.17 Problem. Let T, S be stopping times and Z an integrable random variable.We have

(i) E[ZIFT] = ECZ I's AT], P-a.s. on {TS S}(ii) E[E(ZIFT)I.F.s] = E[ZLFS A sr], P-a.s.

Now we can start to appreciate the usefulness of the concept of stoppingtime in the study of stochastic processes.

2.18 Proposition. Let X = {X Ft; 0 < t < oo} be a progressively measurableprocess, and let T be a stopping time of the filtration IF,I. Then therandom variable XT of Definition 1.15, defined on the set {T < co} EFT, isFT-measurable, and the "stopped process" {X7. At .5'-'1; 0 S t < co } is progres-sively measurable.

PROOF. For the first claim, one has to show that for any B e .4(W) and anyt 0, the event {Xr E B} n {T < t} is in F; but this event can also be writtenin the form {XT Ate B} n {T < t}, and so it is sufficient to prove the progressivemeasurability of the stopped process.

To this end, one observes that the mapping (s, w) i- (T(w) A s, w) of [0, t] x SIinto itself is .4([0, t]) 0 Ft-measurable. Besides, by the assumption of pro-gressive measurability, the mapping

(s, w) i- Xs(w): ([0, t] x 12, .4( [0, t]) ® Ft) -. (Rd, .4(W))

is measurable, and therefore the same is true for the composite mapping

(s, w)i- X Too (w): ([0, t] x f2, M( [0, t]) 0 Ft) -, (Rd, WV)).

2.19 Problem. Under the same assumptions as in Proposition 2.18, and withf(t, x): [0, Go) x Rd -R a bounded, M([l), co)) ®M(W)- measurable function,show that the process Y, = Po f(s, X3) ds; t > 0 is progressively measurable withrespect to {A}, and YT is an Fr-measurable random variable.

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

2.21 Problem. Verify that the class ,F is indeed a o--field with respect to whichT is measurable, that it coincides with {Ae,F;AnIT < tle.Vt 0}, andthat if T is a stopping time (so that both .FT, <FT+ are defined), then A-

2.22 Problem. Verify that analogues of Lemmas 2.15 and 2.16 hold if T andS are assumed to be optional and <FT, .Fs and .FT As are replaced by .FT+,and .FT A S)+, respectively. Prove that if S is an optional time and T is a positivestopping time with S < T, and S < Ton {S < co}, then ,s+ <FT

2.23 Problem. Show that if { Tn}'_, is a sequence of optional times andT = inf, T, then gr-- T+ = nnc°=1 FT+ Besides, if each Tn is a positive stoppingtime and T < T on { T < co}, then we have -g"T+ = nr1=1 "T-

2.24 Problem. Given an optional time T of the filtration Pr., 1, consider thesequence { Tn},`,°=, of random times given by

T(w); on {w; T(w) +oo1

Tn(w) = k

2' on tw;k - 1

2" -< T(w) < -k}2"

for n > 1, k > 1. Obviously Tn > 'Tn+, > T, for every n > 1. Show that each Tis a stopping time, that lim, T = T, and that for every A e FT+ we haveA n { T = (k/2")} e n, k 1.

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

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

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

(2.1)

{ (t, (0) E (0, cc) X f2; X,(w) X,_ (w)} U {(t, co) e [0, CO) X 12; Tn(w) = t }.n=1

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

1.3. Continuous-Time Martingales 11

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

1.3. Continuous-Time Martingales

We assume in this section that the reader is familiar with the concept andbasic properties of martingales in discrete time. An excellent presentation ofthis material can be found in Chung (1974, §§9.3 and 9.4, pp. 319-341) and weshall cite from this source frequently. Alternative references are Ash (1972) andBillingsley (1979). The purpose of this section is to extend the discrete-timeresults to continuous-time martingales.

The standard example of a continuous-time martingale is one-dimensionalBrownian motion. This process can be regarded as the continuous-time ver-sion of the one-dimensional symmetric random walk, as we shall see inChapter 2. Since we have not yet introduced Brownian motion, we shall takeinstead the compensated Poisson process as a continuing example developedin the problems throughout this section. The compensated Poisson process isa martingale which will serve us later in the construction of Poisson randommeasures, a tool necessary for the treatment of passage and local times ofBrownian motion.

In this section we shall consider exclusively real-valued processes X ={X1; 0 < t < col on a probability space (0, P), adapted to a given filtration1,,1 and such that El X,I < co holds for every t > 0.

3.1 Definition. The process {X Ft; 0 < t < col is said to be a submartingale(respectively, a supermartingale) if, for every 0 < s < t < cc, we have, a.s. P:E(X,I.F) X, (respectively, E(Xtig;) < X3).

We shall say that {X 0 < t < col is a martingale if it is both a sub-martingale and a supermartingale.

3.2 Problem. Let T1, T2, ... be a sequence of independent, exponentially dis-tributed random variables with parameter A > 0:

P[7; E dt] = Ae't dt, t > 0.

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

(3.1) N, = max {n > 0; S 1}; 0 t < co.

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

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

> tig,N] = e-1("), a.s. P.

(Hint: Choose Ae AN and a nonnegative integer n. Show that thereexists an event A ea(T,,...,Tn) such that A n {N, = n} = An {N, = n},and use the independence between Tn." and the pair (S, 1A) to establish

fin{r,=.}

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

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

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

= P[A n {N, = n }]. e-'(`-s)(A(t - s))i

J=--0 j!for every integer k > 0.)

3.3 Definition A Poisson process with intensity A > 0 is an adapted, integer-valued RCLL process N = {N A; 0 < t < oo } such that N0 = 0 a.s., and for0 < s < t, N, - N, is independent of and is Poisson distributed with mean2(t - s).

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

M, - At, A; 0 < t < co.

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

3.4 Problem. Prove that a compensated Poisson process {M A; t 0} is amartingale.

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

A. Fundamental Inequalities

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

X, a.s. P,

then we say that "{X <9,; 0 < t < co} is a submartingale with last elementX,0". We have a similar convention in the (super)martingale case.

A straightforward application of the conditional Jensen inequality (Chung(1974), Thm. 9.1.4) yields the following result.

3.6 Proposition. Let {X,, Ft; 0 < t < co} be a martingale (respectively, sub-martingale), and cp: 01-+ IR a convex (respectively, convex nondecreasing) func-tion, such that Elcp(X,)I < co holds for every t _.. 0. Then {cp(X,), ,Fi; 0 < t < co}is a submartingale.

The method used to prove Jensen's inequality and Proposition 3.6 extendsto the vector situation of the next problem.

3.7 Problem. Let {X, = (XP),... ,,q`')), ,97,; 0 < t < col be a vector of mar-tingales, and cp: Rd -> P a convex function with Elcp(X,), < co valid for everyt > 0. Then {co(X,), ,i; 0 t < co} is a submartingale; in particular { II X, I1, .9,;0 < t < co} is a submartingale.

Let X = {X,; 0 < t < co} be a real-valued stochastic process. Consider twonumbers a < /3 and a finite subset F of [0, co). We define the number of up-crossings U,(a, 13; X(co)) of the interval [a, /3] by the restricted sample path{X1; t E F} as follows. Set

T I (CO) = min{t e F; X,(co) a},

and define recursively for j = 1, 2, ...

01(w) = min{t E F; t _>_ ti(co), X,(co) > $),

ti,i(co) = mil* e F; t > ori(co), X,(co) < a).

The convention here is that the minimum of empty set is +co, and we denoteby UF(a, /3; X(w)) the largest integer j for which a(co) < co. If / [0, co) is notnecessarily finite, we define

Ui(a, 13; X(w)) = sup{ UF(a, /3; X(w)); F I, F is finite).

The number of downcrossings 1),(a, /3; X(co)) is defined similarly.The following theorem extends to the continuous-time case certain well-

known results of discrete martingales.

3.8 Theorem. Let {X *-7,; 0 < t < co} be a submartingale whose every path isright-continuous, let [o-, r] be a subinterval of [0, co), and let a < 13, ). > 0 bereal numbers. We have the following results:

(i) First submartingale inequality:

A -P[ sup X, > Al 5 E(X,±).c-[

(ii) Second submartingale inequality:

2- P[ inf X, _< -.1.1 5 E(Xt+) - E(X,).cr.r(iii) Uperossings and downcrossings inequalities:

EU10 1(z /1; X(a))) < E(X`) + laj, EIA,.1(2, 13; X (w))

E(X, - a)'13 - 2 17 -1

(iv) Doob's maximal inequality:

E( suprX,)p < (

P 1

p )P E(Xf), p > 1,cr -

provided X, > 0 a.s. P for every t > 0, and E(Xf) < co.(v) Regularity of the paths: Almost every sample path {X,(w); 0 5 t < cc)} is

bounded on compact intervals; is free of discontinuities of the second kind,i.e., admits left -hand limits everywhere on (0, co); and its jumps are ex-hausted by a sequence of stopping times (Proposition 2.26).

PROOF. Let the finite set F consist of o, T, and a finite subset of [o, 1 n Q.We obtain from Theorem 9.4.1 of Chung (1974): FP[max,EF X, > it] <as well as: liP[min, EF X, < -it] < E(Xt+) - E(X,). By considering an increas-ing sequence {F} T=1 of finite sets whose union is the whole of ([o-, r] n Q) u{6, T }, we may replace F by this union in the preceding inequalities. Theright-continuity of sample paths implies then itP[sup,,,, X, > pi] ._.. E(Xt+)and itP[inf,,,,, X, < -it] 5 E(XT) - E(X,). Finally, we let it i A to obtain(i) and (ii).

Being the limit of random variables of the form UF(2, /3; X(co)) with finiteF, U[c.,i(a, 13; X (a))) is measurable. We obtain (iii), (iv) from Theorems 9.4.2,9.5.4 in Chung (1974) (see also Meyer (1966), pp. 93-94). For (v), we note firstthat the boundedness of (almost all) sample paths on the compact interval[0, n], n > 1, follows directly from (i), (ii); second, we consider the events

Art3 A {we I; U10.](a, 13; X (w)) = co}, n > 1, a < 13.

By virtue of (iii), these have zero probability, and the same is true for the union

A(") = U An)13,rt<13

ri,fleQ

which includes the set

{co e SI; lim Xs(co) < lim Xs(w), for some t e [0, n]}.Sit 51t

Consequently, for every co e il\A(n), the left limit X,_ (w) = limst, Xs(co) existsfor all 0 < t 5 n. This is true for every n 1, so the preceding left limit existsfor every 0 < t < co, a) e (U ,°,3=, A(n))`.

3.9 Problem. Let N be a Poisson process with intensity 2.

(a) For any c > 0,

lim P[ sup (N, - As)1-00

(b) For any c > 0,

lim P[ inf (Ns- As) < -c A-It1 5 1

ost c/27r

(c) For 0 < o- < T, we have

4r2E[ sup (N,- 1 -T.cr<t<T t o-

(Hint: Use Stirling's approximation to show that lim,, (1//Tit)E(N,-10+ =

3.10 Remark. From Problem 3.9 (a) and (b), we see that for each c > 0, thereexists 7', > 0 such that

P[Nr _ A > c - < 3

, V t > Tc.t c.N./27r

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

P[ sup2^12^""

Nt

t 4> s] < E2 2n

82

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

The following result from the discrete-parameter theory will be used re-peatedly in the sequel; it is contained in the proof of Theorem 9.4.7 in Chung(1974), but it deserves to be singled out and reviewed.

3.11 Problem. Let {F,,} ,;°_, be a decreasing sequence of sub-a-fields of F (i.e.,'n+1 g g V n 1), and let {X., n > 1} be a backward submartin-gale; i.e., El X ni < co, X" is F-measurable, and E(XIF,, ) X,,,, a.s. P, forevery n > 1. Then 1 A E(X)> -co implies that the sequence {X n} n'°-1

is uniformly integrable.

3.12 Remark. If IX Ft; 0 t < ool is a submartingale and {t} T=1 is a non-increasing sequence of nonnegative numbers, then {X,n, Ft.; n >_ 1} is a back-ward submartingale.

It was supposed in Theorem 3.8 that the submartingale X has right-continuous sample paths. It is of interest to investigate conditions underwhich we may assume this to be the case.

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

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

3.14 Proposition. Let X = {X ,F,; 0 < t < co} be a submartingale. We havethe following:

(i) There is an event Cr e..97 with P(SI*) = 1, such that for every we Sr:

the limits X,+(w) g urn Xs(w), X,_ A lim Xs(co)st.t sttseQ SEQ

exist for all t > 0 (respectively, t > 0).

(ii) The limits in (i) satisfy

E(Xt+1,Ft) Xt

E(X,I.Ft4'

a.s. P, V t > 0.

a.s. P, V t > 0.

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

PROOF.

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

A(c3 = E Uto,1,Q(ot, /3; X (w)) = co}, n > 1, a <

A(n) = U A.a<I3

ct,fleQ

Then each AVp has probability zero, as does each A("). The conclusionsfollow readily.

(ii) Let {t}c°_, be a sequence of rational numbers in (t, co), monotonicallydecreasing to t > 0 as n co. Then IX1., A; n > 11 is a backward sub-martingale, and the sequence IE(Xj.)1T,_, is decreasing and boundedbelow by E(X,). Problem 3.11 tells us that IX,nr_1 is a uniformly integrable

sequence. From the submartingale property we have IA X, dP 5 fA X,. dP,for every n > 1 and A e .; uniform integrability renders almost sure intoL' -convergence (Chung (1974), Theorem 4.5.4), and by letting n oo weobtain SA X, dP < 1,4 X, dP, for every A e The first inequality in (ii)follows.

Now take a sequence ftnIn`c=i in (0, t) n Q, monotonically increasing tot > 0. According to the submartingale property E[X,I.F,.] > X,. a.s. Wemay let n a) and use Levy's theorem (Chung (1974), Theorem 9.4.8) toobtain the second inequality in (ii).

(iii) Take a monotone decreasing sequence ts,,In°3_, of rational numbers, with0 < s < s < t holding for every n > 1, and limn-. s = s. According tothe first part of (ii), E(X,,A.)> Xs. a.s. Letting n co and using Levy'stheorem again, we obtain the submartingale property E(X,+I,S+) > Xs+a.s. It is not difficult to show, using (i), that P-almost every path Xi+is RCLL.

PROOF OF THEOREM 3.13. Assume that the function EX, is right-continuous;we show that IX,+, A; 0 < t < co as defined in Proposition 3.14 is a modifi-cation of X. The former process is adapted because of the right-continuity of{A}. Given t > 0, let Ign, I be a sequence of rational numbers with q t.nc,D=

Then lim, Xq. = X,+, a.s., and uniform integrability implies that EX,+ =limn, EXq.. By assumption, limn, EXg. = EX and Proposition 3.14 (ii)gives X,± X a.s. It follows that X = X a.s.

Conversely, suppose that {Yet; 0 < t < co} is a right-continuous modifica-tion of X. Fix t > 0 and let Itn In`°_, be a sequence of numbers with to l t. Wehave P[X, = fe X,. = fe,.; n > 1] = 1 and limn gt. = fe a.s. Therefore,limn, X,. = X, a.s., and the uniform integrability of 1X1.1°°=, implies thatEX, = lim,EX,.. The right-continuity of the function t EX, follows.

B. Convergence Results

For the remainder of this section, we deal only with right-continuous pro-cesses, usually imposing no condition on the filtrations {A}. Thus, the de-scription right-continuous in phrases such as "right-continuous martingale"refers to the sample paths and not the filtration. It will be obvious that theassumption of right-continuity can be replaced in these results by the assump-tion of right-continuity for P-almost every sample path.

3.15 Theorem (Submartingale Convergence). Let {X ,51;,; 0 5 t < oo} be aright-continuous submartingale and assume C A sup,,,,E(X,+) < co. ThenXao(w) A lim,o0 X,(w) exists for a.e. w E 0, and EIXeol < 00.

PROOF. From Theorem 3.8 (iii) we have for any n > 1 and real numbers a < 13:ELio,n0, /3; X((0)) 5 (E(Xn+) + lal)/(/3 - a), and by letting n co we obtain,

thanks to the monotone convergence theorem:

C + la!EUE0.00(a, /3; X(w)) 5

# - cc

The events A,,3 gl- {w; U[0,03)(a, /3; X(w)) = col, -co < a < /3 < co, are thusP-negligible, and the same is true for the event A = H.J.<13 A ,p, which con-tains the set {co; lim, X,(w) > limy. X,(w) }. °,13Q

Therefore, for every wen\ A, Xco(co) = lim,._ X,(w) exists. Moreover,

EIX,I = 2E(X7) - E(X,) 5 2C - EX()

shows that the assumption sup,>0 E(X,+) < co is equivalent to the apparentlystronger one sup,>0 EIX,I < co, which in turn forces the integrability of Xco,by Fatou's lemma.

3.16 Problem. Let {X,, A; 0 < t < col be a right-continuous, nonnegativesupermartingale; then X,(w) = lim,, X,(w) exists for P-a.e. (DEO, and{X,, ,F,; 0 < t 5 co} is a supermartingale.

3.17 Definition. A right-continuous, nonnegative supermartingale {Z,, 3,--,;0 < t < col with lim, E(Z,) = 0 is called a potential.

Problem 3.16 guarantees that a potential {Z,, .97,; 0 < t < co } has a lastelement Zco, and Z,,, = 0 a.s. P.

3.18 Exercise. Suppose that the filtration I.Fil satisfies the usual conditions.Then every right-continuous, uniformly integrable supermartingale {X,F,;0 5 t < co} admits the Riesz decomposition X, = M, + Z,, a.s. P, as the sumof a right-continuous, uniformly integrable martingale {M,, A; 0 < t < coland a potential {Z,, .,; 0 < t < col.

3.19 Problem. The following three conditions are equivalent for a nonnegative,right-continuous submartingale {X,, A; 0 < t < col:

(a) it is a uniformly integrable family of random variables;(b) it converges in L', as t -, co;(c) it converges P a.s. (as t -, co) to an integrable random variable Xoc, such

that {X,, ,9;; 0 < t < col is a submartingale.

Observe that the implications (a) =. (b) (c) hold without the assumption ofnonnegativity.

3.20 Problem. The following four conditions are equivalent for a right-continuous martingale {X,, A; 0 5 t < co}:

(a), (b) as in Problem 3.19;(c) it converges P a.s. (as t -, co) to an integrable random variable Xco, such

that {X,, ,F,; 0 5 t < co} is a martingale;

(d) there exists an integrable random variable Y, such that X, = E( a.s.P, for every t > 0.

Besides, if (d) holds and ,c is the random variable in (c), then

E(Y1.973) = X. a.s. P.

3.21 Problem. Let {N .9-;,; 0 5 t < co} be a Poisson process with parameterA > 0. For u e C and i = 1, define the process

X, = exp[iuN, - At(eiu - 1)]; 0 5 t < o0.

(i) Show that {Re(X,), .9-;,; 0 < t < {Im(X,), .9-;,; 0 < t < co} are martin-gales.

(ii) Consider X with u = - i. Does this martingale satisfy the equivalent con-ditions of Problem 3.20?

C. The Optional Sampling Theorem

What can happen if one samples a martingale at random, instead of fixed,times? For instance, if X, represents the fortune, at time t, of an indefatigablegambler (who plays continuously!) engaged in a "fair" game, can he hope toimprove his expected fortune by judicious choice of the time to quit? If noclairvoyance into the future is allowed (in other words, if our gambler is re-stricted to quit at stopping times), and if there is any justice in the world, theanswer should be "no." Doob's optional sampling theorem tells us under whatconditions we can expect this to be true.

3.22 Theorem (Optional Sampling). Let IX .07-t; 0 < t < col be a right-continuous submartingale with a last element X., and let S < T be two optionaltimes of the filtration We have

E(XTIA.) > Xs a.s. P.

If S is a stopping time, then .F's can replace 9.'s+ above. In particular, EXTEX0, and for a martingale with a last element we have EXT = EX0.

PROOF. Consider the sequence of random times

S(w) if S(w) = +co

ifk

S(w) <-

2" 2"

1

2"

and the similarly defined sequences I T,,I. These were shown in Problem 2.24to be stopping times. For every fixed integer n 1, both S,, and 7,, takeon a countable number of values and we also have S,, 5 Tn. Therefore, bythe "discrete" optional sampling Theorem 9.3.5 in Chung (1974) we have

S (w) =

IA XsndP 5 iA X T. dP for every A e An, and a fortiori for every A a .F5+ =n.c0=. g-s, by virtue of Problem 2.23. If S is a stopping time, then S .. S implies..Fs g mss as in Lemma 2.15, and the preceding inequality also holds for everyAu ..Fs .

It is checked similarly that {X5., An; n > 1} is a backward submartingale,with {E(Xs.)},T=1 decreasing and bounded below by E(X0). Therefore, thesequence of random variables {X,.}`°_, is uniformly integrable (Problem 3.11),and the same is of course true for {X Tn}'_, . The process is right-continuous,so XT(ow) = lim, X Tn(co) and Xs(w) = lim, Xsn(w) hold for a.e. co e D.It follows from uniform integrability that XT, Xs are integrable, and that1, Xs dP 5 IA XT dP holds for every A a A + .

3.23 Problem. Establish the optional sampling theorem for a right-continuoussubmartingale {X 5,; 0 < t < oo} and optional times S < T under either ofthe following two conditions:

(i) T is a bounded optional time (there exists a number a > 0, such that T < a);(ii) there exists an integrable random variable Y, such that X, 5 E(YIA) a.s.

P, for every t > 0.

3.24 Problem. Suppose that {X A; 0 < t < co} is a right-continuous sub-martingale and S < T are stopping times of {,F,}. Then

(i) {XT. A t, .; 0 < < Go} is a submartingale;(ii) E[XT,,IA] > Xs," a.s. P, for every t > 0.

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

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

(3.2) E(XT) > E(Xs).

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

(i) If X = IX g",; 0 < t < col is a right-continuous submartingale, then sois )7 = {2t '' Kr+t - X T , g',; 0 < t < Op} .

(ii) If g = {g .; 0 < t < oc } is a right-continuous submartingale withgo = 0, a.s. P, then X = {X, A 1(t_7.) 0, ,F,; 0 < t < co} is also asubmartingale.

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

1.4. The Doob-Meyer Decomposition 21

1

(i) P [sup Z, = -bZ,, a.s. on {Zs < b}.t>s

1

z.'<b}(ii) P [sup Z, to] = P[Z,

bb] + -E[Zslf ]

t>s

3.29 Problem. Let {X ,F,; 0 < t < co} be a continuous, nonnegative super-martingale and T = inf {t 0; X, = 0}. Show that

XT+, = 0; 0 t < co holds a.s. on IT < co}.

3.30 Exercise. Suppose that the filtration {A} satisfies the usual conditionsand let X(") = {X:1", ,F,; 0 < t < co}, n > 1 be an increasing sequence of right-continuous supermartingales, such that the random variable, lim_co X:")is nonnegative and integrable for every 0 < t < co. Then there exists an RCLLsupermartingale X = {X ,F,; 0 5 t < co} which is a modification of theprocess = { Ft; 0 t < co}.

1.4. The Doob-Meyer Decomposition

This section is devoted to the decomposition of certain submartingales as thesummation of a martingale and an increasing process (Theorem 4.10, alreadypresaged by Remark 3.5). We develop first the necessary discrete-time results.

4.1 Definition. Consider a probability space (It P) and a random sequence{A},7=0 adapted to the discrete filtration {,},T=0. The sequence is called in-creasing, if for P-a.e. (c) E S/ we have 0 = Ao(w) < A,(w) < - - - , and E(A) < coholds for every n > 1.

An increasing sequence is called integrable if E(A0D) < co, where As, =A. An arbitrary random sequence { },T=0 is called predictable for the

filtration {,},7_0, if for every n 1 the random variable is ,_1-measurable.Note that if A = {A, F; n = 0, 1, ... } is predictable with E I AI < co forevery n, and if {M, n = 0, 1, ...} is a bounded martingale, then the mar-tingale transform of A by M defined by

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

is itself a martingale. This martingale transform is the discrete-time versionof the stochastic integral with respect to a martingale, defined in Chapter 3.A fundamental property of such integrals is that they are martingales whenparametrized by their upper limit of integration.

Let us recall from Chung (1974), Theorem 9.3.2 and Exercise 9.3.9, that any


submartingale {X,,, ".; n = 0, 1, ... } admits the Doob decomposition X =M + An as the summation of a martingale {M, F.} and an increasingsequence {A, ,}. It suffices for this to take Ao = 0 and A+1 = A -X +E(X.+1 ign) = =o[E1X k+ -X ki, for n 0. This increasing sequenceis actually predictable, and with this proviso the Doob decomposition of asubmartingale is unique.

We shall try in this section to extend the Doob decomposition to suitablecontinuous-time submartingales. In order to motivate the developments, letus discuss the concept of predictability for stochastic sequences in some furtherdetail.

4.2 Definition. An increasing sequence IA, F; n = 0, 1, ...1 is called naturalif for every bounded martingale IM,,, .97;; n = 0, 1, ...1 we have

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

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

4.3 Proposition. An increasing random sequence A is predictable if and only ifit is natural.

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

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

It follows that

(4.3) E[M{A - E(A1,F_1) 1] = E[(M - M-,)A]

+ E[M_11A - E(A1,T,_1) 1]

- E[(M - = 0

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

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

Mk = k > n,E(Mni,k), k = 0, 1, ..., n.

We obtain El An - = 0, whence the desired conclusion.


From now on we shall revert to our filtration {,51;,} parametrized by t e[0, cc) on the probability space (f2, P). Let us consider a process A = IA1;0 < t < col adapted to By analogy with Definitions 4.1 and 4.2, we havethe following:

4.4 Definition. An adapted process A is called increasing if for P-a.e. w a SI wehave

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

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

4.5 Definition. An increasing process A is called natural if for every bounded,right-continuous martingale {M ,;(; 0 < t < col we have

(4.4) E .1 Ms dAs = E 1 Ms_ dA,, for every 0 < t <(0,fl (041

4.6 Remarks.

(i) If A is an increasing and X a measurable process, then with w e SI fixed,the sample path IX,(w); 0 < t < col is a measurable function from [0, cc)into R. It follows that the Lebesgue-Stieltjes integrals

I ,± (w) -4 Xs-± (w) dAs(w)(o ,ti

are well defined. If X is progressively measurable (e.g., right-continuousand adapted), and if I, = - I,- is well defined and finite for all t > 0,then I is right-continuous and progressively measurable.

(ii) Every continuous, increasing process is natural. Indeed then, for P-a.e.E SI we have

fo.o

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

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

(Ms(co) - Ms_(o.))) dA = 0 for every 0 < t < co,

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

(4.4)' E(M,A,) = EJ

Ms_ dAs.

PROOF. Consider a partition H = {to, t, , ... , tk} of [0, t], with 0 = to < --- < t = t, and define

n

Mn = E Mtkl(tk-1,td(S).k=1

The martingale property of M yields

n n n-1E I NINA, = E E 111,k(Atk - Atk_,) = E[ E m,,,A,k -E mtk ' ,Atk]

(0,1] k=1 k=1 k=0

..-1= E(Mtili) -E E Afk(M,,,,, - MO = E(MiAt).

k=1

Now let 111111 A max, , < (tk - tk_, )-* 0, so Mn -* Ms, and use the boundedconvergence theorem for Lebesgue-Stieltjes integration to obtain

(4.5) E(1141A1) = E 1 MsdAs.(o,t]

D

The following concept is a strengthening of the notion of uniform inte-grability for submartingales.

4.8 Definition. Let us consider the class Y(.9'.) of all stopping times T of thefiltration {,,} which satisfy P(T < co) = 1 (respectively, P(T 5 a) = 1 for agiven finite number a > 0). The right-continuous process {X1, A; 0 < t < co}is said to be of class D, if the family {X T} T e y is uniformly integrable; of classDL, if the family {XT} T c 9,. is uniformly integrable, for every 0 < a < oo.

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

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

(4.6) X, = M, + A 0 < t < co

suggested by the Doob decomposition, where {M1, ,97,; 0 < t < co} is a mar-tingale and {A1, A; 0 < t < co} is an increasing process.

Show also that if X is a uniformly integrable martingale, then it is of class D.

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

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

tion of a right-continuous martingale M = {M .51;i; 0 < t < co} and an increas-ing process A = {Al, t < co}. The latter can be taken to be natural;under this additional condition, the decomposition (4.6) is unique (up to indistin-guishability). Further, if X is of class D, then M is a uniformly integrablemartingale and A is integrable.

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

E[ ,(A; - A;')] = E f0,t1

d13, = lim E E [Be.) -"_.co j=1 j- J J-

(

where ri= {t(S),...,t(,")}, n > 1 is a sequence of partitions of [0, t] with= maxi j<,. (ein) -t.(i1) converging to zero as n co, and such that

each 1-In+1 is a refinement of H. But now

EH,p),(B,(;) - Ber)i)] = 0, and thus E[ ,(A; - A7)] = 0.

For an arbitrary bounded random variable we can select R Al to be aright-continuous modification of {EUIFJ, Ft} (Theorem 3.13); we obtainEMA; - AD] = 0 and therefore P(A; = At") = 1, for every t 0. The right-continuity of A' and A" now gives us their indistinguishability.

For the existence of the decomposition (4.6) on [0, co), with X of class DL,it suffices to establish it on every finite interval [0, a]; by uniqueness, we canthen extend the construction to the entire of [0, co). Thus, for fixed 0 < a < co,let us select a right-continuous modification of the nonpositive submartingale

Y, X, - E[XIF], 0 t a.

Consider the partitions 11 = {tr, ei"), , t(1),} of the interval [0, a] of theform tr) = (j/2")a, j = 0, 1, ..., 2^. For every n > 1, we have the Doobdecomposition

= M,7, + A , j = 0, 1, , 2"

where the predictable increasing sequence A(n) is given by

A(,'(:!) = AP)i + EJ-1

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

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

(4.7) Ye?) = /1(,V) - E[A!,n]Ig"00], j = 0, 1, ..., 2".

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

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

We have { T71 < } = {4 > A} E Ay. for j = 1, , r, and { TP) < a) ={ Ala") > 2 }. Therefore, TA!")E 9. On each set { 71")= }, we have E [A(:) I Ard =E[A!,") I gTr], so (4.7) implies

(4.8) YTS", = AV^) - EEA(:)1.°17Tri < A - EEA(an)1.°17Tri

on { 71") < a). Thus

(4.9) I Al7) dP 5 2P[T1") < a] - I Y7TodP.

Replacing A by A/2 in (4.8) and integrating the equality over themeasurable set { < a}, we obtain

Ypn) dP = (A(") - A(PLI)dP/2-L171 <a} f{T71<a}

IIn') <a} A/2 2

A(A(Z1) - A(t) dP > -P [71") < a],T

and thus (4.9) leads to

(4.10) Y (.)d PAT)> A}

A(an) dP < -2f{ <a } YT1"d P - f(Tr <a} 7'

-

The family {XT}TE.q is uniform y integrable by assumption, and thus so is{YT }TE.q. But

P[11") < a] = >A]<A

E(A(:)) E( Yo)

SO

sup P[71") < a] -> 0 as A co.a > 1

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

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

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

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

(4.11) A, = 17, E(A.IA); 0 t a.

4.11 Problem. Show that if {Aw}c°_, is a sequence of integrable randomvariables on a probability space (12, .F, P) which converges weakly in 1) to anintegrable random variable A, then for each 6 -field c .F, the sequenceE[A(")1W] converges to E[Al] weakly in L'.

Let n = U-=, nn. Fort E II, we have from Problem 4.11 and a comparisonof (4.7) and (4.11) that lim, E(,V)) = E(At) for every bounded randomvariable For s, t E n with 0 s < t < a, and any bounded and nonnegativerandom variable we obtain E[(A, - A,)] = ER(A!") - Ar)] 0,

and by selecting = we get AS < A a.s. P. Because II is countable,for a.e. w WE SI the function ti-* At(co) is nondecreasing on II, and right-continuityshows that it is nondecreasing on [0, a] as well. It is trivially seen that Ao = 0,a.s. P. Further, for any bounded and right-continuous martingale gt, Ftl, wehave from (4.2) and Proposition 4.3:

2"

E(0A(:)) E E - A:740j =1

2"

= E E toni [Yon) - Ytnni]J-

2"

= E E [Air - Aepi],

where we are making use of the fact that both sequences {Ai -Y and{Ar -of (4.5):

for t e n, are martingales. Letting n co one obtains by virtue

E f sdA, = E(0,a]

f S dAs,(0,a1

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

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

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

4.12 Definition. A submartingale {X .; 0 < t < oo} is called regular if forevery a > 0 and every nondecreasing sequence of stopping times { gwith T = limn T,, we have limn E(XT,.) = E(XT).

4.13 Problem. Verify that a continuous, nonnegative submartingale is regular.

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

PROOF. Continuity of A yields the regularity of X quite easily, by appealingto the optional sampling theorem for bounded stopping times (Problem3.23(i)).

Conversely, let us suppose that X is regular; then for any sequence{T}n°_, as in Definition 4.12, we have by optional sampling: limn, E(AT.) =limn E(XTn) - E(MT) = E(AT), and therefore AT(w)(w) j AT(.)(w) except forw in a P-null set which may depend on T

To remove this dependence on T, let us consider the same sequence {11}n'.1of partitions of the interval [0, a] as in the proof of Theorem 4.10, and selecta number A > 0. For each interval (tr, tp,), j = 0, 1, ..., 2" - 1 we considera right-continuous modification of the martingale

= E[A A t") < t <

This is possible by virtue of Theorem 3.13. The resulting process {V);0 < t < a} is right-continuous on (0,a) except possibly at the points of thepartition, and dominates the increasing process {A A 211; 0 < t < a}; in par-ticular, the two processes agree a.s. at the points tV, el.!. Because A is anatural increasing process, we have from (4.4)

Vs") dAs = E dAs; j = 0, 1, ..., 2" - 1,yv,

and by summing over j, we obtain

(4.12) E J Vs") dAs = E f Vs"2 dAs,0,0

for any 0 < t < a. Now the process

=

(0,1]

- (A A 241), 0 t < a,0, t = a,

is right-continuous and adapted to {.97, }; therefore, for any E > 0 the randomtime

T(s) = a A inf {0 t 5 a; fir > = a A inf {0 < t < a; 111) - (A A At) >

is an optional time of the right-continuous filtration {.F, }, hence a stoppingtime in X,' (cf. Problem 2.6 and Proposition 2.3). Further, defining for eachn > I the function con( ): [0, a] -+ 11 by (p(t)= tin+)1; ti ") < t < t.11,)1, we have(p.(7,,(E)) e tea. Because en) is decreasing in n, the increasing limit T =

T(E) exists a.s., is a stopping time in 5o, and we also have

T = lim (p.(7,.(0) a.s. P.

By optional sampling we obtain now2 " -1

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

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

and therefore

E[(A A ilo.(T.(0)) - (A A AT,,(0)] = EN(4.1,!(,) - (2 n AT,,(0)]

= E[1{7.(0<a)(61,!(E) (2 A A T,,(e)))]

EP [ Tn(E) < a].

We employ now the regularity of X to conclude that for every E > 0,

P[Q > e].= P[Tn(E)< a] <_!E[(2 n Aon(T.(,))) - (A n AT,(e))] 0

as n oc, where Q,, <1 ,7 1supo n) - (2 A A,)I. Therefore, this last sequence

of random variables converges to zero in probability, and hence also almostsurely along a (relabeled) subsequence. We apply this observation to (4.12),along with the monotone convergence theorem for Lebesgue-Stieltjes integra-tion, to obtain

Ef(A A As)dAs= E f (A A As4dAs, 0 < oo,

(o,ti

which yields the continuity of the path ti-* 2 A At(co) for every 2 > 0, andhence the continuity of At(co) for P-a.e. w

4.15 Problem. Let X = {Xt, Ft; 0 < t < co} be a continuous, nonnegativeprocess with X0 = 0 a.s., and A = {A Ft; 0 < t < co} any continuous, in-creasing process for which

(4.13) E(XT) < E(AT)

holds for every bounded stopping time T of {",}. Introduce the processV, A maxo.(s.,, X5, consider a continuous, strictly increasing function F on[0, oo) with F(0) = 0, and define G(x) g 2F(x) + x Lc' 14' dF(u); 0 < x < oo.Establish the inequalities

E(AT)(4.14) P [ V T e] < ; V > 0

(4.15) P[VT e, AT < 6]E(6 A AT)

; V c > 0, 6 > 0c

(4.16) EF(VT) _< EG(AT)

for any stopping time T of {F }.

4.16 Remark. If the process X of Problem 4.15 is a submartingale, then A canbe taken as the continuous, increasing process in the Doob-Meyer decom-position (4.6) of Theorems 4.10 and 4.14.

4.17 Remark. The corollary

(4.15)' P[ VT > e] < E(6 A AT)+ PEAT > 6]

c

of (4.15) is very useful in the limit theory of continuous-time martingales; it isknown as the Lenglart inequality. We shall use it to establish convergenceresults for martingales (Problem 5.25) and stochastic integrals (Proposition3.2.26). On the other hand, it follows easily from (4.16) that

(4.17) E( VI)1

2

-p

pE(Ac);0 < p < 1

holds for any stopping time T of {g; }.

1.5. Continuous, Square-Integrable Martingales

In order to appreciate Brownian motion properly, one must understand therole it plays as the canonical example of various classes of processes. One suchclass is that of continuous, square-integrable martingales. Throughout thissection, we have a fixed probability space (Q, ..F., P) and a filtration {,F} whichsatisfies the usual conditions (Definition 2.25).

5.1 Definition. Let X = {X gt; 0 < t < oo} be a right-continuous martin-gale. We say that X is square-integrable if EV < co for every t > 0. If, inaddition, X0 = 0 a.s., we write X e.1'2 (or X e .4q, if X is also continuous).

5.2 Remark. Although we have defined JP; so that its members have everysample path continuous, the results which follow are also true if we assumeonly that P-almost every sample path is continuous.

For any X e .Ai 2, we observe that X2 = {V, F; 0 < t < oo } is a nonnega-tive submartingale (Proposition 3.6), hence of class DL, and so X2 has a uniqueDoob-Meyer decomposition (Theorem 4.10, Problem 4.9):

V = M, + At; 0 t < oo

1.5. Continuous, Square-Integrable Martingales 31

where M = {Mt, Ft; 0 < t < co} is a right-continuous martingale and A ={A -A; 0 s t < cc} is a natural increasing process. We normalize these pro-cesses so that Mo = Ao = 0, a.s. P. If X e Jf2, then A and M are continuous(Theorem 4.14 and Problem 4.13); recall Definitions 4.4 and 4.5 for the termsincreasing and natural.

5.3 Definition. For X e./112, we define the quadratic variation of X to be theprocess <X>, -4- A where A is the natural increasing process in the Doob-Meyer decomposition of X2. In other words, <X> is that unique (up toindistinguishability) adapted, natural increasing process, for which <X>0 = 0a.s. and X2 - <X> is a martingale.

5.4 Example. Consider a Poisson process {N A; 0 < t < oo} as in Definition3.3 and assume that the filtration {,,} satisfies the usual conditions (this canbe accomplished, for instance, by "augmentation" of IF,N1; cf. Remark 2.7.10).It is easy to verify that the martingale M, = N, - At, of Problem 3.4 is in///i'2, and <M>, = At.

If we take two elements X, Y of X2, then both processes (X + Y)2 -<X + Y> and (X - Y)2 - <X - Y> are martingales, and therefore so is theirdifference 4X Y - + Y> - <X - Y>].

5.5 Definition. For any two martingales X, Y in 112, we define their cross-variation process <X, Y> by

<X, Y>, = 1[<X + Y>, - <X - Y>,]; 0 < t <

and observe that X Y - <X, Y> is a martingale. Two elements X, Y of .112are called orthogonal if <X, Y>, = 0, a.s. P, holds for every 0 < t < co.

The uniqueness argument in Theorem 4.10 also shows that <X, Y> is, upto indistinguishability, the only process of the form A = A(1) - A(2) with A(j)adapted and natural increasing (j = 1, 2), such that X Y -A is a martingale.In particular, <X, X> = <X>. For continuous X and Y, we give a differentuniqueness argument in Theorem 5.13.

5.6 Remark. In view of the identities

E[(X, - Xs/(Y - Ys/1"-'s] = E[XtYt - X. Ys Fs]

= EUX , Y>, - <X, Y>,1 37.j,

valid P a.s. for every 0 < s < t < co, the orthogonality of X, Y in .112 isequivalent to the statements "X Y is a martingale" or "the increments of Xand Y over [s, t] are conditionally uncorrelated, given ,Fs".

5.7 Problem. Show that < > is a bilinear form on 12, i.e., for any membersX, Y, Z of 12 and real numbers a, )3, we have

(i) <ccX + /I Y, Z> = a <X, Z> + 13 <Y, Z>,(ii) <X, Y> = <Y, X>,

(iii) loc, Y>12 < <X><Y>(iv) For P-a.e. w e SI,

4'r(w) - 4s(w) < 1-[<X>,(w) - <X>.*0) + < Y>,(co) - < Y>sIcon;

0<s<t<oo,where 4, denotes the total variation of A <X, Y> on [0, t].

The use of the term quadratic variation in Definition 5.3 may appear to beunfounded. Indeed, a more conventional use of this term is the following. LetX = {X,; 0 < t < co} be a process, fix t > 0, and let 11 = {to, ti,..., t,,,}, with0 = to < t1 < t2 .< < t, = t, be a partition of [0, t]. Define the p-th variation(p > 0) of X over the partition n to be

V(P)(11) = i I Xt, - Xtk, IP.k=1

Now define the mesh of the partition 11 as 111111 = max, <ic 14 - tk_21 Ifv,(2)(n) converges in some sense as II n II -.0, the limit is entitled to be calledthe quadratic variation of X on [0, t]. Our justification of Definition 5.3 forcontinuous martingales (on which we shall concentrate from now on) is thefollowing result:

5.8 Theorem. Let X be in A. For partitions 11 of [0,t], we havelimiln11-0 V(2)(1) = <X>, (in probability); i.e., for every e > 0, ri > 0 there existso > 0 such that linll < (5 implies

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

The proof of Theorem 5.8 proceeds through two lemmas. The key factemployed here is that, when squaring sums of martingale increments and takingthe expectation, one can neglect the cross-product terms. More precisely, ifX e it2 and 0 <s<t<u<v, then

EUX,, - XJ(X, - X5)] = EIE[X,, - Xul,](X, - X5)} = 0.

We shall apply this fact to both martingales X edt, and X' - <X>. In thelatter case, we note that because

EUX - XJ2 1.°7A = E[X ,2 - 2X E[X 1"] + X,;1.9%]

= E[Xt2 -X,2,I,] = E[ <X> -<X>1],

the terms Xv2 - <X> - (X,42 - <X>.) and (X - XJ2 - (<X>,, - <X>,4) havethe same conditional expectation given g", namely zero, and thus the expecta-tion of products of such terms over nonoverlapping intervals is still zero.

5.9 Lemma. Let X e di2 satisfy IX,1 < K < oo for all s e [0, t], a.s. P. Let11= {to,ti,..., tm}, with 0 = to 5 t1 5 5 t. = t, be a partition of [0, t].Then E[Vt(2)(n)12 6K4.

PROOF. Using the martingale property, we have for 0 < k < m - 1,

2E[ (X,,- 1)2 .7, E[{ (X1, - X1,_1)}j=k+1 j=k+1

=E[(X,2, -

j=k+1

E[X,2,.1,(k] K2

SO

Ak]

Ak]

E[m-i mE E x,_,)2pck - xtk_i>21k=1 j=k+1

(5.1) = E[ Em-1

(X- x1,,_1)2 E Euxt,- x1,_1)24-7,3]k=1 j=k+1

We also have

Ic2ELE (Xk xik_,)21< K4.k=1

m-1

(5.2) E[ (Xtk - Xtk_i)4]< 4K2 E [E (X,k Xtk-1 )2] < 4K4.

k=1 k=1

Inequalities (5.1) and (5.2 imply

E[1' 2'(n)]2 = E[ (Xtk -X, )41k=1

m

+ 2E E E x,)2pck- x(k,)2]k=1 j=k+1

< 610.

5.10 Lemma. Let X e .11'2 satisfy XSI < K < Do for all se[0,t], a.s. P. Forpartitions n of [0, t], we have

lim Ev,(4)(n) = 0.II bil-o

PROOF. For any partition n, we may write

v(4)(H). v,(2)-ff .) 14(x; 111111),where

(5.3) tn,(X; 6) A sup{ I(X,. - Xs)l; 0 r < s t, s -r 6}

is measurable because the supremum can be restricted to rational s and r. The

Holder inequality implies

I)I

H10)1/2.Evt(4)(n) (E0,;(2)(n)]2,1/2(E4(x;

As 111111 approaches zero, the first factor on the right-hand side remainsbounded and the second term tends to zero, by the uniform continuity on[0, t] of the sample paths of X and by the bounded convergence theorem.

PROOF OF THEOREM 5.8. We consider first the bounded case: I X,I < K < oo and<X s> < K hold for all s E [0, a.s. P. For any partition II = {to, tl, t,,,}

as earlier we may write (see the discussion preceding Lemma 5.9 and relation(5.3)):

E(v(2)(n)- <x>1)2 = {(Xtk - X,k_i)2 - - <X>,k_,)}]2

= E EUX,k k-I (XX,, - <X >tk_M2

k=1

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

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

As the mesh of II approaches zero, the first term on the right-hand side of thisinequality converges to zero because of Lemma 5.10; the second term does aswell, by the bounded convergence theorem and the sample path uniform con-tinuity of <X>. Convergence in L2 implies convergence in probability, so thisproves the theorem for martingales which are uniformly bounded.

Now suppose X e Llz is not necessarily bounded. We use the technique oflocalization to reduce this case to the one already studied. Let us define asequence of stopping times (Problem 2.7) for n = 1, 2, ... by

7:, inf It > 0; 1X11 > n or <X>, n }.

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

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

Therefore, for partitions II of [0, t], we have

lim E[ (X,k A Xtk IA T,,)2 <X>t A 7;12 = O.11n11-0 k=i

Since T. T co a.s., we have for any fixed t that P[7,, < t] = 0. Thesefacts can be used to prove the desired convergence of v,(2)(n) to <x>, inprobability. 111

5.11 Problem. Let {X g'-t; 0 5 t < co} be a continuous process with theproperty that for each fixed t > 0 and for some p > 0,

lira vt(P)(n) = L, (in probability),0111-0

where Li is a random variable taking values in [0, co) a.s. Show that for q > p,vi(q)(n) = 0 (in probability), and for 0 <

(in probability) on the event 14 > 01.q < limllnllo V( )(n) = GO

5.12 Problem. Let X be in ./C, and T be a stopping time of IFtl. If <X> T = 0,a.s. P, then we have P[XT = 0; d0 < t < co] = 1.

The conclusion to be drawn from Theorem 5.8 and Problems 5.11 and 5.12is that for continuous, square-integrable martingales, quadratic variation isthe "right" variation to study. All variations of higher order are zero, and,except in trivial cases where the martingale is a.s. constant on an initialinterval, all variations of lower order are infinite with positive probability.Thus, the sample paths of continuous, square-integrable martingales are quitedifferent from "ordinary" continuous functions. Being of unbounded firstvariation, they cannot be differentiable, nor is it possible to define integrals ofthe form Ito Ys(w) dXs(w) with respect to X e dr, in a pathwise (i.e., for everyor P-almost every w en), Lebesgue-Stieltjes sense. We shall return to thisproblem of the definition of stochastic integrals in Chapter 3, where we shallgive Ito's construction and change-of-variable formula; the latter is the coun-terpart of the chain rule from classical calculus, adapted to account for theunbounded first, but bounded second, variation of such processes.

It is also worth noting that for X e..11`2, the process <X>, being monotone,is its own first variation process and has quadratic variation zero. Thus, anintegral of the forml Yid < X>, is defined in a pathwise, Lebesgue-Stieltjes sense(Remark 4.6 (i)).

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

5.13 Theorem. Let X = {X Fi; 0 < t < co} and Y = {X f7i; 0 < t < co} bemembers of dic2 . There is a unique (up to indistinguishability) {A}-adapted,continuous process of bounded variation {A ..Ft; 0 < t < co} satisfying Ao = 0a.s. P, such that {X,Yi - A A; 0 < t < co} is a martingale. This process isgiven by the cross-variation <X, Y> of Definition 5.5.

PROOF. Clearly, A = <X, Y> enjoys the stated properties (continuity is a con-sequence of Theorem 4.14 and Problem 4.13). This shows existence of A. Toprove uniqueness, suppose there exists another process B satisfying the condi-tions on A. Then

M A (X Y - A) - (X Y - B) = B - A

is a continuous martingale with finite first variation. If we define

T = inf It 0: =n },

then IM:n) MIA T, .Ft; 0 5 t < col is a continuous, bounded (hence square-integrable) martingale, with finite first variation on every interval [0, t]. Itfollows from (5.4) and Problem 5.11 that

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

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

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

lim E (xtk - - = <X, Y>, (in probability).11r111-0 k=1

Twice in this section we have used the technique of localization, once in theproof of Theorem 5.8 to extend a result about bounded martingales to square-integrable ones, and again in the proof of Theorem 5.13 to apply a result aboutsquare-integrable martingales to a continuous martingale which was notnecessarily square-integrable. The next definitions and problems develop thisidea formally.

5.15 Definition. Let X = {X 0 < t < col be a (continuous) process withX0 = 0 a.s. If there exists a nondecreasing sequence { 7,,},T=1 of stopping timesof such that A Xt AT , .Ft; 0 < t < co} is a martingale for each n 1

and P[lim,'T = oo] = 1, then we say that X is a (continuous) local martin-gale and write X e di' (respectively, X e icI if X is continuous).

5.16 Remark. Every martingale is a local martingale (cf. Problem 3.24(i)), butthe converse is not true. We shall encounter continuous, local martingaleswhich are integrable, or even uniformly integrable, but fail to be martingales(cf. Exercises 3.3.36, 3.3.37, 3.5.18 (iii)).

5.17 Problem. Let X, Y be in .41`1". Then there is a unique (up to indis-tinguishability) adapted, continuous process of bounded variation <X, Y>satisfying <X, Y>0 = 0 a.s. P, such that X Y - <X, jrioc. If X = Y, wewrite <X> = <X, X>, and this process is nondecreasing.

5.18 Definition. We call the process <X, Y> of Problem 5.17 the cross-variationof X and Y, in accordance with Definition 5.5. We call <X> the quadraticvariation of X.

5.19 Problem.

(i) A local martingale of class DL is a martingale.(ii) A nonnegative local martingale is a supermartingale.

(iii) If M E ./IrI" and S is a stopping time of then E(MD < E <M)s,where Ai! -4

We shall show in Theorem 3.3.16 that one-dimensional Brownian motionIB ..11,; 0 < t < col is the unique member of .11`.1" whose quadratic variationat time t is t; i.e., B,2 - t is a martingale. We shall also show that d-dimen-sional Brownian motion {(Bp), ..., iv)), A; 0 < t < col is characterized bythe condition

<B(`) Bo>, = 5,t, t > 0,

where 64 is the Kronecker delta.

5.20 Exercise. Suppose X e di2 has stationary, independent increments. Then<X>, = t(EXt),t 0.

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

We close this section by imposing a metric structure on di2 and discussingthe nature of both dif2 and its subspace .AZ under this metric.

5.22 Definition. For any X e 112 and 0 < t < co, we define

11X N/E(X().We also set

11X11,, A 1IIXII E=, 2"

Let us observe that the function t 11, on [0, co) is nondecreasing,because X2 is a submartingale. Further, IIX -Y II is a pseudo-metric on .#2,which becomes a metric if we identify indistinguishable processes. Indeed,suppose that for X, Y e.//l2 we have 11X - Y11 = 0; this implies X = Y,, a.s. P,for every n > 1, and thus X, = E(X1";,) = E(Y.1.-Fr) = Y, a.s. P, for every0 < t < n. Since X and Y are right-continuous, they are indistinguishable(Problem 1.5).

5.23 Proposition. Under the preceding metric, //2 is a complete metric space,and 12 a closed subspace of A'2.

PROOF. Let us consider a Cauchy sequence {X(")},,cc_1 `/#2: hmn,m.x MX(n) -X(m)l1 = 0. For each fixed t, IAT1)1'_1 is Cauchy in L2(0, y, P), and so hasan L2-limit X,. For 0 < s < t < cc and A e A, we have from L2-convergenceand the Cauchy-Schwarz inequality that lim,E[1A(X!") - Xs)] = 0,limn E[lA(X") - X,)] = 0. Therefore, E[lAX:")] = E[1AX!")] impliesE[lAXJ = E[lAX,], and X is seen to be a martingale; we canchoose a right-continuous modification so that X e .4/2. We have

II X( ") -X II = 0.

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

1 1P[ sup Pq") - Xti > < - XTI2 = X`"'- X II -0 0Clt<T

as n -> co. Along an appropriate subsequence {NM ., we must have

P[ sup I XP.) - Xti >1 1< y; k> 1

0 <t<T

and the Borel-Cantelli lemma implies that .elqnk) converges to X uniformly on[0, T], almost surely. The continuity of X follows.

5.24 Problem. Let M = {M Fr; 0 < t < oo } be a process in 412 u .41`.'" andassume that its quadratic variation process <M> is integrable: E <M>, < co.Then:

(i) M is a martingale, and M and the submartingale M2 are both uni-formly integrable; in particular, Moc, = Mt exists a.s. P, andEMS = E<M>;

(ii) we may take a right-continuous modification of Z, = E(M.021.fl - Mi2;t 0, which is a potential.

5.25 Problem. Let M e .1r I" and show that for any stopping time T of 1.97,1,

E(S A <M)T)(5.5) P[ max 1M,1 a] + P[ <M>r 6],

0<t<T E2

V E > 0, .5 > 0. In particular, for a sequence proln.°_1 c .A'''"c we have

(5.6) <M(")> max IM(")1 0.T t n-boo

5.26 Problem. Let IM A; 0 < t < co and IN %; 0 t < ocl on (S2,3'7, P)be continuous local martingales relative to their respective filtrations, andassume that .97,0 and are independent. With ff; A u %), show thatIM Yf; 0 < t < col, { N Ye; 0 t < co} and {M,N Yt' 0 < t < co} arelocal martingales. If we define iti9, = ns>,a(ffs u Al), where is the collec-tion of P-negligible events in F, then the filtration {} satisfies the usualconditions, and relative to it the processes M, N and MN are still localmartingales. In particular, <M, N> -a 0.

1.6. Solutions to Selected Problems

1.8. We first construct an example with A The collection of sets of the form{(X,,, Xr2,...)E B}, where B E ge(Rd) 0 ./(Ella) and 0 <t1 < t, < < t0,forms a cr-field, and each such set is in Indeed, every set in 31;tox has such a

1.6. Solutions to Selected Problems 39

representation; cf. Doob (1953), p. 604. Choose Q = [0, 2), y = i([0, 2)), andP(F) = meas (F n [0, 1]); F E 347, where meas stands for "Lebesgue measure."For CO E [0, 1], define X,(w) = 0, V t > 0; for CO E (1, 2), define X,(w) = 0 if t 0 co,

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

We next show that if gl,,;`- and ..97,0 is complete, then A Egi,o. Let N c S2be the set on which X is not RCLL. Then

where

A=(U An NC,

n=1

1

An = n U _ x > 1.m=1 qi,q, Qn10,101

-921<(0.)

2.6. Try to argue the validity of the identity {Hr < t} = UsnQ {XsE F}, for any

t > 0. The inclusion is obvious, even for sets which are not open. Use right-continuity, and the fact that F is open, to go the other way.

2.7. (Wentzell (1981)): For x E Fe, let p(x, r) = inf ilx - yll; ye r}, and consider thenested sequence of open neighborhoods of F given by F,, = {x E ffid; p(x, F) <(1 /n) }. By virtue of Problem 2.6, the times T, n > 1, are optional. Theyform a nondecreasing sequence, dominated by H = Hr, with limit TA limn. Tn

H, and we have the following dichotomy:

On {1/ = 0}: T = 0, V n > 1.On {H > 0 }: there exists an integer k = k(w) > 1 such that

= 0; V 1 n < k, and 0 < T < T,, < H; V n k.

We shall show that T = H, and for this it suffices to establish T > H on> 0, T < col.

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

2.17. For every AET we know that A n < S) belongs to both ..97r (Lemma2.16) and ys (Lemma 2.15), and therefore also to T ,.S = T n ,o6r.s. Con-sequently, I{r.,s}E(Zi.Fr,,$)dP = JAn <S}dP .1,4,{r,s}E(Z1.97r)dP =ZSA 1{T <S} E(Z1217T)dP, so (i) follows.

For claim (ii) we conclude from (i) that

1{T<S}E[E(ZVFT)13"-s] = E[1{r<s}E(ZI.FY1Fs] = E[1{T<s}E(ZISTs,,T)Vor's]

= 1{r<s}E[ZI-97s,7-]+

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

1{s< T}E[E(Z19'01Fs] = !Is< E[E(Z snTI

= !IS< nETZ IFS T1

2.22. We discuss only the second claim, following Chung (1984 For any A e F, wehave

A=CU [An <r < TOL, [A n {S= ao)].

E Q

Now A n {S < r < T} = A n {S < r} n {T > r} is an event in .FT, as is easilyverified, because A n {S < e.g,. On the other hand, A n {S = ao} =[A n {S = oo}] n {T = col is easily seen to be in T. It follows that A E FT.

2.23. T is an optional time, by Lemma 2.11, and so FT+ is defined and contained in.54-7;+ for every n 1. Therefore, .FT+ - 107=1 -5477,+ To go the other way, con-sider an event A such that A n T < t} e .F for every n > 1 and t 0. Obviouslythen, A n {T < t} = A (UIT=1 {Tn < t}) = n'e=i(A {"T < t})e ,F and thusA e ,T+. The second claim is justified similarly, using Problem 2.22.

3.2. (i) Fix s > 0 and a nonnegative integer n. Consider the "trace" a-field g of allsets obtained by intersecting the members of Fs" with the set {Ns = n}.Consider also the similar trace a-field if of a(T,, ,7;,) on {N5 = n}. Agenerating family for g is the collection of sets of the form {N,, < n1, ,

/s1,k <nk, Ns = n}, where 0 < t1 < < t, < s, and each such set is a memberof If . A generating family for if is the collection of all sets of the form{S1 ti , , S_, t_,, Ns = n}, where 1:)_ t1 tk s, and each suchset is a member of . It follows that g = .

Therefore, for every A E Fsig there exists A ea(Ti,..., TO such that An{Ns = = A n {Ns = n}. Using the independence of T,1+1 and (S, 1A) weobtain

firl{N,=n)

= P[{S±, > n A n {S s < S,,}]

= P[{S T,,,, > n A n {S s}]

P[S,, > tl.FsN] dP

=f P[{S > t - u} n A n {S s}],le-1" dut-s

= e- '11` -5) P[{S + u > s} n A n s}],le- A" du

= e-mt-s)P[{S + T+, > s} n A n {S

= e-m")P[A n {N5 =

Summation over n > 0 yields IA P[S,,,.+, > dP = e-M`-s)P(A) for everyde Fs".

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

identity:

k-1XP[Yk > e]

(;1.. 0)-i e'; k 1, 0 > O.3=o 3!

We have, as in (i),

fin{N,=n}

= P[{N, n + n (-) {Ns = n}]

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

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

P[N, - 5 klFsn dP

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

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

+ .1

s ,P[{S > t - IA} n A n {Ns = n}]P(Yk E du)o

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

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

=0 .11

+

.1t-s

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

= P[A n (Ns = n }] E e-A0-0(Act -J=0 i!

Adding up over n > 0 we obtain

J.kP[N, - klei.sndP = P(A) E e-m

(A(t -t-s)

J=0

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

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

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

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

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

=

JX' dP f X; dP dP.

sfix,..,> A) {x,;>-).}

Therefore, {X,;},`,°=, is a uniformly integrable sequence. On the other hand,

0 > fn<_A) XdP = E(X) -fXdP > E(X) - I X,dP

= E(X) - E(X,) X , dP, for n > m.f{x<-,1.)

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

sup PC,IdP < 2.n>m f{x<-A}

Consequently, for these choices of m and A we have:

sup X;. dP < E, and thus pr 1,;°_, is also uniformly integrable.^>^. f(x,-,> A)

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

(b) (c): Let be the L'-limit of {Xr; 0 < t < co }. For 0 < s < t and A Ewe have IA Xs dP < f X, dP, and letting t co we obtain the submartingaleproperty IA Xs dP < L X, dP; 0 < s < co, A Fs.

(C) (a): For 0 t < c o and A > 0, we have fix,,A} X, dP fixo. A} X0 dP,which converges to zero, uniformly in t, as A I co, because P[X, > <(1 /A)EX, < (1 / A)EX,.

3.20. Apply Problem 3.19 to the submartingales ...97,; 0 t < co } to obtain theequivalence of (a), (b), and (c). The latter obviously implies (d), which in turngives (a). If (d) holds, then jA Y dP = f A X, dP, V A E A. Letting t co, we obtainSA Y dP = fA Xco dP. The collection of sets A E 347. for which this equality holdsis a monotone class containing the field U,ZO A. Consequently, the equalityholds for every A E Fc, which gives E( Yl..9700) = Xeo, a.s.

3.26. The necessity of (3.2) follows from the version of the optional sampling theoremfor bounded stopping times (Problem 3.23 (i)). For sufficiency, consider 0 < s <t < co, A E ..Fs and define the stopping time S(co) A s lA (a)) + t lAc(a)). The conditionE(X,) > E(X s) is tantamount to the submartingale property E[X,1A] > E[Xs1A].

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

co co

n 3,--(T+0+,n=1 n=1

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

(i) With 0 < s < t < co, Problem 3.23 implies

E[fCrigs] E[XT, -XTIT+.] X T+s XT Xs, a.s. P.

(ii) Let S, < S2 be bounded stopping times of {A}; then for every t 0,

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

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

EXs = Eg(Ss_ T)v 0 Eg(Si- Dv 0 = EXst.

Another application of Problem 3.26 shows that X is a submartingale.

3.28. (Robbins and Siegmund (1970)): With the stopping time

T= inf {t E [S, 00); Z, = b },

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

ZsdP = ZT AtdPArn{Z.<10) fArn{Z.<6}

= b- P[A {zs < b, T t}] + Z,1{T>odP.SAn{4<b}

The integrand Z,1{T>t} is dominated by b, and converges to zero as t co byassumption; it develops then from the dominated convergence theorem that

An{2.<b}ZsdP = b P[A n (Zs < b,T < cc]

= b f P[T < col.Fs]dP,An{Z,<b)

establishing the first conclusion. The second follows readily.

4.9. (a) According to Problem 3.23 (i) we have

4.11.

JX

X T dP < X,,dP and P[XT > AE(

] <A

T)<

E(X

A

)

for every a > 0, A > 0, TE and therefore

lim sup f XT dP = 0.).- Te {xr>.1.}

(b) It suffices to show the uniform integrability of {MT} T5,. Once again,Problem 3.23 (i) yields MT = E(M,71,T) a.s. P, for every TE 99, and the claimthen follows easily, just as in the implication (d) =. (a) of Problem 3.20.

This latter problem, coupled with Theorem 3.22, yields the representationXT = E(Xcol'aFT) a.s. P, V T E .9' for every uniformly integrable martingale X,which is thus shown to be of class D.

For an arbitrary bounded random variable on P),

EC E(ie")1W)] = E[E(IW) E(.4(")1W)] = E[A() E(IW)],

which converges to E[A E(11)] = E[E(A1W)].

4.15. Define the stopping times H,= inf { 0; X, > e}, S = inf {t 0; A, L. .5} (Prob-lem 2.7) and T=TAnAH,. We have

tP[VTn > c] < E[XT 1{V,. >0] < E(XT,,) < E(AT) < E(AT)

and (4.14) follows because '1;, T T n H,, a.s. as n co. On the other hand, we have

E(As,T) E( A AT)P[VT < (5] P[VT As L

(5

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

EF(VT) = J P(VT L u)dF(u)A AT)

+ P(AT _u)}dF(u)

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

1

= E[2F(AT)+ AT f -dF(U) I = EG(AT)4r U

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

5.11. Let H = {to, , t,n}, with 0 = to < t1 < < t. = t, be a partition of [0, t]. Forq > p, we have

v,(q)(n) < v,(P)(n) max IX,- X1<k<m

The first term on the right-hand side has a finite limit in probability, andthe second term converges to zero in probability. Therefore, the product con-verges to zero in probability. For the second assertion, suppose that 0 < q < p,P(L, > 0) > 0 and assume that v,o)(n) does not tend to oo (in probability) as111111 -0. Then we can find ö > 0, 0 < K < co, and a sequence of partitions{TIn}n',1 such that P(A)L SP(L, > 0), where

A= {L, > 0, V(q)(11) K); n I.

Consequently, with n = {too,. , t;,t}, we have

v(P)(11.) < Km?-9(X; 1111"o on A,,; n > 1.

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

5.12. Because <X> is continuous and nondecreasing, we have P[<X>T,, = 0;0 < t < oo] = 1. An application of the optional sampling theorem to the con-tinuous martingale M A X2 - <X> yields (Problem 3.24 (i)): 0 = EMT,,, =E[XL,- EXh,, which implies P[XT A, = 0] = 1, for every0 < t < oo. The conclusion follows now by continuity.

5.17. There are sequences {Sn}, {T "} of stopping times such that S. 00, T,,T 00, andXI") A Xo.,S, Yi(n) -4 Yr, T are {,,}-martingales. Define

Rn A Sn A Tn A inf It 0: I X,I = n or I = nl,

1.7. Notes 45

and set Xr = Xin R., g(n) = li R, Note that R T cc a.s. Since XP) = R.,and likewise for Y("), these processes are also {,,}-martingales (Problem 3.24),and are in because they are bounded. For m > n, XP) = X:'",,),, and so

- <g(')>t A R. = an.)2 - (')>t n R.

is a martingale. This implies a*>, = <X(m)>,,, R.,. We can thus decree <X>, A<X(")>, whenever t < R and be assured that <X> is well defined. The process(X> is adapted, continuous, and nondecreasing and satisfies <X>0 = 0 a.s.Furthermore,

- R = (X;n))2 - <g(")>r

is a martingale for each n, so X2 - <X> edlc.'. As in Theorem 5.13, we maynow take <X, Y> = -14:[<X + Y> - <X - Y>].

As for the question of uniqueness, suppose both A and B satisfy the conditionsrequired of (X, Y>. Then M A XY- A and N A XY-B are in .1/', so justas before we can construct a sequence {R} of stopping times with R,, j cc suchthat MP) A M, . and N,P) A Al, . are in Consequently Mr - NP) =

- Art, ,,E.1r2, and being of bounded variation this process must beidentically zero (see the proof of Theorem 5.13). It follows that A = B.

5.24. If ME 412, then E(W) = E<M>; VO < t < co. If M EProblem 5.19 (iii) gives E(MS) < E <M> for every stopping time S; it followsthat the family {Ms}se 9, is uniformly integrable (Chung (1974), Exercise 4.5.8),i.e., that M is of class D and therefore a martingale (Problem 5.19 (i)).

In either case, therefore, M is a uniformly integrable martingale; Problem 3.20now shows that Moo = limn. M, exists, and that E(Mj,F,) = M, holds a.s. P,for every t > 0. Fatou's lemma now yields

(6.1) E(M!) = E Clim AV) < lim E(A112)= lim E<M>, = E<M>,too 1-co

and Jensen's inequality: M< < E(M.021F,), a.s. P, for every t > 0. It follows thatthe nonnegative submartingale M2 has a last element, i.e., that {AV, A;0 < t < cc} is a submartingale. Problem 3.19 shows that this submartingale isuniformly integrable, and (6.1) holds with equality. Finally, Z, = E(M2.1327,) -AV is now seen to be a (right-continuous, by appropriate choice of modification)nonnegative supermartingale, with E(Z,) = E(M,02) - E(M,2) converging to zeroas t co.

5.25. Problem 5.19 (iii) allows us to apply Remarks 4.16, 4.17 with X = M2, A = <M>.

1.7. Notes

Sections 1.1, 1.2: These two sections could have been lumped togetherunder the rubric "Fields, Optionality, and Measurability" after the manner ofChung & Doob (1965). Although slightly dated, this article still makesexcellent reading. Good accounts of this material in book form have beenwritten by Meyer [(1966); Chapter IV], Dellacherie [(1972); Chapter III and


to a lesser extent Chapter IV], Dellacherie & Meyer [(1975/1980); ChapterIV], and Chung [(1982); Chapter 1]. These sources provide material on theclassification of stopping times as "predictable," "accessible," and "totallyinaccessible," as well as corresponding notions of measurability for stochasticprocesses, which we need not broach here.

A new notion of "sameness" between two stochastic processes, called syn-onimity has been introduced by Aldous. It was expounded by Hoover (1984)and was found to be useful in the study of martingales.

A deep result of Dellacherie [(1972), p. 51] is the following: for everyprogressively measurable process X and F e ,4(R), the hitting time Hr ofExample 2.5 is a stopping time of {Ft}, provided that this filtration is right-continuous and that each o--field is complete.

Section 1.3: The term martingale was introduced in probability theory byJ. Ville (1939). The concept had been created by P. Levy back in 1934, in anattempt to extend the Kolmogorov inequality and the law of large numbersbeyond the case of independence. Levy's zero-one law (Theorem 9.4.8 andCorollary in Chung (1974)) is the first martingale convergence theorem. Theclassic text, Doob (1953), introduced, for the first time, an impressively com-plete theory of the subject as we know it today. For the foundations of thediscrete-parameter case there is perhaps no better source than the relevantsections in Chapter 9 of Chung (1974) that we have already mentioned; fulleraccounts are Neveu (1975), Chow & Teicher (1978), Chapter 11, and Hall& Heyde (1980). Other books which contain material on the continuous-,parameter case include Meyer [(1966); Chapters V, VI], Dellacherie & Meyer[(1975/1980); Chapters V-VIII], Liptser & Shiryaev [(1977), Chapters 2, 3]and Elliott [(1982), Chapters 3, 4].

Section 1.4: Theorem 4.10 is due to P. A. Meyer (1962, 1963); its proof waslater simplified by K. M. Rao (1969). Our account of this theorem, as well asthat of Theorem 4.14, follows closely Ikeda & Watanabe (1981).

Section 1.5: The study of square-integrable martingales began with Fisk(1966) and continued with the seminal article by Kunita & Watanabe (1967).Theorem 5.8 is due to Fisk (1966). In (5.6), the opposite implication is alsotrue; see Lemma A.1 in Pitman & Yor (1986).


CHAPTER 2

Brownian Motion

2.1. Introduction

Brownian movement is the name given to the irregular movement of pollen,suspended in water, observed by the botanist Robert Brown in 1828. Thisrandom movement, now attributed to the buffeting of the pollen by watermolecules, results in a dispersal or diffusion of the pollen in the water. Therange of application of Brownian motion as defined here goes far beyonda study of microscopic particles in suspension and includes modeling of stockprices, of thermal noise in electrical circuits, of certain limiting behavior inqueueing and inventory systems, and of random perturbations in a variety ofother physical, biological, economic, and management systems. Furthermore,integration with respect to Brownian motion, developed in Chapter 3, givesus a unifying representation for a large class of martingales and diffusionprocesses. Diffusion processes represented this way exhibit a rich connectionwith the theory of partial differential equations (Chapter 4 and Section 5.7).In particular, to each such process there corresponds a second-order parabolicequation which governs the transition probabilities of the process.

The history of Brownian motion is discussed more extensively in Section 11;see also Chapters 2-4 in Nelson (1967).

1.1 Definition. A (standard, one-dimensional) Brownian motion is a continuous,adapted process B = IB A; 0 < t < co }, defined on some probability space(n, .S, P), with the properties that Bo = 0 a.s. and for 0 < s < t, the incrementA - B., is independent of 377s and is normally distributed with mean zeroand variance t - s. We shall speak sometimes of a Brownian motion B ={13 ,Ft; 0 < t < T} on [0, T], for some T > 0, and the meaning of thisterminology is apparent.

48 2. Brownian Motion

If B is a Brownian motion and 0 = to < t, < < t < co, then the incre-ments {A, - /311_,}7=1 are independent and the distribution of Btu -Btudepends on t; and only through the difference ti - tj_i; to wit, it isnormal with mean zero and variance ti - . We say that the process B hasstationary, independent increments. It is easily verified that B is a square-integrable martingale and , = t, t 0.

The filtration {,F,} is a part of the definition of Brownian motion. However,if we are given {B,; 0 < t < co} but no filtration, and if we know that B hasstationary, independent increments and that B, = B, - Bo is normal withmean zero and variance t, then {B ,F13; 0 < t < co} is a Brownian motion(Problem 1.4). Moreover, if {.97,} is a "larger" filtration in the sense that

for t > 0, and if B, -B., is independent of ys whenever 0 < s < t,then {B Ft; 0 < t < co} is also a Brownian motion.

It is often interesting, and necessary, to work with a filtration {ffl} whichis larger than 1,,B1. For instance, we shall see in Example 5.3.5 that thestochastic differential equation (5.3.1) does not have a solution, unless we takethe driving process W to be a Brownian motion with respect to a filtrationwhich is strictly larger than {Fr }. The desire to have existence of solutionsto stochastic differential equations is a major motivation for allowing { inDefinition 1.1 to be strictly larger than {.F,11.

The first problem one encounters with Brownian motion is its existence.One approach to this question is to write down what the finite-dimensionaldistributions of this process (based on the stationarity, independence, andnormality of its increments) must be, and then construct a probability measureand a process on an appropriate measurable space in such a way that weobtain the prescribed finite-dimensional distributions. This direct approachis the one most often used to construct a Markov process, but is rather lengthyand technical; we spell it out in Section 2. A more elegant approach forBrownian motion, which exploits the Gaussian property of this process, isbased on Hilbert space theory and appears in Section 3; it is close in spiritto Wiener's (1923) original construction, which was modified by Levy (1948)and later further simplified by Ciesielski (1961). Nothing in the remainder ofthe book depends on Section 3; however, Theorems 2.2 and 2.8 as well asProblem 2.9 will be useful in later developments.

Section 4 provides yet another proof for the existence of Brownian motion,this time based on the idea of the weak limit of a sequence of random walks.The properties of the space C[0, co) developed in this section will be usedextensively throughout the book.

Section 5 defines the Markov property,, which is enjoyed by Brownianmotion. Section 6 presents the strong Markov property, and, using a proofbased on the optional sampling theorem for martingales, shows that Brownianmotion is a strong Markov process. In Section 7 we discuss various choicesof the filtration for Brownian motion. The central idea here is augmentationof the filtration generated by the process, in order to obtain a right-continuousfiltration. Developing this material in the context of strong Markov processesrequires no additional effort, and we adopt this level of generality.

2.2. First Construction of Brownian Motion 49

Sections 8 and 9 are devoted to properties of Brownian motion. In Section 8we compute distributions of a number of elementary Brownian functionals;among these are first passage times, last exit times, and time and level ofthe maximum over a fixed time-interval. Section 9 deals with almost sureproperties of the Brownian sample path. Here we discuss its growth as t co,its oscillations near t = 0 (law of the iterated logarithm), its nowhere differ-entiability and nowhere monotonicity, and the topological perfectness of theset of times when the sample path is at the origin.

We conclude this introductory section with the Dynkin system theorem(Ash (1972), p. 169). This result will be used frequently in the sequel wheneverwe need to establish that a certain property, known to hold for a collectionof sets closed under intersection, also holds for the a-field generated by thiscollection. Our first application of this result occurs in Problem 1.4.

1.2 Definition. A collection g of subsets of a set n is called a Dynkin system if

(i) Q e g,(ii) A, B e g and B s A imply A\B e 9,

(iii) {A};,°=, 9 and Al A2 c imply U;;3=1 Ae 9.

13 Dynkin System Theorem. Let (6' be a collection of subsets of f2 which isclosed under pairwise intersection. If 9 is a Dynkin system containing (6, then9 also contains the a-field a(W) generated by (6.

1.4 Problem. Let X = {X1; 0 < t < co} be a stochastic process for whichX0, X,1- Xt., ..., X,. - X,._, are independent random variables, for everyinteger n > 1 and indices 0 = t, < t, < < t. < co. Then for any fixed0 < s < t < cc, the increment X, -Xs is independent of .97,x.

2.2. First Construction of Brownian Motion

A. The Consistency Theorem

Let R)c''') denote the set of all real-valued functions on [0, co). An n-dimensionalcylinder set in Or°31 is a set of the form

(2.1) C to) e FRE'''); (a)(ti ), , w(t,,)) e Al,

where t, e [0, cc), i = 1, n, and A e ,I(R"). Let (6 denote the field of allcylinder sets (of all finite dimensions) in RE''), and let 1/(R)0')) denote thesmallest a-field containing (6.

2.1 Definition. Let T be the set of finite sequences t = (t1,... , t) of distinct,nonnegative numbers, where the length n of these sequences ranges over the set


of positive integers. Suppose that for each t of length n, we have a probabilitymeasure Q, on (R", ROW)). Then the collection {12t},ET is called a family offinite-dimensional distributions. This family is said to be consistent providedthat the following two conditions are satisfied:

(a) if s = (t11, tif tin) is a permutation of t = (t, , t2, t), then for anyAl e MR), i = 1, , n, we have

121(A1 x A2 x x A,,) = x Ai2 x x Ain);

(b) if t = t2,..., t) with n 1, s = t2,..., tn_, ), and A e M(R"), then

Q,(A x = Q2(A).

If we have a probability measure P on (RE'''), M(RE'''))), then we can definea family of finite-dimensional distributions by

(2.2) Qt(A) = P[0) e Rio''); (w(t,), . . . , co(t))e A],

where A e ,I(Rn) and t = (t1,..., t)e T This family is easily seen to be con-sistent. We are interested in the converse of this fact, because it will enable usto construct a probability measure P from the finite-dimensional distributionsof Brownian motion.

2.2 Theorem (Daniell (1918), Kolmogorov (1933)). Let {Qt} be a consistentfamily of finite-dimensional distributions. Then there is a probability measureP on (IRE", al(RE'''))), such that (2.2) holds for every te T

PROOF. We begin by defining a set function Q on the field of cylinders W. If Cis given by (2.1) and t = (t1, t2, t)e T, we set

(2.3) Q(C) = Q,(A), C e 6.

2.3 Problem. The set function Q is well defined and finitely additive on W, withQ(Ro'')) = 1.

We now prove the countable additivity of Q on W, and we can then drawon the Caratheodory extension theorem to assert the existence of the desiredextension P of Q to .4(18E0 °°)). Thus, suppose {/3,},71L, is a sequence of disjointsets in W with B A I also in W. Let Cm =v B\UT=i 13,,, SO

Q(B) = Q(C.) + X 121130k=1

Countable additivity will follow from

(2.4) lim Q(Cm) = 0.m-

Now Q(Cm) = Q(C,+,) + Q(B,,i) Q(C,,,), so the limit in (2.4) exists.Assume that this limit is equal to e > 0, and note that n:=1 C, = 0.


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

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

for some Am eM(Rm), and the finite sequence t A (t1, ..., t,)e T is an exten-sion of the finite sequence t,,,_, A (ti,..., tm_, ) e T, m > 2. This may be accom-plished as follows. Each Ck has a form

Ck = lw e RE'''); (w(t, ), ... , w(t,k))e ilmk 1; Am, e .4(Ork),

where tmk = (t1,..., t,de T. Since C, Ck, we can choose these representa-tions so that t is an extension of t and Amk+, Am,, x 08mk+1 Mk. Define

D, = lw; w(t,)eRI,...,Dmi_, = Ico;()(t 1), ... ,co(t,,,j_1))e gr.-11

and Dm, = C1, as well as

Dm. +, = {a);(co(ti),..., w(tm,), a(t,,,i+i))e Am, x RI,...,

D,2_ 1 = lw;(w(t 1), . . . , w(t,,), w(t, +1), . . . , w(t,2_1))e A,, x Rm2-"11-11

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

n:=1 Cm= 0.

2.4 Problem. Let Q be a probability measure on (R", .4(R")). We say thatA e R(111") is regular if for every probability measure Q on (R", ,4(R")) and forevery C > 0, there is a closed set F and an open set G such that FgA Gand Q(G\F) < E. Show that every set in M(01") is regular. (Hint: Show that thecollection of regular sets is a a-field containing all closed sets.)

According to Problem 2.4, there exists for each in a closed set Fm Am suchthat Q, (Am \Fm) < e/2m. By intersecting Fm with a sufficiently large closedsphere centered at the origin, we obtain a compact set Km such that, with

E,,, A {we 0:8[°');(to(t 1), 'Mt.)) G Km),

we have Em D,,, and

Q(Dm \Em) = Q, (Am \Km) < .

The sequence {Em} may fail to be nonincreasing, so we define

and we have

where

Em = nm Ek,k=1

Em = la) e RE°.'); (o)(ti ), ... , w(tm))e Km},

Km = (K1 x Rm .' ) r-) (K2 x glm -2 ) r) - - (-) (Km_i x gl) rm Km,

which is compact. We can bound Q, (1Z.) away from zero, since

Q, (gm) = Q(Em) = Q(Dm) - Q(D,\Em)

= Q(Dm)- QU (Dm\Ek))k=i

Q(Dm)- Q(U (Dk\Ek))k=1

m cs - O.

k=1 2

Therefore, K. is nonempty for each m, and we can choose (x(r),..., x(7))e Km.Being contained in the compact set K1, the sequence {x(r)1,,,°°=2 must havea convergent subsequence fx(imol,`,°_, with limit x1. But lx(ro, x(2m01;,°_2 iscontained in K2, so it has a convergent subsequence with limit (x2, x2).Continuing this process, we can construct (x x2, ...)eR x R x -, such that(x1, , x.) E Km for each m. Consequently, the set

S = {co e Rt°'°°): w(ti) = xi, i = 1, 2, ... }

is contained in each Em, and hence in each Dm. This contradicts the fact thatn,-.,1 Dm = 0. We conclude that (2.4) holds.

Our aim is to construct a probability measure P on (S/F) A (Rt'''),R(Rt°''))) so that the process B = IB .F13; 0 < t < col defined by B,(w)co(t), the coordinate mapping process, is almost a standard, one-dimensionalBrownian motion under P. We say "almost" because we leave aside therequirement of sample path continuity for the moment and concentrateon the finite-dimensional distributions. Recalling the discussion followingDefinition 1.1, we see that whenever 0 = so < s1 < s2 < < s, the cumu-lative distribution function for must be

(2.5) x)Xi x2

p(s 1; 0, Y i)P(s2 - s1;Y19Y 2) -

.p(s. - s.-i; Y.--i, Y.) dY. dY2 dYi

for (x1, , x)e R", where p is the Gaussian kernel

(2.6) p(t; x, y) A1 e-cx-ypi2r, t > 0, x, y e R.

.2irtThe reader can verify (and should, if he has never done so!) that (2.5) isequivalent to the statement that the increments {B., -BSS }7=1 are inde-pendent, and Bs, -1135 is normally distributed with mean zero and variance

- Sj_i.Now let t = (t1, t2, t), where the ti are not necessarily ordered but

are distinct. Let the random vector (Bt,, B,2, ,Bt,,) have the distributiondetermined by (2.5) (where the ti must be ordered from smallest to largest toobtain (s1, .. . , s) appearing in (2.5)). For A E .4(R"), let Qt(A) be the probabilityunder this distribution that (B,1312, - , An) is in A. This defines a family offinite-dimensional distributions {Qt}t T.

2.5 Problem. Show that the just defined family {Qt},e T is consistent.

2.6 Corollary to Theorem 2.2. There is a probability measure P on (01(0''),,4([111°''))), under which the coordinate mapping process

Moo) = co(t); co eRt°''), t > 0,

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

B. The Kolmogorov-entsov Theorem

Our construction of Brownian motion would now be complete, were it notfor the fact that we have built the process on the sample space FEE°'"') of allreal-valued functions on [0, co) rather than on the space C[0, co) of continuousfunctions on this half-line. One might hope to overcome this difficulty byshowing that the probability measure P in Corollary 2.6 assigns measure oneto C[0, co). However, as the next problem shows, C[0, co) is not in the a-field,4(0110'')), so P(C[0, co)) is not defined. This failure is a manifestation ofthe fact that the a-field gi(011°.')) is, quite uncomfortably, "too small" for aspace as big as 01[0'`°); no set in .4(0110'"')) can have restrictions on uncountablymany coordinates. In contrast to the space C[0, co), it is not possible todetermine a function in 0:11°'') by specifying its values at only countablymany coordinates. Consequently, the next theorem takes a different approach,which is to construct a continuous modification of the coordinate mappingprocess in Corollary 2.6.

2.7 Exercise. Show that the only M(RE°''))-measurable set contained in C[0, co)is the empty set. (Hint: A typical set in a(fle'')) has the form

E = {co e RE°''); (co(t ,), co(t 2), . . .) e AI,

where A e a(fps x 01 x )).

2.8 Theorem (Kolmogorov, entsov (1956a)). Suppose that a process X ={X1; 0 < t < T} on a probability space (LI, 31%, P) satisfies the condition

(2.7) EIX, - XX' < CI t - sl", 0 s, r 5 T,

for some positive constants a, fi, and C. Then there exists a continuous modification1 = {gt; 0 5 t < T} of X, which is locally Holder-continuous with exponent y

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

(2.8) P[co; sup It(w) s(w)i = 1,o<i-s<huo It - slY

,,te [0, T1

where h(w) is an a.s. positive random variable and 6 > 0 is an appropriateconstant.

PROOF. For notational simplicity, we take T = 1. Much of what follows isa consequence of the ebygev inequality. First, for any e > 0, we have

E IX, -XIsP[IX, - XsI c] < Cc'lt - "

Ex

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

P[142. - 4-1)/2.1 > 2-Y1 < C 2-n( 1 +fl-",

and consequently,

(2.9) [P max 142. - X(k_i On I 2-Yn1 Sk<2^

= PL2

I Xk/2" X(k-1)/2^1 > 2- Yn]k=1

< C2-"("Y).

The last expression is the general term of a convergent series; by the Borel-Cantelli lemma, there is a set n* e with P(S1*) = 1 such that for each w e Q *,

(2.10) max I X,02^(w) - 4_1)/2.(w)1 < 2-", V n n*(w),1 sic <2.

where n*(w) is a positive, integer-valued random variable.For each integer n > 1, let us consider the partition D = {(k/2");k = 0,

1, , 2"} of [0, 1], and let D = U `°=, D, be the set of dyadic rationals in [0, 1].We shall fix co e n* n > n*(w), and show that for every m > n, we have

(2.11) I Xt(w) - Xs(w) I 2 E v t, se D 0 < t - s < 2j=n+1

For m = n + 1, we can only have t = (k/r), s = ((k - 1)/2'), and (2.11)follows from (2.10). Suppose (2.11) is valid for m = n + 1, M - 1. Takes < t, s, te Dm, consider the numbers t1 = max{u eDm_i; u 5 t} and s1 =min fu e Dm_i; u > s }, and notice the relationships s 5 s1 < t1 < t, s1 - s2-14, t - t1 5 2'. From (2.10) we have I Xs, (w) - Xs(c0)1 < 2 Ym, Xt(w) -

(w) I < 2-7m, and from (2.11) with m = M - 1,M -1

I Xit(w) - Xsi(w)I < 2 Ej=n+1

We obtain (2.11) for m = M.

We can show now that { X,(w); t E DI is uniformly continuous in t for everyto en*. For any numbers s, t e D with 0 < t - s < h(w) g 2-"a", we selectn > n*(w) such that 2-("+1) < t -s < 2-". We have from (2.11)

(2.12) I Xi(w) - Xs(w)I 2 2-Yi bit - slY, 0 < t - s < h(C0),j=n+1

where .5 = 2/(1 - 2-Y). This proves the desired uniform continuity.We define 2 as follows. For w n*, set 21(w) = 0, 0 < t < 1. For w e n*

and t e D, set 2,(co) = X,(w). For w E fr and t E [0, 1] n DC, choose a sequence{s},,`°_, D with s t; uniform continuity and the Cauchy criterion implythat {X,(w)}T=1 has a limit which depends on t but not on the particular se-quence {s},T=1 g D chosen to converge to t, and we set 2,(w) = Xs.(co).The resulting process X is thereby continuous; indeed, 2 satisfies (2.12), so(2.8) is established.

To see that 2 is a modification of X, observe that 2, = X, a.s. for teD;for t e [0, 1] n DC and {s}'_, D with sn t, we have X, in probabilityand X, -51, a.s., so = a.s.

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

(2.13) El X, - X512 < C Ilt -for some positive constants ; f3, and C. Show that the conclusion of Theorem2.8 holds, with (2.8) replaced by

(2.14) +; supI ii(w) - XS(w)1

< .5] = 1.Ilt - sll.

s,re[0,T[a

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

E IB, - clt - sr.

2.11 Corollary to Theorem 2.8. There is a probability measure P on (41[0,-),a(11V0.))), and a stochastic process W = {W ; t > 0} on the same space,such that under P, W is a Brownian motion.

PROOF. According to Theorem 2.8 and Problem 2.10, there is for each T > 0 amodification WT of the process B in Corollary 2.6 such that WT is continuouson [0, T]. Let

= {w; WT (w) = B,(w) for every rational t e [0, T] },

so P(S1T) = 1. On n A n T =1 nT, we have for positive integers T1 and "2,

WT (w) wtT2((0), for every rational t e [0, T1 A T2].

Since both processes are continuous on [0, T, A T2], we must have W,Ti (co) --W,T2(co) for every t e[0, T, A T2], to e n. Define W,(w) to be this commonvalue. For co 0 n, set W,(w) = 0 for all t > 0.

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

2.3. Second Construction of Brownian Motion

This section provides a succinct, self-contained construction of Brownianmotion. It may be omitted without loss of continuity.

Let us suppose that {B ..F,; t > 0} is a Brownian motion, fix 0 < s < t < co,and set 0 A (t + s)/2; then, conditioned on Bs = x and B, = z, the randomvariable Be is normal with mean it A (x + z)/2 and variance a2 A (t - s)/4.To verify this, observe that the known distribution and independence of theincrements B Be - Bs, and B, -Bo lead to the joint density

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

S ; y, z) dx dy dz

1

exp{ (Y 12)2 } dx dy dz= p(s; 0, x)p(t - s; x, z)

a .,/ 2it 2a2

in the notation of (2.6), after a bit of algebra. Dividing by

P[Bse dx, B, e dz] = p(s; 0, x)p(t - s; x, z) dx dz,

we obtain

1

(3.1) P[B0+.0/2 E dy IA = x, B, = z] = e-(Y-P)212a2 dy.a .,/ 2it

The simple form of this conditional distribution for B(,+S) /2 suggests that wecan construct Brownian motion on some finite time-interval, say [0, 1], byinterpolation. Once we have completed the construction on [0, 1], a simple"patching together" of a sequence of such Brownian motions will result ina Brownian motion defined for all t > 0.

To carry out this program, we begin with a countable collection IV,");k e 1(n), n = 0, 1, ... } of independent, standard (zero mean and unit variance)normal random variables on a probability space (Q,,5--., P). Here 1(n) is the setof odd integers between 0 and 2"; i.e., /(0) = {1}, /(1) = {1}, /(2) = 11, 31, etc.For each n > 0, we define a process B(n) = IBI"); 0 < t 5 1} by recursion andlinear interpolation, as follows. For n > 1, /3,172,, , will agree with B172-.91,k = 0, 1, ..., 2"-`. Thus, for each value of n, we need only specify B,13., fork e 1(n). We set

2.3. Second Construction of Brownian Motion 57

Bo) = 0, /V) =

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

B(7+ s)/ 2 /1 + 6i;k").

We shall show that, almost surely, $10 converges uniformly in t to a continuousfunction B and {B ,,11; 0 < t < 1) is a Brownian motion.

Our first step is to give a more convenient representation for the processesBoo, n = 0, 1, .... We define the Haar functions by I-11°)(t) = 1, 0 < t < 1, andfor n 1, k e I(n),

Hr)(t) =

2(n-1)/2,

201-10

k - 1< t < r'

rk +r

1-< t <'

0, otherwise.

We define the Schauder functions by

ISk")(t) = t He)(u) du, 0 < t < 1, n 0, k e 1(n).o

Note that S(i())(t) = t, and for n > 1 the graphs of S,1") are little tents of height2-( " +1)/2 centered at kon and nonoverlapping for different values of k e 1(n).It is clear that BP) = Vi()),S(,°)(t), and by induction on n, it is easily verified that

(3.2) 13:.)(c0)= E E ;:n)(w)s,(r)(t), 0 < t < 1, n > 0.m=0 k /(m)

3.1 Lemma. As n co, the sequence of functions {Br(w); 0 < t < 1), n > 0,given by (3.2) converges uniformly in t to a continuous function 113,(w); 0 < t < 1),for a.e. w En.

PROOF. Define bn = maxk e 1(n) I k ") I. For x > 0

(3.3) PNri > x] -2 e-"2/2 du

e -x2/22-u C"2/2 du = - -

7i x X TC X

which gives

P[bn > = Pr > 2" P[VI n]k e I

, n > 1.

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

E E idn'snoi < E n2-("+"2 < co;n=n(w) k E 1(n) n=n(w)

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

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

CO

<f,g> = E <f,1-11,")>n=0 kel(n)

applied to f = and g = 1[0,4 yields

00

(3.4) E > sk.)(01.)(s)= S A t; 0 s, t s 1.n=0 k e 1(n)

3.2 Theorem. With {.13} "1`°_, defined by (3.2) and 13, = lim,13P), the process{B <F,B; 0 < t < 1} is a Brownian motion on [0, 1].

PROOF. It suffices to prove that, for 0 = to < t1 < < t < 1, the increments{B,, - /31,_1};=1 are independent, normally distributed, with mean zero andvariance ti - ti_, . For this, we show that for A.ie 11, j = 1, ... , n and i = j- 1,

. .[(3.5) E exp i E Ai(/31, - 131.,_,) = ri exp --1 A4t- - t-_,) .2 'J.'.i=1 .i=1

Set An+, = 0. Using the independence and standard normality of the randomvariables {d")}, we have from (3.2)

E[expl-i E -

= E[expt E E ckno - aosint,)}]m=0 kE/(m) j=1

= fl 11 E exp - 4") E (Ai+, - aosr(t.,)m=0 k 1(m) 1=1

1 n= n n exp [ E (Ai+, -m=0 ,E,r(m) 2 pi }2]

n n= exp [ - E E - - E E Sint,)S,(,m)(t;)].

2 pi (=i m=0 kcl(m)

Letting M co and using (3.4), we obtain

2.4. The

E[exp

Space C[0, co], Weak Convergence,

n

li E A.,(B,, -13,,,)}]=.1=1

n -1 n

= exp { - E E (2.,+, -1.,)(2,,,J=1 i.J+1

n-1= exp {- E (A.,+, - a..,)(-).,,,)t,

pi.-1

= exp E (A. +1 - apt.

= exp -

E[exp

1,12t2

and Wiener Measure 59

n

{ - i E (Aii., - '.;)B,, }1j=1

1 n

- aot, - - E (Ai+, - 2,)2t.,}2 J=1

1

- 2 (Aj+i - ili)2 til

O

3.3 Exercise. Prove Theorem 3.2 without resort to the Parseval identity (3.4),by completing the following steps.

(a) The increments {Al. - V_10.}1.:1 are independent, normal randomvariables with mean zero and variance l/r.

(b) If 0 = to < t1 < < t 5 1 and each t; is a dyadic rational, then theincrements {B,, -Bpi }T=1 are independent, normal random variables withmean zero and variance (tj -

(c) The assertion in (b) holds even if {t3 }.7,_1 is not contained in the set ofdyadic rationals.

3.4 Corollary. There is a probability space (II, , P) and a stochastic processB = {B ..5/7!; 0 < t < co} on it, such that B is a standard, one-dimensionalBrownian motion.

PROOF. According to Theorem 3.2, there is a sequence (a,,,Pof probability spaces together with a Brownian motion XA n I0

2,

on each space. Let S2 = S21 x S22 x , = .F2 ' , and P = Pi xP2 x . Define B on S2 recursively by

= V1), 0 < t < 1,

Bn Xj( n < t < n + 1.

This process is clearly continuous, and the increments are easily seen to beindependent and normal with zero mean and the proper variances.

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

The sample spaces for the Brownian motions we built in Sections 2 and 3were, respectively, the space W.') of all real-valued functions on [0, co)and a space S2 rich enough to carry a countable collection of independent,

standard normal random variables. The "canonical" space for Brownianmotion, the one most convenient for many future developments, is C[0, co),the space of all continuous, real-valued functions on [0, co) with metric

co

(4.1) p(a) , co2) =1E max 0(01(0 - w2(t)I A 1).

n=1 2 0<t<n

In this section, we show how to construct a measure, called Wiener measure,on this space so that the coordinate mapping process is Brownian motion.This construction is given as the proof of Theorem 4.20 (Donsker's invarianceprinciple) and involves the notion of weak convergence of random walks toBrownian motion.

4.1 Problem. Show that p defined by (4.1) is a metric on C[0, co) and, underp, C[0, co) is a complete, separable metric space.

4.2 Problem. Let W(A) be the collection of finite-dimensional cylinder sets ofthe form (2.1); i.e.,

(2.1)' C= {Co G C[0, GO; (0)(ti ),...,(2)(4))e AI; n> 1, A EAR"),

where, for all i = 1, n, tie [0, co) (respectively, tie [0, t]). Denote by 5(%)the smallest a-field containing (6(A).

Show that = .4(C[0, co)), the Borel a-field generated by the open sets inC[0, co), and that = (.11(C[0, co))) A .4,(C[0, co)), where cp,: C[0, co)C[0, co) is the mapping (cp,a))(s) = co(t A s); 0 < s < o o.

Whenever X is a random variable on a probability space (SI, F, P) withvalues in a measurable space (S, M(S)), i.e., the function X: SI S is . F / .4(S)-measurable, then X induces a probability measure PX-1 on (S, .4(S)) by

(4.2) PX-1(B)= P {wen; X(co)e B}, B e M(S).

An important special case of (4.2) occurs when X = {Xi; 0 < t < co} is acontinuous stochastic process on (Q, .F, P). Such an X can be regarded asa random variable on (n, .F, P) with values in (C[0, oo),M(C[0, co))), andP X-1 is called the law of X. The reader should verify that the law of acontinuous process is determined by its finite-dimensional distributions.

A. Weak Convergence

The following concept is of fundamental importance in probability theory.

4.3 Definition. Let (S, p) be a metric space with Borel a-field .4(S). Let {Pn},;°_1be a sequence of probability measures on (S, M(S)), and let P be anothermeasure on this space. We say that {Pn}n'_1 converges weakly to P and writePn P, if and only if

2.4. The Space C[0, co], Weak Convergence, and Wiener Measure 61

lim f(s)dP(s) = f f(s)dP(s)n,for every bounded, continuous real-valued function f on S.

It follows, in particular, that the weak limit P is a probability measure, andthat it is unique.

4.4 Definition. Let {(S.1, .93, P)}`°_, be a sequence of probability spaces, andon each of them consider a random variable X with values in the metric space(S, p) . Let P) be another probability space, on which a random variableX with values in (S, p) is given. We say that {Xn}'_, converges to X in distri-bution, and write Xn 4 X, if the sequence of measures {PnXn-1 }nc°=, convergesweakly to the measure PX -1.

Equivalently, X 4 X if and only if

lim Ef(Xn) = Ef(X)n-.co

for every bounded, continuous real-valued function f on S, where En and Edenote expectations with respect to Pn and P, respectively.

Recall that if S in Definition 4.4 is Old, then x X if and only if thesequence of characteristic functions pn(u) A E. exp{i(u, Xn)} converges toqi(u) A E exp{i(u, X)}, for every u e 11". This is the so-called Cramer-Wolddevice (Theorem 7.7 in Billingsley (1968)).

The most important example of convergence in distribution is that providedby the central limit theorem. In the Lindeberg-Levy form used here, thetheorem asserts that if { fl}nc"Li is a sequence of independent, identically distri-buted random variables with mean zero and variance 62, then {Sn} defined by

1

`Sn= L 40-v n k =1

converges in distribution to a standard normal random variable. It is this factwhich dictates that a properly normalized sequence of random walks willconverge in distribution to Brownian motion (the invariance principle of Sub-section D).

4.5 Problem. Suppose {Xn}nc°_1 is a sequence of random variables taking valuesin a metric space (S, , pi) and converging in distribution to X. Suppose (S,,p,)is another metric space, and cp: Si - S, is continuous. Show that Yn A cp(Xn)converges in distribution to Y qi(X).

13. Tightness

The following theorem is stated without proof; its special case S = IR is usedto prove the central limit theorem. In the form provided here, a proof can


be found in several sources, for instance Billingsley (1968), pp. 35-40, orParthasarathy (1967), pp. 47-49.

4.6 Definition. Let (S, p) be a metric space and let 1-1 be a family of probabilitymeasures on (S,M(S)). We say that 1-1 is relatively compact if every sequenceof elements of II contains a weakly convergent subsequence. We say thatII is tight if for every E> 0, there exists a compact set K g S such thatP(K) 1 - c, for every P ell.

If {X,,}QE A is a family of random variables, each one defined on a prob-ability space (14, g; Pi) and taking values in S, we say that this family isrelatively compact or tight if the family of induced measures {PX,,-1}E A hasthe appropriate property.

4.7 Theorem (Prohorov (1956)). Let 11 be a family of probability measures ona complete, separable metric space S. This family is relatively compact if andonly if it is tight.

We are interested in the case S = C[0, co). For this case, we shall providea characterization of tightness (Theorem 4.10). To do so, we define for eachw e C[0, co), T > 0, and S > 0 the modulus of continuity on [0, T]:

(4.3) mT(to,S) L4 max lw(s) - w(t)I.Is-115_6c)..s,t<T

4.8 Problem. Show that mT (co, (5) is continuous in co e C[0, co) under themetric p of (4.1), is nondecreasing in S, and limo,tomT(a),(5) = 0 for eachco e C[0, co).

We shall need the following version of the Arzeld-Ascoli theorem.

4.9 Theorem. A set A g C[0, co) has compact closure if and only if the follow-ing two conditions hold:

(4.4) sup Ico(0)1 < co,weit

(4.5) lim sup mT(co, S) = 0 for every T > 0.blo coeA

PROOF. Assume that the closure of A, denoted by A, is compact. Since A iscontained in the union of the open sets

G,, = {w e C[0, co); Ico(0)1 < n}, n = 1, 2, ...

it must be contained in some particular G, and (4.4) follows. For E > 0, letKb = Ico e A ; mT (co, .5) > E }. Each Kb is closed (Problem 4.8) and is containedin A, so each K,, is compact. Problem 4.8 implies n b>0 Kb = 0, so for some(5(E) > 0, we must have K," = 0. This proves (4.5).

2.4. The Space C[0, x], Weak Convergence, and Wiener Measure 63

We now assume (4.4), (4.5) and prove the compactness of A. Since C[0, op)is a metric space, it suffices to prove that every sequence iconbf-i g A hasa convergent subsequence. We fix T > 0 and note that for some (5, > 0, wehave mT(co,(5,) < 1 for each w e A; so for fixed integer m > 1 and t e (0, T]with (rn - 1)(51 < t < //161 A T, we have from (4.5):

,-i1001 < 1(0(0)1 + E ko(k(1) - co((k - 0(51)1 + w(t) - w((m - 061)1

k=1

kw(C)1 + m.

It follows that for each r e Q +, the set of nonnegative rationals, {w,,(r)}Th isbounded. Let Iro, r,, r2,... I be an enumeration of Q +. Then choose {4)1,7_1,a subsequence of {w,,} ;°_, with o$,°)(ro) converging to a limit denoted w(r0).From 14)}'_,, choose a further subsequence {(1)(n1)},7-1 such that co;,1)(r1) con-verges to a limit co(ri). Continue this process, and then let {(7),,}°°_, = 14)1,7_,be the "diagonal sequence." We have ain(r) co(r) for each re Q+

Let us note from (4.5) that for each E > 0, there exists SW > 0 such that16.(s) - a whenever 0 < s, t < T and Is - tI < (5(a). The same in-equality, therefore, holds for w when we impose the additional conditions, t e Q+ . It follows that w is uniformly continuous on [0, T] n 12÷ and so hasan extension to a continuous function, also called w, on [0, T]; furthermore,lw(s) - w(t)I < a whenever 0 < s, t < T and Is - tI < Ma). For n sufficientlylarge, we have that whenever t e [0, T], there is some rk e Q+ with k < n andIt -rk S(a). For sufficiently large M > n, we have Ith,,,(rJ) - co(ri)1 5 a forall j = 0, 1, , n and m > M. Consequently,

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

< 3E, Vm> M,0 <t < T.

We can make this argument for any T > 0, so {(7)1,7_, converges uniformlyon bounded intervals to the function we C[0, co). I=1

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

(4.6) lim sup P,,[(0; > A] = 0,At n> 1

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

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

Let us now assume (4.6) and (4.7). Given a positive integer T and n > 0,We choose > 0 so that

sup P[(o; 10)(0)1 > 1/2'.n >1

We choose (5k > 0, k = 1, 2, ... such that

sup P,,[(2.); mT (co, bk) >1 qi2T+k+1.

r1.1

Define the closed sets

1

AT = {(0,10)(0)1 A, mT (co, Sk) k = 1, 2, ...}, A= n AT,T=1

so Pn(AT) > 1 - E,T=o 17/27--f-k+i = 1 - 17/2T and PP(A) > 1 - n, for every n 1.

By Theorem 4.9, A is compact, so {P},°,p_1 is tight. 1=1

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

(i) sup,i EIXr1" A M < oo,(ii) supm,i El X; m) - X.1m)12 < CT1t s11+13; V T > 0 and 0 < s, t < T

for some positive constants a, /3, v (universal) and CT (depending on T > 0).Show that the probability measures Pm P(X(m))-1; m > 1 induced by these

processes on (C[0, oo), .4(C [0, co))) form a tight sequence.(Hint: Follow the technique of proof in the Kolmogorov-Centsov Theorem2.8, to verify the conditions (4.6), (4.7) of Theorem 4.10).

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

(4.8) lim f f(co)dP,,(w) = f f(w)dP(w).n CIO , ) Cio . 00)

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

C. Convergence of Finite-Dimensional Distributions

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

2.4. The Space C[0, co], Weak Convergence, and Wiener Measure 65

(0.,:y7, Pa)), we can ask whether V") X in the sense of Definition 4.4. Wecan also ask whether the finite-dimensional distributions of {X(")}x_i converge

to those of X, i.e., whether

(X7), , )0")) 4 (x,,, xt2,..., xtd).

The latter question is considerably easier to answer than the former, sincethe convergence in distribution of finite-dimensional random vectors can beresolved by studying characteristic functions.

For any finite subset { t1, , td} of [0, co), let us define the projection mappingtd: C[0, CO) Rd as

= (co(t ), co(td)).

If the function f: Rd IR is bounded and continuous, then the composite map-ping f 0 C[0, co) -.CI enjoys the same properties; thus, X(n) 4: Ximplies

lim En fM7), , lim En(f 0n-,co n

= E(f 0 = Ef(X,,, , Xid).

In other words, if the sequence of processes {X(")}`°_1 converges in distributionto the process X, then all finite-dimensional distributions converge as well.The converse holds in the presence of tightness (Theorem 4.15), but not ingeneral; this failure is illustrated by the following exercise.

4.14 Exercise. Consider the sequence of (nonrandom) processes

1q") = nt 1[0, 1/2n]. ,(t1+ (1 - nt) 1,112n, lin](t);

0 < t < oo, n > 1 and let X, = 0, t > 0. Show that all finite-dimensional distri-butions of X(n) converge weakly to the corresponding finite-dimensional distri-butions of X, but the sequence of processes {X(")}'_, does not converge indistribution to the process X.

4.15 Theorem. Let {X(")}°°_, be a tight sequence of continuous processes withthe property that, whenever 0 < t, < < td < co, then the sequence of randomvectors X1,1,))1°11 converges in distribution. Let Pn be the measureinduced on (C[0, co), Ac[o, co))) by V"). Then {Pn},T=1 converges weakly toa measure P, under which the coordinate mapping process W,(w) w(t) onC[0, co) satisfies

(X:7), , X:d")) 4 (w,,,..., w,), 0 < ti < < td < oo, d 1.

PROOF. Every subsequence {X( "°} of {X(")} is tight, and so has a furthersubsequence {g(")} such that the measures induced on C[0, co) by {g(")}converge weakly to a probability measure P, by the Prohorov theorem 4.7. Ifa different subsequence {X( ")} induces measures on C[0, oo) converging toa probability measure Q, then P and Q must have the same finite-dimensionaldistributions, i.e.,

P [co e C[0, co); (w(t,), ..., w(td))e A] = Q[co e C[0, cc); (co(ti ), . . . , co(td)) E A],

0 < t 1 < t 2 < < td < cc, A e M(l), d > 1.

This means P = Q.Suppose the sequence of measures {Pn};°_, induced by {x(^)}-_, did not

converge weakly to P. Then there must be a bounded, continuous functionf: C[0, co) -> R such that lim .1 f(co)P,,(dco) does not exist, or else this limitexists but is different from f f(co)P(dco). In either case, we can choose asubsequence {Pn},T,...1 for which lim, i f(w)P,,(dco) exists but is different fromf f(w)P(dw). This subsequence can have no further subsequence {P},,'_1 withPn 4.' P, and this violates the conclusion of the previous paragraph. Li

We shall need the following result.

4.16 Problem. Let {X(")},T-1, {Y(n)},f-1, and X be random variables withvalues in a metric space (S, p); we assume that for each n > 1, X(n) and Y(") aredefined on the same probability space. If X(") - X and p(X("), Y(")) --> 0 inprobability, as n - co, then Y(n) 4 x as n --> co.

D. The Invariance Principle and the Wiener Measure

Let us consider now a sequence gilT,i of independent, identically distributedrandom variables with mean zero and variance a2, 0 < o-2 < co, as well asthe sequence of partial sums So = 0, Sk = El=i , k > 1. A continuous-timeprocess Y = { Y; t 0} can be obtained from the sequence {Sk},T=0 by linearinterpolation; i.e.,

(4.9) Y, = ST,1 + (t - TtKp.,, t 0,

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

(4.10) X: n) = 1 Y. t > 0.aJ

2.4. The Space C[0, co], Weal. Convergence, and Wiener Measure 67

Note that with s = k/n and t = (k + 1)/n, the increment Xr -Xr =(1/o-ji)k+i is independent of = 1, , Furthermore, Xr -has zero mean and variance t - s. This suggests that {X:"); t > 01 is approxi-mately a Brownian motion. We now show that, even though the randomvariables are not necessarily normal, the central limit theorem dictates thatthe limiting distributions of the increments of X(n) are normal.

4.17 Theorem. With {VI defined by (4.10) and 0 < t1 < < td < co, wehave

where {B t 0} is a standard, one-dimensional Brownian motion.

PROOF. We take the case d = 2; the other cases differ from this one only bybeing notationally more cumbersome. Set s = t1, t = t2. We wish to show

(X!"), XI")) 4 (Bs, Be).

Since

XI")1

Slim]

we have by the Cebygev inequality,

P[1

X:")o-,/n

1

> < 082n

as n co. It is clear then that

1(Xs"', Xr) S( isn], SitI) -* 0 in probability,

tT

so, by Problem 4.16, it suffices to show

n(Sp1, Sim) --0 (B3, Be).

n

From Problem 4.5 we see that this is equivalent to proving

1 itn]E E -4 (Bs, Be - Bs).

o-F j=1 j=[sn]+1

The independence of the random variables {i}.71_1 'mplies

iu l[sn]lim E [exp E +

iv1[4isn j

iu lim E expn-.00 to-N/n.fri n CO

(4.11) = lim E expiv

+1

provided both limits on the right-hand side exist. We deal with limn_o,E[exp{(iu/6 .1r-t)E;s:}]; the other limit can be treated similarly. Since

Isrq

L, C.10"f j =1 o- ,A[sn] J=1

and, by the central limit theorem, (fs/o- ,A[sn]j)E.Ig converges in distri-bution to a normal random variable with mean zero and variance s, we have

iu

n-vco

lim E [exp 1[511 e -"2s/2o-,/n J=1

-0 in probability,

Similarly,

lim E[exp iv = e-v2"-s)/2.n-oo 6 1;1 j= Fil+1

Substitution of these last two equations into (4.11) completes the proof.

Actually, the sequence {X(")} of linearly interpolated and normalized randomwalks in (4.10) converges to Brownian motion in distribution. For the tightnessrequired to carry out such an extension (recall Theorem 4.15), we shall needtwo auxiliary results.

4.18 Lemma. Set Sk = EI)=1 where { .,}72,1 is a sequence of independent,identically distributed random variables, with mean zero and finite variance62 > 0. Then, for any s > 0,

lim lim 1 P[ max ISJI > = O.610 n-co 0 1 .1.11161 4-1

PRooF. By the central limit theorem, we have for each S > 0 that(1/6En61 + 1 )SEnol+, converges in distribution to a standard normal ran-dom variable Z, whence (1/6 ,/n6)S1n61 +1 4 Z. Fix A > 0 and let {(pk}i.,°:=1 be asequence of bounded, continuous functions on 1 1 1 with p i , j 1(_co, Wehave for each i,

lim P[ISin,q+i AciiTtS] < lirn Ecpkl Slna1 +1J = E(pk(Z).n-.aD n -co NPU5

Let k co to conclude

(4.12) lim P[I,Sin6+11 Ao-,1/41(5] < P[IZI > A] < 7EIZ13, A > 0.

We now define i = min{ j 1; I.SJI > co-,Ft}. With 0 < 6 < E2/2, we have(imitating the proof of the Kolmogorov inequality; e.g., Chung (1974), p. 116):

2.4. The Space C[0, op], Weak Convergence, and Wiener Measure 69

[(4.13) P max ISJ1> Ea Nit;0 5:1<no]+1

P [IS[[no]+11 a Nfl(E - ,./2(5)][[no]

+ 1 P[ISPIO1+11 < a ji(e - ,./26)1T = j] P[T = .n-j=i

But if T = j, then IS[nall-il< ajz(t - .126) implies 1Si -Spop-il> 6 ,./2n6.By the ebygev inequality, the probability of this event is bounded above by

1 1 [ma+

2nbaE[(S; Swil+i)21T = =

2nSo-2 E

Returning to (4.13), we may now write

P[max ISJI > E6j11

0 <j.[[nE5.]+1

< fn(v - 26)] + iP[t < Ifi51]

PCIS[[..3+11 6.,F 1

4E -126)] + P[ max ISJI0 5j5 [[nr3. +1

1 5 j < Pl.

from which follows

P[ max ISJI > win] 5 2P [1 ST(014-11 0\Fi(e -Setting A = (E - ,./Y,(5)/,./3 in (4.12), we see that

> ECI 1li-1 ,

,F5lim

1- P[ max ISJI > to-jd <2

E1Z13,n-m 6 0 5/5[n51+1 (E - N/245)3

and letting 6.1 0 we obtain the desired result.

4.19 Lemma. Under the assumptions of Lemma 4.18, we have for any T > 0,

lim lim P[ max ISf+k - Sk > cajd = 0.6.1.o J5[[no]]-1-1

Tnn+1

PROOF. For 0 2 be the unique integer satisfyingT/m < b < T /(m - 1). Since

[[nT1] + 1 Tlim

+ 1 6[[0]]< m,

We have [[nT1] + 1 < an6]] + 1)m for sufficiently large n. For such a large

n, suppose 1Si +1, - Ski > .11-1 for some k, 0 5 k < TnT1] + 1, and some j,1 < j < [[n.511 + 1. There exists then a unique integer p, 0 5 p m - 1, suchthat

(Tn.51 + 1)p < k < (Tn.51 + 1)(p + 1).

There are two possibilities for k + j. One possibility is that

(Tn.51 + 1)p < k + j (11n61 + 1)(p + 1),

in which case either iSk - Sq,l+opi > ice INAz, or else iSk+; - Sanol+opi >lerrin. The second possibility is that

(Tn.511 + 1)(p + 1) < k + j < (Tn611 + 1)(p + 2),

in which case either iSk - S(Qn,31+1)pl > iEcr \At, IS(Qnol+i)p - S([[no1+1)(p+1)1 >

jeQNIP-1, or else IS rr(0,31+1)(p+1)

But

Sk+ii > icer\Fi. In conclusion, we see that

1max ISJ,k - Ski > errin

1 . j<[[nol+10 5_k 51[171+1

m (1

g- U max I Spi-pqnaii+i) - S panO11+1)1 > -3EGr AIp=0 1 ..j .[[no11+1

[ 1

P max ISi+ pano]] +1) S 1)&61+1)1 > -3E6r NA?1 5..1[[11.511+1

1= p[ max IS.I > -ErrNAti,1 ..1<ft,31+1

_I 3

and thus:

P[max Si+k -Sk I > Err\Ad (m + 1)P[ max S.I > Err\Ail

1 [[Ifoll+1 1 S[[1u31+1_I 3

0 It.[[n711+1

Since m < (TM) + 1, we obtain the desired conclusion from Lemma 4.18.

We are now in a position to establish the main result of this section, namelythe convergence in distribution of the sequence of normalized random walksin (4.10) to Brownian motion. This result is also known as the invarianceprinciple.

4.20 Theorem (The Invariance Principle of Donsker (1951)). Let (n, P) bea probability space on which is given a sequence gitti of independent, identi-cally distributed random variables with mean zero and finite variance a2 > 0.Define X(n) = PO"); t > 0} by (4.10), and let Pn be the measure induced by X(n)on (C[0, co), .4(C[0, co))). Then {Pn}n'=1 converges weakly to a measure P*,

2.5. The Markov Property 71

under which the coordinate mapping process W(co) w(t) on C[0, co) is astandard, one-dimensional Brownian motion.

PROOF. In light of Theorems 4.15 and 4.17, it remains to show that {X(")}x_1is tight. For this we use Theorem 4.10, and since X' = 0 a.s. for every n, weneed only establish, for arbitrary s > 0 and T > 0, the convergence

(4.14) lim sup P[ max PC.!") - XP) I > El = 0.64.0 n> 1 Is-tl<3

0 <s,t < T

We may replace supn >1 in this expression by lim, since for a finite numberof integers n we can make the probability appearing in (4.14) as small as wechoose, by reducing 6. But

P[ max

0 <s,i< T

and

1X1n) - x:")I > 61= fi[ max I Ys - > so- ,A11,rib

0 <s,t <nT

max I Ys - Y,I < max YS - Ygi < max ISi+k - Ski,Is-ti <ntS]j+1 1 <j S1n511+1

0 <s,t <nT 0 <ni1+1 0 <n11+1

where the last inequality follows from the fact that Y is piecewise linear andchanges slope only at integer values oft. Now (4.14) follows from Lemma 4.19.

4.21 Definition. The probability measure on (C[0, co), ,I(C[0, co))), underwhich the coordinate mapping process No)) w(t), 0 < t < co, is a standard,one-dimensional Brownian motion, is called Wiener measure.

4.22 Remark. A standard, one-dimensional Brownian motion defined on anyprobability space can be thought of as a random variable with values inC[0, co); regarded this way, Brownian motion induces the Wiener measureon (C[0, co), ,I(C[0, co))). For this reason, we call (C[0, co), R(C[0, co)), Ps),where pi, is Wiener measure, the canonical probability space for Brownianmotion.

2.5. The Markov Property

In this section we define the notion of a d-dimensional Markov processand cite d-dimensional Brownian motion as an example. There are severalequivalent statements of the Markov property, and we spend some timedeveloping them.

A. Brownian Motion in Several Dimensions

5.1 Definition. Let d be a positive integer and it a probability measure on(Rd, .4(01d)) Let B = {B F; t _. 0} be a continuous, adapted process withvalues in Rd, defined on some probability space (SI, ..571, P). This process is calleda d-dimensional Brownian motion with initial distribution t, if

(i) P[B0 e 11 = it(F), V F e ,4(Rd);(ii) for 0 < s < t, the increment A - B3 is independent of g' and is normally

distributed with mean zero and covariance matrix equal to (t - s)Id, whereId is the (d x d) identity matrix.

If it assigns measure one to some singleton {x}, we say that B is a d-dimensional Brownian motion starting at x.

Here is one way to construct a d-dimensional Brownian motion withinitial distribution t. Let X(coo) = coo be the identity random variable on(01d9

(old), it), and for each i = 1, ... , d, let /II° = lir, AB"); t > 01 be astandard, one-dimensional Brownian motion on some (0"), .9""), P")). On theproduct space

(Rd nu) not) ,4(01d) ®y(1) ® ... ® F(d), x p(1 x ... x p(d),

define

Bt(w) A X(coo) + (R1)(coi )9 ... 9 Rd)(cod))9

and set .9; = g-,B. Then B = {B1, Ft; t > 0} is the desired object.There is a second construction of d-dimensional Brownian motion with

initial distribution t, a construction which motivates the concept of Markovfamily, to be introduced in this section. Let P"), i = 1, ... , d be d copies ofWiener measure on (C[0, co), .(C[0, co))). Then P° A P(1) x x P(d) is ameasure, called d-dimensional Wiener measure, on (C[0, oo)d, .4(C [0, 03)d)).Under P°, the coordinate mapping process Bt(co) A co(t) together with thefiltration WI is a d-dimensional Brownian motion starting at the origin. Forx e Rd, we define the probability measure Px on (C[0, oo)d,M(C[0, co)d)) by

(5.1) Px(F) = P°(F - x), F e .4(C[0, oo)a),

where F -x = {we C[0, co)d; u)() + x e F}. Under Px, B A {B ,,B; t > 0}is a d-dimensional Brownian motion starting at x. Finally, for a probabilitymeasure it on (Rd, 2(Rd)), we define P" on a(c[o, co)d) by

(5.2) P4(F) = J Px(F)u(dx).;la

Problem 5.2 shows that such a definition is possible.

5.2 Problem. Show that for each F e g(C[0, co)d), the mapping x i- Px(F) is1(Rd)/.4([0, 1])-measurable. (Hint: Use the Dynkin System Theorem 1.3.)

5.3 Proposition. The coordinate mapping process B = IB gor,,B; t > 01 on(C[0, 09)d, .4(C[0, cor), P") is a d-dimensional Brownian motion with initialdistribution u.

5.4 Problem. Give a careful proof of Proposition 5.3 and the assertionspreceding Problem 5.2.

5.5 Problem. Let {B, = gr.,; 0 < t < oc,} be a d-dimensionalBrownian motion. Show that the processes

MP) A BP) - Bg), .9;; 0 t < oo, 1 , = 654.; 1 <j < d. Furthermore, the vector of martingales M = (M(1), , M(d)) is inde-pendent of .yo.

5.6 Definition. Given a metric space (S, p), we denote by AS)" the completionof the Borel o-field .4(S) (generated by the open sets) with respect to thefinite measure ji on (S, .4(S)). The universal o--field is ail(S) A 11,,,R(S)P, wherethe intersection is over all finite measures (or, equivalently, all probabilitymeasures) u. A ail(S)/.4(R)-measurable, real-valued function is said to beuniversally measurable.

5.7 Problem. Let (S, p) be a metric space and let f be a real-valued functiondefined on S. Show that f is universally measurable if and only if for everyfinite measure g on (S, a(S)), there is a Borel-measurable function g: Ssuch that pfx ES; f(x) g (x)} = 0.

5.8 Definition. A d-dimensional Brownian family is an adapted, d-dimensionalprocess B = {B t > 0} on a measurable space (S/,.F), and a family ofprobability measures {Px Ix c Re, such that

(i) for each F e the mapping Px(F) is universally measurable;(ii) for each x e Rd, /3' [Bo = x] = 1;(iii) under each Px, the process B is a d-dimensional Brownian motion

starting at x.

We have already seen how to construct a family of probability measures{Px} on the canonical space (C[0, oor,.4(C[0, °o)")) so that the coordinatemapping process, relative to the filtration it generates, is a Brownian motionstarting at x under any Px. With = .4(C[0, cor), Problem 5.2 showsthat the universal measurability requirement (i) of Definition 5.8 is satisfied.Indeed, for this canonical example of a d-dimensional Brownian family, themapping xi- Px(F) is actually Borel-measurable for each F e ,F. The reasonwe formulate Definition 5.8 with the weaker measurability condition is toallow expansion of .F to a larger o--field (see Remark 7.16).

B. Markov Processes and Markov Families

Let us suppose now that we observe a Brownian motion with initial distributiony up to time s, 0 < s < t. In particular, we see the value of Bs, which we call y.Conditioned on these observations, what is the probability that B, is insome set f e MO:id)? Now B, = (B, - Bs) + Bs, and the increment B, -Bs isindependent of the observations up to time s and is distributed just as B,_,is under P°. On the other hand, Bs does depend on the observations; indeed,we are conditioning on B, = y. It follows that the sum (B, - Bs) + Bs isdistributed as B,_, is under P. Two points then become clear. First, knowledgeof the whole past up to time s provides no more useful information about B,than knowing the value of Bs; in other words,

(5.3) P"[B, e F I.Fs] = P"[B, e Fl Bs], 0 < s < t, F e a(Rd).

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

(5.4) P"[B,e rIB, = y] = PY [B,_, e fa o s < t, re.4(01d).

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

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

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

in the notation of (4.2).

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

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

P"[X,e ri,Fs] = PP[X,e rixj, P"-a.s.

Our experience with Brownian motion indicates that it is notationally andconceptually helpful to have a whole family of probability measures, ratherthan just one. Toward this end, we define the concept of a Markov family.

5.11 Definition. Let d be a positive integer. A d-dimensional Markov familyis an adapted process X = {X Ft; t > 0} on some (S2,,), together with afamily of probability measures {Px}x, Re on (SI, F), such that

(a) for each F e .9 ", the mapping x i-3 Px(F) is universally measurable;(b) Px[X0 = x] = I, V x e Rd;


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

Px e F .Fs] = Px [XH, e r I xs], Px-a.s;

(d) for x e Rd, s, t 0 and F e *Rd),

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


The following statement is a consequence of Problem 5.9 and the discussionpreceding it.

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

C. Equivalent Formulations of the Markov Property

The Markov property, encapsulated by conditions (c) and (d) of Definition 5.11,can be reformulated in several equivalent ways. Some of these formulationsamount to incorporating (c) and (d) into a single condition; others replacethe evaluation of X at the single time s + t by its evaluation at multiple timesafter s. The bulk of this subsection presents those formulations of the Markovproperty which we shall find most convenient in the sequel.

Given an adapted process X = {X t 0} and a family of probabilitymeasures {Px gld on (S/, g;), such that condition (a) of Definition 5.11 issatisfied, we can define a collection of operators { Ut}t, 0 which map bounded,Borel-measurable, real-valued functions on R' into bounded, universallymeasurable, real-valued functions on the same space. These are given by

(5.7) (Utf )(x) A Ex f (X

In the case where f is the indicator of F E M(Rd), we have Exf(X,) = Px [X, e F],and the universal measurability of U, f follows directly from Definition 5.11 (a);for an arbitrary, Borel-measurable function f, the universal measurability ofEU is then a consequence of the bounded convergence theorem.

5.13 Proposition. Conditions (c) and (d) of Definition 5.11 can be replaced by:

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

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

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

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

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


x e Rd, s 0: Px[Xt+serixs] = a(Xs), a.s. Px, and from condition (c):Px[X,e I" L. s] = a(Xs), a.s. Px, which would then establish (e).

However, we only know that Utl r( ) is universally measurable. This means(from Problem 5.7) that, for given s, t > 0, x e Rd, there exists a Borel-measurablefunction g: Rd --, [0, 1] such that

(5.8)

whence

(5.9)

(u, 1r) (y) = g(y), for PxXs 1-a.e. y e Rd,

(Utir)(Xs) = g(Xs), a.s. Px.

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

(5.9) gives

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

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

Px[X,,se rix, = y] = g(y) for PxX;-1-a.e. ye Rd,

and this in turn yields (d), thanks to (5.8).

this

5.14 Remark on Notation. For given w e SI, we denote by X.(w) the functiont 1- X,(w). Thus, X. is a measurable mapping from 0,39 into ((Rd)1°'),ge(oRdyo,c.),)( the space of all Rd-valued functions on [0, co) equipped withthe smallest a-field containing all finite-dimensional cylinder sets.

5.15 Proposition. For a Markov family X, (II, .x), {Px}xe [Iv, we have:

(c') For x e Rd, s> 0 and F e A(Rdy°,03)),

Px [Xs+. e Figs] = Px[Xs+. e FI Xs], Px-a.s;

(d') For x e Rd, s > 0 and F e.4((Rd)1° "))),

Px[Xs+.e FIX, = y] = PY[X.e F], PxXs-l-a.e. y.

(Note: If f E .4(Rd) and F = {w e(Rd)1°''); w(t)ET }, for fixed t 0, then (c')and (d') reduce to (c) and (d), respectively, of Definition 5.11.)

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

F = {0.) e(Old)(°'x'); w(to) e ro, . ., co(tn_1)er,,_1, w(tn)er},

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


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

= P[Xsero, ..., Xs+,._,ern_i, Xs+,. 6 rnIxs], P-a.s.

We prove this statement by induction on n. For n = 0, it is obvious. Assumeit true for n - 1. A consequence of this assumption is that for any bounded,Borel-measurable cp: IRd" -, 01,

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

Now (c) implies that

(5.13) P[Xsero, ..., Xs+tn_ie rn_i, Xs+,. e rn I gs]

= Ex[1{x.e 1-0,...,xs*,._1 er_,}Px[Xs+t e rnigs+tn_jigs]

= Ex [1(x. e ro,...,x.._, er_,}Px[Xs+t,, 6 TnIxs+t,,_,] Igs].

Any gx.+, -measurable random variable can be written as a Borel-measurablefunction of Xs+,._i (Chung (1974), p. 299), and so there exists a Borel-measurablefunction g: Rd -- [0, 1], such that Px[Xs+,. e 1-.1Xs+/_, ] = g(Xs+in_i), a.s. P.Setting cp(x0, ... , x_1) g'-- 100... 1,-;_i(x_,)g(xn_i), we can use (5.12) toreplace gs by Q(Xs) in (5.13) and then, reversing the previous steps, to obtain(5.11). The proof of (d') is similar, although notationally more complex.

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

(5.14) Xs,(co) = X, (Cisco); V co e SI, s, t > 0.

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

(5.15) (Osco)(t) = co(s + t), t > 0.



When the shift operators exist, then the function Xs+.(w) of Remark 5.14is none other than X.(0sw), so {X.eF} = BS {X. e F}. As F ranges over" {X. a F} ranges over 'cx. Thus, (c') and (d') can be reformulatedas follows: for every F and s > 0,

(c")

(d")

Px[tV1FI.Fs] = Px[tV1FIXs],

Px[0;1 FIX, = = PY [F], PXXs I a.e. y.

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

(e") Px[VFIF] = Px.(F), Px-a.s.

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

5.16 Theorem. Let X = {X t > 0} be an adapted process on a measurablespace (52,,), let it-Px xe Rd be a family of probability measures on (SI, .F), andlet {0,},0 be a family of .97/,-measurable shift-operators satisfying (5.14).Then X, (1/,.F), {Px }Re is a Markov family if and only if (a), (b), and (e") hold.

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

px[G osiFixs] px[Gixipx[os-iFixj, px_a.s.

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

We close this section with additional examples of Markov families.

5.18 Problem. Suppose X = {X t > 0} is a Markov process on (CI, P)and gyp: [0, co) Rd and T: [0, co) L(l V,OV), the space of linear transforma-tions from Rd to Rd, are given (nonrandom) functions with (p(0) = 0 and T(t)nonsingular for every t > 0. Set Y, = (p(t) + T(t)X Then Y = Yi, t 0}

is also a Markov process.

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

lx}xRd is a

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


2.6. The Strong Markov Property and the Reflection Principle 79

5.20 Definition. A Poisson family with intensity A > 0 is a process N = IN ,Ft;t > 01 on a measurable space (Si, .F) and a family of probability measures{ px}xER, such that

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

with intensity A.

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

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

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

2.6. The Strong Markov Property and theReflection Principle

Part of the appeal of Brownian motion lies in the fact that the distribution ofcertain of its functionals can be obtained in closed form. Perhaps the mostfundamental of these functionals is the passage time 'I', to a level b e H, definedby

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

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

We shall first obtain the probability density function of Tb by a heuristicargument, based on the so-called reflection principle of Desire Andre (Levy(1948), p. 293). A rigorous presentation of this argument requires use ofthe strong Markov property for Brownian motion. Accordingly, after somemotivational discussion, we define the concept of a strong Markov family andprove that any Brownian family is strongly Markovian. This will allow usto place the heuristic argument on firm mathematical ground.

A. The Reflection Principle

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

p0 [Tb < t] pOr-L 1 b < t, B, > b] + P°[Tb < t, B, < b].

Now P° [Tb < t, A > b] = P° [B, > b]. On the other hand, if Tb < t andB, < b, then sometime before time t the Brownian path reached level b, andthen in the remaining time it traveled from b to a point c less than b. Becauseof the symmetry with respect to b of a Brownian motion starting at b, the"probability" of doing this is the same as the "probability" of traveling fromb to the point 2b - c. The heuristic rationale here is that, for every pathwhich crosses level b and is found at time t at a point below b, there is a"shadow path" (see figure) obtained from reflection about the level b whichexceeds this level at time t, and these two paths have the same "probability."Of course, the actual probability for the occurrence of any particular path iszero, so this argument is only heuristic; even if the probability in question werepositive, it would not be entirely obvious how to derive the type of "symmetry"claimed here from the definition of Brownian motion. Nevertheless, thisargument leads us to the correct equation

P ° [ Tb < t, B, < b]

2b -

= P°[Tb < t, B, > b] = P°[B, > IA,

which then yields

(6.2) P° [Tb < t] = 2P° [B, > b] =7t1bt -1/2

0,

e-x212 dx.

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

(6.3) P°[Tbe d[] =,J2irt3

e-b2/2`dt; t > 0.. ,/27rt3

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

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

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

B. Strong Markov Processes and Families

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

(i) Pu[Xoe 1] =µ(r), V I" e .4(01°);(ii) for any optional time S of {Ft}, t 0 and I" eR(Rd),

r[Xs+, erls+] = PP[xs÷terixs], P" -a.s. on {S < co}.

6.3 Definition. Let d be a positive integer. A d-dimensional strong Markovfamily is a progressively measurable process X = {X ,F,; t > 0} on some(C/, F), together with a family of probability measure {Px}xeRa on (SZ, g"),such that:

(a) for each F e.F, the mapping x 1-* Px(F) is universally measurable;(b) Px[Xo = x] = 1, V x elle;(c) for x e Rd, t > 0, r eM(Rd), and any optional time S of {Ft},

Px[Xs+teri,s+] = Px[xs+ieriXs], Px-a.s. on {S < co};

(d) for x e Rd, t > 0, F e .4(Rd), and any optional time S of {A},

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

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

PPLXs+terls+i = PilXs+teriXs] = 0, P" -a.s. on {S = co},

and so the restriction S < co in these conditions is unnecessary.

6.5 Remark. An optional time of I.F,1 is a stopping time of {,t+} (Corollary1.2.4). Because of the assumption of progressive measurability, the randomvariable Xs appearing in Definitions 6.2 and 6.3 is A,-measurable (Pro-position 1.2.18). Moreover, if S is a stopping time of {F,}, then Xs is Fs-measurable. In this case, we can take conditional expectations with respect toA on both sides of (c) in Definition 6.3, to obtain

Px[Xs+i E FIFO = Px[Xs+IEFIXA, Px-a.s. on IS <

Setting S equal to a constant s > 0, we obtain condition (c) of Definition 5.11.Thus, every strong Markov family is a Markov family. Likewise, every strongMarkov process is a Markov process. However, not every Markov familyenjoys the strong Markov property; a counterexample to this effect, involvinga progressively measurable process X, appears in Wentzell (1981), p. 161.

Whenever S is an optional time of IA and u > 0, then S + u is a stoppingtime of (Problem 1.2.10). This fact can be used to replace the constant sin the proof of Proposition 5.15 by the optional time S, thereby obtainingthe following result.

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

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

Px[Xs+.E FIAJ = Px[Xs+.E PIXs], Px-a.s. on {S < oo 1;

(d') for x E Rd, F E a((Rdr,-)), and any optional time S of {A},

Px[xs+. E FIRS = y] = PY [X. E F], PxXS-1-a.e. y.

Using the operators IU,1, in (5.7), conditions (c) and (d) of Definition 6.3can be combined.

6.7 Proposition. Let X = {X ,E; t > 0} be a progressively measurable processon (52,,9), and let {Px Re be a family of probability measures satisfying (a)and (b) of Definition 6.3. Then X, (52, 3T), kPx, 1 xERd is strong Markov if andonly if for any {3 }- optional time t > 0, and x e Rd, one of the followingequivalent conditions holds:

(e) for any r E a(OBd),

Px[X,EFIAJ = r)(Xs), Px-a.s. on {S < col;

(e') for any bounded, continuous f: Rd ->

Ex[f(Xs+f)IFs+] = (U1.1)(Xs), Px-a.s. on {S <

PROOF. The proof that (e) is equivalent to (c) and (d) is the same as the proofof the analogous equivalence for Markov families given in Proposition 5.13.Since any bounded, continuous real-valued function on Rd is the pointwise

limit of a bounded sequence of linear combinations of indicators of Borel sets,(e') follows from (e) and the bounded convergence theorem. On the other hand,if (e') holds and F g_ Rd is closed, then 1r is the pointwise limit of {f} °11,where f(x) = [1 - np(x,F)] v 0 and p(x, n = - y E Fl. Each f isbounded and continuous, so (e) holds for closed sets F. The collection of setsF e .4(Ra) for which (e) holds forms a Dynkin system, so, by Theorem 1.3, (e)holds for all F e M(Rd).

6.8 Remark. If X = { X ; t > 0}, (R.F), is a strong Markovfamily and tt is a probability measure on (Rd, .4(Rd)), we can define a proba-bility measure P" by (5.2) for every F e.517, and then X on (SI, , P") is a strongMarkov process with initial distribution it. Condition (ii) of Definition 6.2 canbe verified upon writing condition (e) in integrated form:

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

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

It is often convenient to work with bounded optional times only. Thefollowing problem shows that stating the strong Markov property in termsof such optional times entails no loss of generality. We shall use this fact inour proof that Brownian families are strongly Markovian.

6.9 Problem. Let S be an optional time of the filtration {Ft} on some (C/,..F, P).

(i) Show that if Z1 and Z2 are integrable random variables and Z1 = Z2on some ,975.4.-measurable set A, then

E[Z,1,53-5+] = E[Z21F.j, a.s. on A.

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

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

(Hint: Use Problem 1.2.17(i)).(iii) Show that if (e) (or (e')) in Proposition 6.7 holds for every bounded

optional time S of then it holds for every optional time.

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

= 0, on {S = s}.

In other words, 0 is defined so that whenever S(w) < co, then

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

In particular, we have {Xs+. e E} = {X. e E}, and (c') and (d') are, respec-tively, equivalent to the statements: for every x e Rd, Fegicx, and any optionaltime S of {A},

(c")

(d")

px[os-iFIAA Px-a.s. on {S < col;

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

Both (c") and (d") can be captured by the single condition:

(e") for x e Rd, F e 'co', and any optional time S of {A},px[vFigFs+] = pxs,-kr,) Px-a.s. on {S < co}.

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

6.10 Theorem. Let X = IX ,Ft; t > 01 be a progressively measurable processon (SI, F), let {Px}xE Rd be a family of probability measures on (e, F), and let{0,}s>o be a family of F/g-measurable shift operators satisfying (5.14). ThenX , (Si, {Px} x E Rd is a strong Markov family if and only if (a), (b), and (e") hold.

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

(e"') For all x e gl, any bounded, gc,X-measurable random variable Y, andany optional time S of {gt}, we have

Ex[Y 0 Osigs+] = Exs(Y), Px-a.s. on {S < oo}.

(Note: If we write this equation with the arguments filled in, it becomes

Ex [ Y 0 OsIgs+] (w) = Y(w')Pxs(-)(')(dw'), Px-a.e. w e {S <

where (Y Os)(w") g Y10s(.-)(w")).)

C. The Strong Markov Property for Brownian Motion

The discussion on the strong Markov property for Brownian motion willrequire some background material on regular conditional probabilities.

6.12 Definition. Let X be a random variable on a probability space (e,F, P)taking values in a complete, separable metric space (5, M(S)). Let be asub-a-field of F. A regular conditional probability of X given is a functionQ: SZ x a(S) -> [0, 1] such that

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


Under the conditions of Definition 6.12 on X, (S2, .y, P), (S, AS)), and , aregular conditional probability for X given W exists (Ash (1972), pp. 264-265,or Parthasarathy (1967), pp. 146-150). One consequence of this fact is thatthe conditional characteristic function of a random vector can be used todetermine its conditional distribution, in the manner outlined by the nextlemma.

6.13 Lemma. Let X be a d-dimensional random vector on (CI, 37, P). Supposeis a sub-a-field of F and suppose that for each w e SI, there is a function

p(w; -): Rd C such that for each u e Rd,

cp(w; u) = E[ei(".x)ifl(w), P-a.e. w.

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

(p(p; u) = f el("P'(dx),Rd

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

P[X el- l5](w) = P'(F), P-a.e. w.

PROOF. Let Q be a regular conditional probability for X given , so for eachfixed u a Rd we can build up from indicators to show that

(6.4) cp(w; u) = E[ei("m151(w) = I ei(u.x)Q(a); dx), P-a.e. w.Rd

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

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

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

Suppose B = IB A; t > 0 }, (52, 37), {Px}Egid is a d-dimensional Brownianfamily. Choose u a Rd and define the complex-valued process

M, A exp [i(u, B,) +2-1102 ], t > 0.

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


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

are martingales on 4,, Pl.

PROOF. For 0 < s < t, we have

PEA '1,1.9';] = Ex[Ms exp (i(u, A -A tS) + II u112) F1

-= Ms Ex [exp (i(u, A - A) + t S 11U112)] = Ms,

where we have used the independence of B, -Bs and Fs, as well as (6.5). Takingreal and imaginary parts, we obtain the martingale property for IR .; t ..._. 01and II A; t > 01. 0

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

PROOF. We verify that a Brownian family B = {B ..Ft; t .. 0 }, (SI, ..F), NP{ 1x ,.ER.satisfies condition (e) of Proposition 6.7. Thus, let S be an optional time of{F,}. In light of Problem 6.9, we may assume that S is bounded. Fix x e Rd.The optional sampling theorem (Theorem 1.3.22 and Problem 1.3.23 (i))applied to the martingales of Lemma 6.14 yields, for Px-a.e. w e a

Ex [exp(i(u, Bs+,))1..Fs,](w) = exp [i(u, Bs,,s (w)) -2

II ul12]) .

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

0We can carry this line of argument a bit further to obtain a related result.

6.16 Theorem. Let S be an a.s. finite optional time of the filtration {. } for thed-dimensional Brownian motion B = {B A; t > 0}. Then with Wt -'4- Bs+t - Bs,the process W = {147i, Fri; t > 0} is a d-dimensional Brownian motion, indepen-dent of Fs+.

PROOF. We show that for every n 1, 0 < to < < t < oo, and ui , ... ,u e Rd, we have a.s. P:

(6.6) E [exp (i kEi (uk, "tk - w,,,_,)) Fs, = fl exp]k=1

-1

t-Ali 4-1)0402 ;

thus, according to Lemma 6.13 and (6.5), not only are the increments{W,,, - W,k_, },",,_1 independent normal random vectors with mean zero andcovariance matrices (tk - 4_04, but they are also independent of the a-fieldFs+ This substantiates the claim of the theorem.

We prove (6.6) for bounded, optional times S of {,F, }; the argument givenin Solution 6.9 can be used to extend this result to a.s. finite S. Assume (6.6)holds for some n, and choose 0 < to < < t,, < t,.,. Applying the equalityin the proof of Theorem 6.15 to the optional time S + t, we have

(6.7) ElexPri(uni-i, Wt., Wt)71:F(s+i)+1

= Elexp[i(t4d+1, Bs+t,,)71,-*;-(s+to+1 exp [ -i(u+,, Bs+,)]

exp[-i(t.+1 - t.)11/1+1112], P-a.s.

Therefore,

n+1

E [exp (i E (uk, w,k - wtk_i))k=1

-Fs+

= E [exp Ci E (uk, wt, - Wtk_i ))k=1

E lexp(i(ud.+1, W,*, - Wt))1,(s+,)+ 1 -Fs+

= exp [ -2 (tn+ i - tn) II 140-1 1122

.E exp i E (4,, wtk - Wtk_ 1)

k=1

n+1 1

= ri exp - -2

(tk - tk_ 1 ) 11 ukil 2 , P-a.s.,k=1

which completes the induction step. The proof that (6.6) holds for n = 1 isobtained by setting t = 0 in (6.7).

In order to present a rigorous derivation of the density (6.3) for the passagetime Tb in (6.1), a slight extension of the strong Markov property for right-continuous processes will be needed.

6.17 Proposition. Let X = {X t > 0 }, {Px}x. Rd be a strong Markovfamily, and the process X be right-continuous. Let S be an optional time of {,}and T an °T's., -measurable random time satisfying T(w) S(w) for all wen.Then, for any xe Rd and any bounded, continuous f : Rd R,

(6.8) Ex f(XT) I-Fs+ ha)) = (Ur(.)-s(a)(Xs(A(0)), Px-a.e. co e IT <

PROOF. For n > 1, let

S + -1 (P"(T - S)1 + 1), if T < oo,= 2"

oo, if T = co,

so that Tn = S + k2-" when (k - 1)2-" < T -S < k2 -". We have Tn1 T on{T < co}. From (e') we have for k > 0,

Ex [f (s+k2-.)1.975+] = (Uk/2.f )(Xs), Px-a.s. on {S < co},

and Problem 6.9 (i) then implies

Ex [ f (X T)1.-Cr's+]((o) = (UT,,(.)-smf)(Xs(o((0)), Px-a.e. co e {T < co}.

The bounded convergence theorem for conditional expectations and the right-continuity of X imply that the left-hand side converges to Ex[f(Xr)l,s+] (w)as n -, co. Since (U, f )(y) = EV (X,) is right-continuous in t for every y e Rd,the right-hand side converges to (UT,,o_smf)(Xs(,)(0)).

6.18 Corollary. Under the conditions of Proposition 6.17, (6.8) holds for everybounded, .4([18°)/M(P0-measurable function f. In particular, for any F e .4(0V)we have for Px-a.e. w e{T < co):

r [XT E F I Fs+ ] (0) = (UT(.)-s(w11r)(Xs(w)(0)).

PROOF. Approximate the indicator of a closed set F by bounded, continuousfunctions as in the proof of Proposition 6.7. Then prove the result for anyF e R(W), and extend to bounded, Borel-measurable functions.

6.19 Proposition. Let {B .97,; t > 0} be a standard, one-dimensional Brownianmotion, and for b 0, let Tb be the first passage time to b as in (6.1). Then Tbhas the density given by (6.3).

PROOF. Because { -B ,57;,; t > 0} is also a standard, one-dimensional Brownianmotion, it suffices to consider the case b > 0. In Corollary 6.18 set S = Tb,

T =

ft if S < t,oo if S > t,

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

1

P °[Tb < t, B, < b] = f P°[137- e I- 1 Fs+] dP" = -2 P° [Tt, < t].(rb<i}

It follows that

P°['Ti, < t] = P° [Tb < t, Et, > b] + P°[7j, < t, B, < b]

= P° [B, > b] + I P° [T,, < t],

and (6.2) is proved.

6.20 Remark. It follows from (6.2), by letting t -, co, that the passage timesare almost surely finite: P° [Tb < co] = 1.

2.7. Brownian Filtrations 89

6.21 Problem. Recall the notions of Poisson family and Poisson process fromDefinitions 5.20 and 1.3.3, respectively. Show that the former is a strongMarkov family, and the latter is a strong Markov process.

2.7. Brownian Filtrations

In Section 1 we made a point of defining Brownian motion B = {B A; t > 0}with a filtration {Ft} which is possibly larger than {,-,8} and anticipatedsome of the reasons that mandate this generality. One reason is related tothe fact that, although the filtration {..Fel} is left-continuous, it fails to beright-continuous (Problem 7.1). Some of the developments in later chaptersrequire either right or two-sided continuity of the filtration {A}, and so inthis section we construct filtrations with these properties.

Let us recall the basic definitions from Section 1.1. For a filtration {9;; t > 0}on the measurable space (0.., ,), we set A+ = nE>0 A,, for t > 0, ._ =a(U,, A) for t > 0, moo_ = "0, and .94-x, = 6(Uto-5-4-t). We say that {A} isright- (respectively, left-) continuous if Ft+ = . (respectively, ._ = .Ft) holdsfor every 0 < t < co. When X = {X..F,x;t > 0} is a process on (SI, ..F), thenleft-continuity of {,x} at some fixed t > 0 can be interpreted to mean thatX, can be discovered by observing X,, 0 < s < t. Right-continuity meansintuitively that if X, has been observed for 0 < s < t, then nothing more canbe learned by peeking infinitesimally far into the future. We recall here that.Ftx = o-(Xs; 0 < s < t).

7.1 Problem. Let {X ..Fx; 0 < t < co} be a d-dimensional process.

(i) Show that the filtration {Fix., } is right-continuous.(ii) Show that if X is left-continuous, then the filtration {.97x} is left-

continuous.(iii) Show by example that, even if X is continuous, {Fix} can fail to be

right-continuous and 19,, I can fail to be left-continuous.

We shall need to develop the important notions of completion and augmen-tation of .7-fields, in the context of a process X = {X Fix; 0 < t < co} withinitial distribution p on the space (0, ',I PI, where Pu[X0 en = p(F);re M(Rd). We start by setting, for 0 t co,

Xf A {F g 0; 3 G eg-x with F g G, P"(G) = o}.

'V will be called "the collection of fm-null sets" and denoted simply by .Aco.

7.2 Definition. For any 0 < t < co, we define

(i) the completion:." ''-- o-(F,x u XII, and(ii) the augmentation:. " A o-(,x u XI

of the a-field ,97,x under P. For t = oo the two concepts agree, and we setsimply FP -4 o-GFccx u

The augmented filtration 1.3-711 possesses certain desirable properties, whichwill be used frequently in the sequel and are developed in the ensuing problemsand propositions.

7.3 Problem. For any sub-a-field / of define /1' = u .A(g) and

= {F 3GE1 such that FAGE.Aig }.

Show that /" = Yt. We now extend Pg by defining Pg(F) A Pg(G) wheneverFE1", and GE1 is chosen to satisfy FAGE.Aig. Show that the probabilityspace (Q, Pg) is complete:

F El", Pg(F)= 0, D g_ F DElg.

7.4 Problem. From Definition 7.2 we have ..F,P Fp, for every 0 < t < co.Show by example that the inclusion can be strict.

7.5 Problem. Show that the a-field FP of Definition 7.2 agrees with

g;f0 A a (U g;g)-t>o

7.6 Problem. If the process X has left-continuous paths, then the filtration1,11 is left-continuous.

A. Right-Continuity of the Augmented Filtrationfor a Strong Markov Process

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

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

PROOF. Let (SI, Pg) be the probability space on which X is defined.Fix s > 0 and consider the degenerate, {,F,,x}-optional time S = s. With0 < to < t, < < tn < s < t,,,, < < t. and ro, F, in *Rd), thestrong Markov property gives

[X,0 E ro, Xt,,,E F, fix],

= 1{Xt0Ero, Pg [X tn Ern, G rdxs],

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

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

Let us take now FEX _c .91; we have P[F137,x+] = IF, a.s. I", so IFhas an 37sx-measurable version which we denote by Y. Because G A { Y = 1} e.f and F AG g_ {1, Y} we have Feg: and consequently ..sf,g:; s > 0.

Now let us suppose that FE. for every integer n > 1 we haveFes1,11,,,), as well as a set Gne.Fx,- ,-(1/n) such that F A G EXP. We defineG A n:=1U c°=m G, and since G= n := m U ,T= m G,, for any positive integer M,we have GeFs'f,_ ,°"Fr. To prove that Fe,f, it suffices to show F A GeXP.Now

03 00

G\F g. (U G)\F = U (G\F) e X"-n =1 n=1

On the other hand,

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

co r co CO CO

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

It follows that F e <Fs'', so <F:, g_ .Fs and right-continuity is proved.

7.8 Corollary. For a d-dimensional, left-continuous strong Markov processX = {X ; t > 0} with initial distribution p, the augmented filtration 1.Fflis continuous.

7.9 Theorem. Let B = {B .F,B; t > 0} be a d-dimensional Brownian motionwith initial distribution p on (f2, .9";c0B, Pl. Relative to the filtration {,F,"},{B t > 0} is still a d-dimensional Brownian motion.

PROOF. Augmentation of a-fields does not disturb the assumptions of Definition5.1.

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

Since any d-dimensional Brownian motion is strongly Markov (Theorem6.15), the augmentation of the filtration in Theorem 7.9 does not affect thestrong Markov property. This raises the following general question. Suppose{X ,,x; t > 0} is a d-dimensional, strong Markov process with initial distri-

bution u on (SI, Flo`, P4). Is the process {X Ft"; t > 0} also strongly Markov?In other words, is it true, for every optional time S of {Fil}, t > 0 andF e R(W), that

(7.1) 1"[XS+t E r 1 -91+ ] = 1"[Xs+t E r 1 Xs], PP-a.s. on {S < co }?

Although the answer to this question is affirmative, phrased in this generalitythe question is not as important as it might appear. In each particular case,some technique must be used to prove that {X ,97,x; t > 0} is strongly Markovin the first place, and this technique can usually be employed to establish thestrong Markov property for {X F,P; t > 0) as well. Theorems 7.9 and 6.15exemplify this kind of argument for d-dimensional Brownian motion. None-theless, the interested reader can work through the following series of exercisesto verify that (7.1) is valid in the generality claimed.

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

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

7.12 Exercise. Suppose that T is an optional time of {Fix }. For fixed positiveinteger n, define

T. =

2 "'

on IT = col

on {ic2" 2"- 1 < T < k}

-

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

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

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

then (7.1) holds for every optional time S of {Fr }.

This completes our discussion of the augmentation of the filtration generatedby a strong Markov process. At first glance, augmentation appears to be arather artificial device, but in retrospect it can be seen to be more useful andnatural than merely completing each a -field Fix with respect to PTM. It is morenatural because it involves only one collection of PP-null sets, the collectionwe called ,A14, rather than a separate collection for each t > O. It is more usefulbecause completing each a-field Fix need not result in a right-continuousfiltration, as the next problem demonstrates.

7.14 Problem. Let {Bt; t > 01 be the coordinate mapping process on (C [0, 044(C[0, oo))), P° be Wiener measure, and Fi denote the completion ofunder P°. Consider the set

F = lw e C[0, co); w is constant on [0, s] for some e > 01.

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

B. A "Universal" Filtration

The difficulty with the filtration Ign, obtained for a strong Markov processwith initial distribution ft, is its dependence on it. In particular, such a filtrationis inappropriate for a strong Markov family, where there is a continuum ofinitial conditions. We now construct a filtration which is well suited for thiscase.

Let {X 0}5 (S2, 2,1), f DXJxeOld be a d-dimensional, strong Markov

family. For each probability measure on (Rd, ARd)), we define PP as in (5.2):

PP(F)= f Px(F)p(dx), V F e ,

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

(7.3) A A n FtP, 0 < t < 00,

where the intersection is over all probability measures it on (Rd, M(Rd)). Notethat Fix 0 < t < CO for any probability measure jt on (Rd, gi(Rd));therefore, if {X t > 0} and {X t z 0} are both strongly Markovianunder Po, then so is {X gt; t > 0}. Because the order of intersection isinterchangeable and {Ft"} is right-continuous, we have

= n n =n n gp= n =s>t p s>t

Thus } is also right-continuous.

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

PROOF. It is easily verifed that, under each Px,{B ,; t 0} is a d-dimensionalBrownian motion starting at x. It remains only to establish the universalmeasurability condition (i) of Definition 5.8. Fix F e Ao. For each probabilitymeasure it on (Rd,M(Rd)), we have F e..F", so there is some G e.: F! withF AG eAr". Let N e satisfy FAG N and Pv(N)= 0. The functionsg(x) g PX(G) and n(x) g PX(N) are universally measurable by assumption.Furthermore,

94

n(x)ii(dx) = r(N)= 0,

2. Brownian Motion

so n = 0, The nonnegative functions hi (x) Px(F\G) and h2(x)Px(G\F) are dominated by n, so h, and h2 are zero ft-a.e., and hence h, andh2 are measurable with respect to .4(Rd)", the completion of AR") under /.1.Set f(x) Px(F). We have f(x) = g(x) + hi(x) - h2(x), so f is also .4(Rd)"-measurable. This is true for every it; thus, f is universally measurable.

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

C. The Blumenthal Zero-One Law

We close this section with a useful consequence of the results concerningaugmentation.

7.17 Theorem (Blumenthal (1957) Zero-One Law). Let IB A; t ,),{Px })C Rd be a d-dimensional Brownian family, where is given by (7.3). IfF e go, then for each x e Rd we have either Px(F) = 0 or Px(F) = 1.

PROOF. For F ego and each x e Rd, there exists G E .a)7011 such that Px(F A G) = 0.But G must have the form G = {Boer} for some T eM(Rd), so

Px(F) = r(G) = Px {Bo e F} = 1,-(x).

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

7.19 Problem. Let Wt, .Ft; 0 < t < co} be a standard, one-dimensionalBrownian motion on (S1, P), and define

Sb = inf It 0; Wt > bl; b O.

(i) Show that for each b > 0, P['T,, 0 Sb] = 0.(ii) Show that if L is a finite, nonnegative random variable on (SI, P)

which is independent of Fm, then {TL 0 SL} E.9 and P[7i, SL] = 0.

2.8. Computations Based on Passage Times

In order to motivate the strong Markov property in Section 2.6, we derivedthe density for the first passage time of a one-dimensional Brownian motionfrom the origin to b 0 0. In this section we obtain a number of distributions

2.8. Computations Based on Passage Times 95

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

A. Brownian Motion and Its Running Maximum

Throughout this section, { W .Ft; 0 < t < (S/, 37;), {Px}xcgg will be a one-dimensional Brownian family. We recall from (6.1) the passage times

Tb = inf {t 0; W, = b}; b e

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

(8.1) M, = max Ws.

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

2(2b - a) { (2b - 2)

(8.2) P°[W, e da, M, e db] = exp - da db.2irt3 2t

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

(U,,1(_,1)(b) A Pb[W,, a] = 2b - a]

A (Ut-s1[2b-a,0)(b); 0 < s < t.

Corollary 6.18 then yields

P°[Wt < al<Frb+] = (Ut- Tb1(-co,a1)(b) (Ut- Tb1[26-a,c0)(b)

= P° [Wt 2b - a.s. P° on {T6 < t }.

Integrating both sides of this equation over {Tb < t} and noting that{Tb 5 t} = {M, > b}, we obtain

P° [ W, a, M, b] = P°[W, 2b - a, b]

= P° [W,1

26 - a] = e-x2/2t dx.2rct 26-a

Differentiation leads to (8.2).

8.2 Problem. Show that for t > 0,

(8.3) P° [Mt e db] = P° [IWt 1 e db] = P° [Mt -W, e db]

2e-b212' db; b > 0.

/2nt

8.3 Remark. From (8.3) we see that

(8.4) P° t] = P° [Mt > 1)] =2 n'

Cx2/2 dx; b > 0./2,7c J

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

(8.5) P°[Tbe dt] = e- 1'212' dt; b > 0, t > 0.b,

2nt3

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

(8.6) Eoe-arb e-b,f2; b > 0, a > 0.

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

8.4 Exercise. Derive (8.6) (and consequently (8.5)) by applying the optionalsampling theorem to the {. }-martingale

(8.7) X, = exp{.1W, - IA2t}; 0 t < co,

with A = > 0.

The following simple proposition will be extremely helpful in our study oflocal time in Section 6.2.

8.5 Proposition. The process of passage times T = {T, .97,-.4.; 0 < a < co} hasthe property that, under P° and for 0 < a < b, the increment Tb -T, is inde-pendent of FT.., and has the density

P° [Tb - 'Tae dt] =b -a

, e-(6-0212t dt; 0 < t < co./ 2nt3

In particular,

(8.8) Eo[e-ct(Tb-T.)1,T.+] e-(b-a) .12a; 0.

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

B. Brownian Motion on a Half-Line

When Brownian motion is constrained to have state space [0, oo), one mustspecify what happens when the origin is reached. The following problemsexplore the simplest cases of absorption and (instantaneous) reflection.

8.6 Problem. Derive the transition density for Brownian motion absorbed atthe origin {W, To, ,Ft; 0 < t < 00 }, by verifying that

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

-41- [p(t; x, y) - p(t; x, -y)] dy; t > 0, x, y > O.

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

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

A [p(s; x, y) + p(s; x, - y)] dy; s > 0, t _.. 0 and x, y _. O.

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

(8.11) P° [Y,+, e dy I Y, = x] = p +(s; x, y) dy; s > 0, t > 0 and x, y .. 0.

Conclude that under P° the processes IWI and Y have the same finite-dimensional distributions.

The surprising equivalence in law of the processes Y and I WI was observedby P. Levy (1948), who employed it in his deep study of Brownian local time(cf. Chapter 6). The third process M appearing in (8.3) cannot be equivalentin law to Y and I WI, since the paths of M are nondecreasing, whereas thoseof Y and I WI are not. Nonetheless, M will turn out to be of considerableinterest in Section 6.2, where we develop a number of deep properties ofBrownian local time, using M as the object of study.

C. Brownian Motion on a Finite Interval

In this subsection we consider Brownian motion with state space [0, a], wherea is positive and finite. In order to study the case of reflection at bothendpoints, consider the function co: IR -+ [0, a] which satisfies co(2na) = 0,cp((2n + 1)a) = a; n = 0, ± 1, ±2, ..., and is linear between these points.

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

X

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

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

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

(8.12) Px[W, a dy, To A > t] = j p_(t; x, y + 2na) dy.n=-3c

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

Px[T - To t] =

We have then

P°[T2n° < t].

(8.13) lim Px[T < t] = 0; 0 < t < GO.n-oo

For any y e (0, co), we have from Corollary 6.18 and the symmetry ofBrownian motion that

Px[W, = Px[W, -yl",.+] on {T < t },

and so for any integer n > 0,

(8.14) 13' [W, y, T t] = Px[W, - y, T. t] = Px[W, -y, an t].

Similarly, for y - 09, a), we have

Px[W, = Px[W, > 2a -y1,97,+] on {an 5_ t },

whence

(8.15) Px[W, y, an t] = Px[W, 2a - y, an 5_ t]

= Px[W, > 2a - y, tn_i < t]; n > 1.

We may apply (8.14) and (8.15) alternately and repeatedly to conclude, for0 < y < a, n > 0:

Px CW1 to = Px[W, < -y - 2na],

Px[Wi )7, an = Px[Wr y- 2na],and by differentiating with respect to y, we see that

(8.16) Px[Hie dy, to < t] = p(t; x, -y - 2na)dy,

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

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

nn = inf {t > p,,_1; 14/, = 0}, p = inf { t > re.; W = a}; n = 1, 2, ....We may proceed as previously to obtain the formulas

(8.18) lim Px[p t] = 0; 0 < t <11-.09

(8.19) Px[W, e dy, p t] = p(t; x, -y + (2n + 1)a) dy; 0 < y < a, n 0,

(8.20) Px [ W, e dy, nn < t] = p(t; x, y + 2na) dy; 0 < y < a, n > 0.

It is easily verified by considering the cases To < Ta and To > T. thatr_1 V p_, = cr. A n and on v 7r,, = to A p,,; n 1. Consequently,

Px[Wte dy, r_1 ^ Pn-i < t] = 13' [W, e dy, -c_, < t] + Px[W, e dy, Pn-i < t]

(8.21) - Px[W, e dy, on A nn < t],and

(8.22) Px[Wtedy, a A 7c. t] = Px[W, e dy, a t] + Px[Wte dy, 7t t]

- Px [W te dy, A p t].

Successive application of (8.21) and (8.22) yields for every integer k 1:

(8.23)

Px[141,e dy, To A Po < t] = E IPx[Wte dy, t_1 t] + Px[Wte dY, Pa-i < t]n=1

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

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

Now we let k tend to infinity in (8.23); because of (8.13), (8.18) the last termconverges to zero, whereas using (8.16), (8.17) and (8.19), (8.20), we obtain fromthe remaining terms:

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

= E p_(t; x, y + 2na)dy; 0 < y < a, t > O.

8.11 Exercise. Show that for t > 0, 0 < x < a:

1(8.24) Px[ To A e dt] = ,1 E + x) exp

(2na

\/2/rt3 n=

x)2

(2na + a - x)2 1 ]+ (2na + a - x) ex p jl 2t

dt.

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

(8.25)1 (2na + x)2

Px [To E dt, To < To] = (2na + x) exp 1 dt,\/27rt3 n=-co 2t

(8.26)

e dt, 7; < To] = E (2na + a x) exp(2na + a - x)2

dt.1 e°

2nt3 2t

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

sinh((a - x) 2a)

sinh(a)sinh(x,/2a)

sinh(a.,./244).

We leave the verification of these identities as a problem. Note that by adding(8.27) and (8.28) we obtain the transform of (8.24):

cosh ((x - 2) 2a))(8.29) Ex[e-'('°^2.°)] =

cosh (a- f2a)

(8.27)

(8.28)

Ex [e'T°1(ro<r)]

Ex [e'Tal{Ta <ro}]

2

This provides an independent verification of (8.24).

8.12 Problem. Derive the formulas (8.27), (8.28) by applying the optionalsampling theorem to the martingale of (8.7).

8.13 Exercise. Show that for a > 0, 0 < x < a:

<7;;]a -x =x

a

8.14 Problem. Show that Ex(To A T.) = x(a - x); 0 < x < a.

D. Distributions Involving Last Exit Times

Proposition 8.1 coupled with the Markov property enables one to computedistributions for a wide variety of Brownian functionals. We illustrate themethod by computing some joint distributions involving the last time beforet that the reflected Brownian motion Y of Problem 8.8 is at the origin. Notethat such last exit times are not stopping times.

8.15 Proposition. Define

(8.30) 0, A sup{0 < s 5 t; Ws = Mt}.

Then for a e IR, b a +, 0 < s < t, we have:

(8.31)

e[W,eda, Mte db, 0,e ds] -b(b - a)

expb2 (b -a )2

db ds.

n ,/ s3 (t - s)3 2s 2(t - s)

PROOF. For b > 0, s > 6 > 0, x > 0, a < b and 0 < s < t, we have

(8.32) P°Lb

< Ms < b + 6, Ws e b - dx, max W. < b, 147,e da].,.c,< P°[b < M, < b + 6, 0, < s, Ws e b - dx, W, e da]

< P°[b < Ms < b + 6, Ws e b - dx, max W < b + E, W, edd.s5 s,

Divide by 6 and let 6 i 0, s i 0 (in that order). The upper and lower bounds inthe preceding inequalities converge to the same limit, which is

(8.33) P°[Mte db, 0, < s, Ws e b - dx, 141,e da]

= P°[Mse db, Ws e b - dx, max W. < b, W, e da]Ss.,,,

= P° [Ms e db, Ws e b - dx] Pb-x[M,_, b, 141,_se da]

b + x= _ [expt (x + p+)2 (2b - a)21

lc / s3 (t - s) 2a2 2t f

- exp { (x + /24]2 a2}dx da db,

2o:2 2f

where we have used (8.2) and

b(t - s) + (a - b)sa

n, S(t - s)-.L , =t t

In terms of 1(z) A (1/,/27r)Sz_co e-x212 dx we may now evaluate the integrals

..1:(b + x)exp (x ± 211-1-)} dx = a2e-P2t/2°2

2a2

+ -s (b + (b - a))a.12fr - 40 (-111-),t a

and so integrating out x in (8.33) and using the equality

(8.34)p2+

+ +(b + (b - a))2

=.b2 (b - a)2

2a2 2t 2s 2(t - s)'

we arrive at the formula

2 -P° [M, e db, s,

2iit 3E da] - [0( 1)(26 a) exp

(2b

o- 2t

a)21.

- (- l'±=)2

a exp {- cf -}1 da dbo-

Note that a / as( - 1.4/a) = 112o-((b/s) ± (b - a)/(t - s)), and so

-a p( e db, 0, s, W,E da]as

b(b - a) { b2 (b - a)2}exp da db ds.

Tr,/ s3(t - s)3 2s 2(t - s)

8.16 Remark. If we define A inf {0 < s < t; WS = AI to be the first time Wattains its maximum over [0, t], then (8.32) is still valid when A is replaced byA. Thus, et and A have the same distribution, and since Of < e we see thatP°[A = A] = 1. In other words, the time at which the maximum over [0, t]is attained is almost surely unique.

8.17 Problem. Show that for b > 0, 0 < s < t:

[M, E db, 0,e ds] = b e'212s db ds,its3(t - s)

whence

P° [et e ds] = , P°[M,E dbl =b 2

= 12s dbds

.

s(t - s)

In particular, the conditional density of M, given A does not depend on t.We say that 0, obeys the arc-sine law, since

2P°[0, < s] - arcsin -; 0 < s < t, t > 0.

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

(8.35) y, sup{0 s t; W = 0}.

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

dsP°[y, e ds] - ; 0 < s < t.

/ s(t - s)

(Hint: Use Problem 8.8.)

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

2.9. The Brownian Sample Paths 103

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

2-2ab ub2

(2nu(s - u)(t - s))312exp - a

2u(s - u) 2(t-2da db ds du;

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

2.9. The Brownian Sample Paths

We present in this section a detailed discussion of the basic absolute propertiesof Brownian motion, i.e., those properties which hold with probability one(also called sample path properties). These include characterizations of "bad"behavior (nondifferentiability and lack of points of increase) as well as "good"behavior (law of the iterated logarithm and Levy modulus of continuity) of theBrownian paths. We also study the local maxima and the zero sets of thesepaths. We shall see in Section 3.4 that the sample paths of any continuousmartingale can be obtained by running those of a Brownian motion accordingto a different, path-dependent clock. Thus, this study of Brownian motion hasmuch to say about the sample path properties of much more general classesof processes, including continuous martingales and diffusions.

A. Elementary Properties

We start by collecting together, in Lemma 9.4, the fundamental equivalencetransformations of Brownian motion. These will prove handy, both in thissection and throughout the book; indeed, we made frequent use of symmetryin the previous section.

9.1 Definition. An Rd-valued stochastic process X = {X,; 0 < t < co} is calledGaussian if, for any integer k > 1 and real numbers 0 < t, < t2 < < t, < co,the random vector (X,,, X,,, , X,,) has a joint normal distribution. If thedistribution of (X,,,,, , X,,,k) does not depend on t, we say that theprocess is stationary.

The finite-dimensional distributions of a Gaussian process X are determinedby its expectation vector m(t) EX,; t > 0, and its covariance matrix

p(s, t) A E[(X, - m(s))(X, - m(t))T]; s, t 0,

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

9.2 Remark. One-dimensional Brownian motion is a zero-mean Gaussianprocess with covariance function

(9.1) p(s,t) = s t; s, t 0.

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

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

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

(9.2) lim -Wt = 0, a.s.t-,a) t

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

9.4 Lemma. When W = {W, 0 < t < co} is a standard Brownian motion,so are the processes obtained from the following "equivalence transformations":

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

(9.3) =1

Wc, 0 < t <

(ii) Time-inversion: Y = { ..F,1; 0 < t < co } defined by

(9.4)ftWil, 0 < t < oo,

= 10 t = O.

(iii) Time-reversal: Z = {Z ..57",,z; 0 < t < co} defined for T > 0 by

(9.5) Z, = WT - WT_t 0 < I < 'T.

(iv) Symmetry: -W = -W gi; 0 < t < col.

PROOF. We shall discuss only part (ii), the others being either similar orcompletely evident. The process Y of (9.4) is easily seen to have a.s. continuouspaths; continuity at the origin is a corollary of Problem 9.3. On the other hand,Y is a zero-mean Gaussian process with covariance function

1E(Ys Y,) = st (-s A

1t) = S A t; S, t > 0

and the conclusion follows from Remark 9.2.

9.5 Problem. Show that the probability that Brownian motion returns to theorigin infinitely often is one.

B. The Zero Set and the Quadratic Variation

We take up now the study of the zero set of the Brownian path. Define

(9.6) 2's (t, OE [0, CO x SI; W,(w) = 0 1,

and for fixed we f2, define the zero set of W.(w):

(9.7) Y, s {0 < t < co; W(w) = 0 }.

9.6 Theorem. For P-a.e. wen, the zero set

(i) has Lebesgue measure zero,(ii) is closed and unbounded,(iii) has an accumulation point at t = 0,(iv) has no isolated point in (0, co), and therefore(v) is dense in itself.

PROOF. We start by observing that the set .2° of (9.6) is in ,4([0, co)) 0 .F,because W is a (progressively) measurable process. By Fubini's theorem,

E[meas(.2°,)] = (meas x P)(t) = f P[W, = 0]dt = 0,

and therefore meas(Y,)) = 0 for P-a.e. w e f2, proving (i); here and in thesequel, meas means "Lebesgue measure." On the other hand, for P-a.e. w e f2the mapping t W(w) is continuous, and s, is the inverse image underthis mapping of the closed set {0 }. Thus, for every such w, the sets, isclosed, is unbounded (Problem 9.5), and has an accumulation point at t = 0(Problem 7.18).

For (iv), let us observe that {w en.; 2°,,, has an isolated point in (0, co)} canbe written as

(9.8) Ua,beQ

0 <a<b<co

fa.) e SI; there is exactly one s e (a, b) with Ws(w) = 01

where Q is the set of rationals. Let us consider the family of almost surelyfinite optional times

/3, inf{s > t; Ws = 0}; t > O.

According to (iii) we have fio = 0, a.s. P; moreover,

= inf{s > A((4); Ws(w) = 0}

= /33(w) + inf {s > 0; Ws+13,(0)(w) - = = /3,(w)

for P-a.e. w ea because { Ws+p, - 0 < s < co} is a standard Brownianmotion (Theorem 6.16). Therefore, for 0 < a b}

has probability zero, and the same is then true for the union (9.8).

9.7 Remark. From Theorem 9.6 and the strong Markov property in the formof Theorem 6.16, we see that for every fixed b e R and P-a.e. w e C2, the level set

.2",,,(b) {0 t < oo; Wi(w) = b}

is closed, unbounded, of Lebesgue measure zero, and dense in itself.

The following problem strengthens the result of Theorem 1.5.8 in the specialcase of Brownian motion.

9.8 Problem. Let {11},T=1 be a sequence of partitions of the interval [0, t] withlimn, 111-1n11 = 0. Then the quadratic variations

v,(2)(1-1.) A I Wks - WC,12k=1

of the Brownian motion W over these partitions converge to t in L2, as n cc.If, furthermore, the partitions become so fine that E`°_.1 1111,11 < co holds, thepreceding convergence takes place also with probability one.

C. Local Maxima and Points of Increase

As discussed in Section 1.5, one can easily show by using Problem 9.8 that foralmost every we Si, the sample path W.(w) is of unbounded variation on everyfinite interval [0, t]. In the remainder of this section we describe just howoscillatory the Brownian path is.

9.9 Theorem. For almost every we O., the sample path W.(w) is monotone in nointerval.

PROOF. If we denote by F the set of w ell with the property that W(w) ismonotone in some interval, we have

F = U {(0e0; W.(w) is monotone on [s,t] },e Q

0 < s<t<co

since every nonempty interval includes one with rational endpoints. There-fore, it suffices to show that on any such interval, say on [0, 1], the path W(w)is monotone for almost no w. By virtue of the symmetry property (iv) ofLemma 9.4, it suffices then to show that the event

A -4 {w e W.(w) is nondecreasing on [0, l] }

is in and has probability zero. But A = nr-in-1

A, where

An n {a) ea - wi,n(w) 0} ei=o

has probability P (A ) = P[W0_01,, - 0] = 2 -". Thus, P(A) <P(A) = 0.

In order to proceed with our study of the Brownian sample paths, we needa few elementary notions and results concerning real-valued functions of onevariable.

9.10 Definition. Let f: [0, co) -Fl be a given function. A number t > 0 iscalled

(i) a point of increase of size 6, if for given .5 > 0 we have f(s) < f(t) < f(u)for every s e [(t - S) +, t) and u E (t, t + 6]; a point of strict increase of size& if the preceding inequalities are strict;

(ii) a point of increase, if it is a point of increase of size 6 for some S > 0; apoint of strict increase, if it is a point of strict increase of size 6 for some6 > 0;

(iii) a point of local maximum, if there exists a number 6 > 0 with f(s) < f(t)valid for every s e[(t - t + (5]; and a point of strict local maxi-mum, if there exists a number 6 > 0 with f(s) < f(t) valid for everyS E - (5)+ t + oh{t}

9.11 Problem. Let f: [0, co) ER be continuous.

(i) Show that the set of points of strict local maximum for f is countable.(ii) If f is monotone on no interval, then the set of points of local maximum

for f is dense in [0, co).

9.12 Theorem. For almost every wen, the set of points of local maximum forthe Brownian path W(w) is countable and dense in [0, co), and all local maximaare strict.

PROOF. Thanks to Theorem 9.9 and Problem 9.11, it suffices to show that theset

A = {w e SI; every local maximum of W.(w) is strict}

includes an event of probability one. Indeed, A includes the (countable) inter-section of events of the type

(9.9) A,,,...,,4 A e max 14/,(w) - max Wt(w) 0 0 },13(5t4 ittst2

taken over all quadruples (t, , t2, t3, t4) of rational numbers satisfying 0 < t1 <t2 < t3 < t,< co. Therefore, it remains to prove that for every such quadruple,the event in (9.9) has probability one. But the difference of the two randomvariables in (9.9) can be written as

(Wt, - + min [147t2(w) - W(w)] + max [W(w) - Wt,(u))],titst2 t35..tst4

and the three terms appearing in this sum are independent. Consequently,

P[A,, ta] f° P[N 1,1i- 0 x + y] P[ min (147,2 - e dx]Jo -00

P[ max (W, - W,3)edy] = 1

because P[W,, - Wi3 x + y] = 1. 0Let us now discuss the question of occurrence of points of increase on the

Brownian path. We start by observing that the set

A = {(t,00)e [0, co) x f2; t is a point of increase of W(co)}

is product-measurable: A e.4([0, oo)) ®F. Indeed, A can be written as thecountable union A = Um =1 A(m), with

A(m) {(t, co)e [0, co) x LI; max Ws(w) = WW(w) = min Ws(co)},0-(1/m))-st tsst+0./m)

and each A(m) is in B([0, co)) 0 F. We denote the sections of A by

A, -A {co e i2; (t, co)e A}, A. g {t e [0, co); (t, co)e Al,

and A,(m), A.(m) have a similar meaning. For 0 < t < co,

P[At(m)] < P[Ws+t -H7t. 0; V s e [0, l/m]] = 0

because {W +, - s > 0} is a standard Brownian motion (Problem 7.18);now A, = Um =1 A,(m) gives also

(9.10)

as well as

P(A,) = 0; 0 < t < co

meas(A.) dP = (meas x P)(A) = f P(At) dt = 0

from Fubini's theorem. It follows that P [co e meas(A.) = 0] = 1. The ques-tion is whether this assertion can be strengthened to P [co e = 0] = 1,or equivalently

(9.11) P[co e i2; the path W(co) has no point of increase] = 1.

That the answer to this question turns out to be affirmative is perhaps one ofthe most surprising aspects of Brownian sample path behavior. We state thisresult here but defer the proof to Chapter 6.

9.13 Theorem (Dvoretzky, Erdos, & Kakutani (1961)). Almost everyBrownian sample path has no point of increase (or decrease); that is, (9.11)holds.

9.14 Remark. We have already seen that almost every Brownian path has adense set of local maxima. If T(w) is a local maximum for W.(co), then one

might imagine that by reflection (replacing W,(w) - WT(,,,)(w) by -(W,(w) -w,(,,,)(w)) for t > T(w)), one could turn the point T(w) into a point of increasefor a new Brownian motion. Such an approach was used successfully at thebeginning of Section 6 to derive the passage time distribution. Here, however,it fails completely. Of course, the results of Section 6 are inappropriate in thiscontext because T(co) is not a stopping time. Even if the filtration {.:97,} is right-continuous, so that {w e SI; W(w) has a local maximum at t} is in for eacht > 0, it is not possible to define a stopping time T for {F,} such that W.(0))has a local maximum at T(w) for all w in some event of positive probability.In other words, one cannot specify in a "proper way" which of the numeroustimes of local maximum is to be selected. Indeed, if it were possible to do this,Theorem 9.13 would be violated.

9.15 Remark. It is quite possible that, for each fixed t > 0, a certain propertyholds almost surely, but then it fails to hold for all t > 0 simultaneouslyon an event whose probability is one (or even positive!). As an extreme andrather trivial example, consider that P[co e W,(w) # 1] = 1 holds for every0 < t < co , but P [a) W,(co) # 1, for every t c [0, co)] = 0. The point hereis that in order to pass from the consideration of fixed but arbitrary t to theconsideration of all t simultaneously, it is usually necessary to reduce the latterconsideration to that of a countable number of coordinates. This is preciselythe problem which must be overcome in the passage from (9.10) to (9.11),and the proof of Theorem 9.13 in Dvoretzky, Erdos & Kakutani (1961) isdemandingt because of the difficulty of reducing the property of "being a pointof increase" for all t > 0 to a description involving only countably many co-ordinates. We choose to give a completely different proof of Theorem 9.13 inSubsection 6.4.B, based on the concept of local time. We do, however, illustratethe technique mentioned previously by taking up a less demanding question,the nondifferentiability of the Brownian path.

D. Nowhere Differentiability

9.16 Definition. For a continuous function f: [0, co) R, we denote by

f(t + h) - f(t)h

(9.12) f(t) = lim+

the upper (right and left) Dini derivates at t, and by

f(t + h) - f(t)h

(9.13) D +f(t) = limh-40+

the lower (right and left) Dini derivates at t. The function f is said to be

See, however, Adelman (1985) for a simpler argument.

differentiable at t from the right (respectively, the left), if D+ f(t) and D f(t)(respectively, D- f(t) and D_ f(t)) are finite numbers and equal. The functionf is said to be differentiable at t > 0 if it is differentiable from both the rightand the left and the four Dini derivates agree. At t = 0, differentiability isdefined as differentiability from the right.

9.17 Exercise. Show that for fixed t e [0, co),

(9.14) P[co eft D+ No)) = co and D÷W,(w)= -cc] = 1.

9.18 Theorem (Paley, Wiener & Zygmund (1933)). For almost every w E Si,the Brownian sample path W,(w) is nowhere differentiable. More precisely, theset

(9.15) {wet; for each t e [0, co), either D+14(,(w) = co or D,147,(w) =

contains an event F e F with P(F) = 1.

9.19 Remark. At every point t of local maximum for W.(w) we haveWg(o)) < 0, and at every point s of local minimum, D,W5(w) > 0. Thus, the

"or" in (9.15) cannot be replaced by "and." We do not know whether the setof (9.15) belongs to "cow.

PROOF. It is enough to consider the interval [0, 1]. For fixed integers j > 1,k > 1, we define the set

(9.16) Aik = U n {wen; iw,,,,(w) - wt(coito.0,11 he[0.11k1

Certainly we have

{we); -co < D+147,(w) 147,(w) < co, for some t e [0, 1]} = U U1=1 k=1

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

Let us fix a sample path co e Aik, i.e., suppose there exists a numbert e [0,1] with 1147,(w) - 14/,(w)1 < jh for every 0 < h < 1/k. Take an integern > 4k. Then there exists an integer i, 1 < i < n, such that (i - 1)/n < t <i/n, and it is easily verified that we also have ((i + v)/n)) - t < (v + 1) /n1/k, for v = 1, 2, 3. It follows that

I W1 +11/n(w) - Wi/n(a))1 W, +11 /l(w) - Wt(0)1 + Wirn(w) - W(w)1

2j j 3j< - -n n n

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

2.9. The Brownian Sample Paths

[(i + 2)/n, (i + 3)/n]. Indeed,

3j 2jWo+2)/n(0) - W(i+l)in(W)1 WII+1K+1)In

n n n= 15 ,

1W(i+3)In(0) - WO-F 2)1n(a* WtI +IWO+ 2)In Hi d <41:1- = .n n n

111

Therefore, with

Cr A3 2V ± 1

(.) GO; I W(i+v)In(W) W(i+v-1)/n(0)1 j}=1 n

we have observed that AJk g_ U7-,1 On) holds for every n > 4k. But now

j1(Wy+v)In W(t+v-1)1n) Zv; V = 1, 2, 3

are independent, standard normal random variables, and one can easily verifythe bound P[IZI < a] < a. It develops that

(9.17) P(Cr)05/3

) <n3/2

; = 1, 2, ..., n.

We have AJk C upon taking

(9.18) C An

n u on) G.F,n=4k i=1

and (9.17) shows us that P(C) < inf> 4k P(U7=1 Cin)) = 0. 0

9.20 Remark. An alternative approach to Theorem 9.18, based on local time,is indicated in Exercise 3.6.6.

9.21 Exercise. By modifying the preceding proof, establish the followingstronger result: for almost every cu e S2, the Brownian path 14/.(u)) is nowhereHolder-continuous with exponent y > I. (Hint: By analogy with (9.16), considerthe sets

(9.19)AJk A {w e Lr2; I Wt,(w) - W(w)1 < jhY for some t e [0,1] and all h e [0,1 /1c]l

and show that each is included in a P-null event.)

E. Law of the Iterated Logarithm

Our next result is the celebrated law of the iterated logarithm, which describesthe oscillations of Brownian motion near t = 0 and as t o o. In preparationfor the theorem, we recall the following upper and lower bounds on the tailof the normal distribution.

9.22 Problem. For every x > 0, we have

(9.20)x

1 + xj.co

1-x2/2 < e -"2 < -e-x212.

x x

9.23 Theorem. (Law of the Iterated Logarithm (A. Hin6in (1933))). For almostevery w e O., we have

Wt((0)(i) lim = 1,.%/

Wi(w)

2t log log(1 /t)

(iii) lim = 1,t-. 2t log log t

(ii) lim Wt(w)

.,./2t log log(l/t)

(iv) lira Wi(w) = 1.t-. .\/2t log log t

1,

9.24 Remark. By symmetry, property (ii) follows from (i), and by time-inversion,properties (iii) and (iv) follow from (i) and (ii), respectively (cf. Lemma 9.4).Thus it suffices to establish (i).

PROOF. The submartingale inequality (Theorem 1.3.8 (i)) applied to the ex-ponential martingale {X 5c; 0 t < oo} of (8.7) gives for A > 0,11 > 0:

A(9.21) P max W

s- -s > / 3 = P max Xs > elfl < e-lfl.

0.,5..t 2 ri._.5.i

With the notation h(t)- 72t log log(l/t) and fixed numbers 0, .5 in (0, 1), wechoose A --= (1 + .5)0-"h(0"), 13 = lh(0"), and t = 0" in (9.21), which becomes

1P[ max (Ws - > <(n log(1/0))' ;

n > 1.o<s<en

The last expression is the general term of a convergent series; by the Borel-Cantelli lemma, there exists an event Ste,, e of probability one and aninteger-valued random variable No, so that for every w e 0 we have

max [Ws(w)1 + s0-"h(0")1< 1-

2h(0"); n 110,(o))

o<s<0. 2

Thus, for every t e V]:

W(w) < max Ws(co) < (1 + -2) h(19") (1 + -2) O-112h(t).ct <sson

Therefore,

sup < (1 ± -6) n> Noo(w)+1On <t On h(t) 2

holds for every w e Sleo, and letting n T oo we obtain

-lim "ih(d° < (1 +2-6)0-112,

t)a.s. P.

By letting 610, 0 i 1 through the rationals, we deduce

W(9.22) lim - < 1.' a.s. P.

tt.o h(t) -

In order to obtain an inequality in the opposite direction, we have to employthe second half of the Borel-Cantelli lemma, which relies on independence.We introduce the independent events

A = { Won - We., .11 -0 h(0")}; n= 1, 2, ... ,

again for fixed 0 < 0 < 1. Inequality (9.20) with x = .,./2 log n + 2 log log(1/0)provides lower bounds on the probabilities of these events:

P(iln)= P[Wen -We -1 > xi> e- x212

>const.

- n >1

,./0" - On+1 .,./27c(x + 1/x) n log 0

Now the last expression is the general term of a divergent series, and the secondhalf of the Borel-Cantelli lemma (Chung (1974), p. 76, or Ash (1972), p. 272)guarantees the existence of an event n,e.F with Pod = 1 such that, for everyco e 00 and k 1, there exists an integer m = m(k, co) ._ k with

(9.23) 14/0-((o) - Wo-,,(0) -11 - 011(0m).

On the other hand, (9.22) applied to the Brownian motion -W shows thatthere exist an event SI* e g" of probability one and an integer-valued randomvariable N*, so that for every w e SP

(9.24) - W,..i(w) < ._ 401/2h(0"); n .. N*(w).

From (9.23) and (9.24) we conclude that, for every w e n, n C/* and everyinteger k > 1, there exists an integer m = m(k, w) > k v N *(w) such that

WO-(w) ,./1 -0 - 4j9.h(r)By letting m -4 oo, we conclude that lima° (Wt/h(t)) _. . ,/1 -0 -140 holdsa.s. P, and letting 010 through the rationals we obtain

Wlim -- > 1; a.s. P.,lo h(t)

D

We observed in Remark 2.12 that almost every Brownian sample path islocally Holder-continuous with exponent y for every y e(0, I), and we also sawin Exercise 9.21 that Brownian paths are nowhere locally Holder-continuousfor any exponent y > f. The law of the iterated logarithm applied to{Wt+h - Wh; 0 < h < co} for fixed t 0 gives

- I W(9.25) lim t+h ,

-_ W ti = co, P-a.s.h 1. 0 \ /it

Thus a typical Brownian path cannot be "locally Hi5Ider-continuous with

exponent y = 4" everywhere on [0, co); however, one may not conclude from(9.25) that such a path has this property nowhere on [0, co); see Remark 9.15and the Notes, Section 11.

F. Modulus of Continuityt

Another way to measure the oscillations of the Brownian path is to seek amodulus of continuity. A function g) is called a modulus of continuity for thefunction f: [0, T] --) 11-,R if 0 s < t T and It - sl 6 imply If(t) - f(s)Ig(5), for all sufficiently small positive S. Because of the law of the iteratedlogarithm, any modulus of continuity for Brownian motion on a boundedinterval, say [0, 1], should be at least as large as ,./2.5 log log(1/6), but becauseof the established local Holder-continuity it need not be any larger than aconstant multiple of SY, for any y00, 1/2). A remarkable result by P. Levy(1937) asserts that with

(9.26) g(6) .\./2.5 log(1/5); 6 > 0,

cg(6) is a modulus of continuity for almost every Brownian path on [0, 1] ifc > 1, but is a modulus for almost no Brownian path on [0, 1] if 0 < c < 1.We say that g in (9.26) is the exact modulus of continuity of almost everyBrownian path. The assertion just made is a straightforward consequence ofthe following theorem.

9.25 Theorem (Levy modulus (1937)). With g: (0, 1] -) (0, oo) given by (9.26),we have

(9.27) P1

max I Wt - WS1 = 11 = 1.g(u) (:).s<t<1t-sb

PROOF. With n > 1, 0 2e-x2121,./27t(x + 1/x) andx = .\/(1 - 0)2n log 2. It develops easily that for n > 1, we have > a2-"(1-0)where a > 0, and thus

[P max I Wia. - Wu- 1)/21 (1 - 0)1/2g(2') < exp( - a2").1 < j<2^

By the Borel-Cantelli lemma, there exists an event no e with P(09) = 1 and

This subsection may be omitted on first reading; its results will not be used in the sequel.

an integer-valued random variable No such that, for every co e Cle, we have

1

ima x I Wi12,,(w) - 0/2.(01 0; n > Ne(w).sjs

Consequently, we obtain

lim1,

max 1141, - Wsl - 0,alo 9(0) oss<i< t

t-sSo

and by letting 010 along the rationals, we have

1

lim max I147, -W I > 1, a.s. P.alo 9(0) oss<is

t-s<d

For the proof of the opposite inequality, which is much more demanding,we select 0 e (0, 1) and 6 > ((1 + 0)I(1 - 0)) - 1, and observe the inequalities

[ 1

0max

2. g(ICZ.-(9.28) P .

)n, I Wp2. - Wonl _>_ 1 + E

k=j-i2.92,01

kE P max I Wa, r-Wod (1 + E)g( )1k=1 0 < i<i+k< 2. 2"

E[2.0]

< 2" E P ki2 > (1 +4n

k=i Ik2-"The probability in the last summand of (9.28) is bounded above, thanks to(9.20), by a constant multiple of n-112(k2')" "2, and

2,0+1

x(1"2 dx =(2" + 1)v

k=1 0 V

where v = 1 + (1 + 6)2. Therefore,

P[ max 1W.i/2" Wi/2.I > 1 + 61 <cons!.

2-nn,0 < j< 2. 9(k /2")k=j-i52.°

with p = (1 - 0) (1 + e)2 - (1 + 0), a positive constant by choice of E. Againby the Borel-Cantelli lemma, we have the existence of an event no e g" withP(00) = 1, and of an integer-valued random variable N, such that

(9.29) 2-(1-e)No 1wi < _; v co EDOe

and

(9.30) maxWi/2.(0 - Wi/24(0)1 < 1 + E; n > No(w),

0 < i< j< 2. g(k/2n)k=i--ts2^'

9.26 Problem. Consider the set D = U;;D=14 of dyadic rationals in [0, 1], withD = {kr"; k = 0, 1, ...,2^}. For every w e Slo and every n N8(w), theinequality

(9.31) No)) - Ws(w)I < (1 + s)[2 g(2-j) + g(t -j=n+1

is valid for every pair (s, t) of dyadic rationals satisfying 0 < t - s < 2-"" -8).(Hint: Proceed as in the proof of Theorem 2.8 and use the fact that g() isstrictly increasing on (0, 1/e].)

Returning to the proof of Theorem 9.25, let us suppose that the dyadicrationals s, t in (9.31) are chosen to satisfy the stronger condition

(9.32) 2-("+"(' -6) < 6 L_ t- s< 2-n(1-8).

But then (9.29) implies 2-n(1-8) < 1/e; n > No(w), and because g is increas-ing on (0, 1/e] we have

E g(2-j) cg(2-n-1) 2 9 2 g ( )j=n+1 -

holds for an appropriate constant c > 0. We may conclude from (9.31), (9.32),and the continuity of W.(w) that for every w elle and n Ne(w),

1

max Wi(w) - W.,(012c

(1 + s)[1 + 2-8(n+11/2

gly 0<s<t51 1 -0 JJ

t-s=8

holds for all 6 e [2-("+1)(1-8),2-"" -0)). Letting n co, we obtain

1

lim max 1147,(w) - Ws(w)I 1 + c,.510 9(0) o<s<t<

v.-n=6

and because g is strictly increasing on (0, 1/e] we may replace the conditiont - s = 6 by t - s < 6 in the preceding expression. It remains only to let 010(and hence simultaneously a 4.0) along the rationals, to conclude that

1lim max 1W(w) - Ws(w)l < 1; a.s. P.64,0 061 o<s<isi

t -ssa

The proof is complete.


1.4. For fixed 0 5 s < t < (x), and arbitrary integer n 1 and indices 0 = so < s, << sn= s, the a-field er(Xo, Xso. , Xs) = cr(Xo, X,, - Xs - x._,) is

independent of X, - X,. The union of all a-fields of this form (over n > 1,

(s1,... , s"_ 1) as described) constitutes a collection '' of sets independent ofX, - X5, which is closed under finite intersections. Now 9, the collection of allsets in .3 which are independent of X, -X is a Dynkin system containing W.From Theorem 1.3 we conclude that sx = a(W) is contained in g. In fact,"-sx g.

2.3. Suppose that there exist integers n > 1, m > 1, index sequences t = (t1,... , tn),s = (s1,...,s.), and Borel sets A ER(R"), BE /(1r) so that CE (If admits bothrepresentations C = {w E 0810 ((Mt! , 0)(t,,))E Al = {CO E 0810 (0)(S1), ,

0..)(4,))E B). It has to be shown that

(2.3)' Qt(A) = Q!(B)

Case 1 : m = n, and there is a permutation (i 1_ , ik) of (1,...,n) so that si =1 < j n.

In this case, A = {(x1,... , X.) E 0:1"; (x,,,... , Xi.) E BI. Both sides of (2.3)' are measureson At CR"), and to prove their identity it suffices to verify that they agree on sets of theform A = Al x x A, .4(08) (hence B = A,1 x x But then (2.3)' is justthe first consistency condition (a) in Definition 2.1.

Case 2: m > n and {ti,...,t} g_Without loss of generality (thanks to Case 1) we may assume that I.; = 1 S j < n.

Then we have B = A x le with k = m -n > 1, and (2.3)' follows from the second con-sistency condition (b) of Definition 2.1.

Case 3: None of the above holds.We enlarge the index set, to wit: {q1,...,q,} = {t1,... ,t"} u {s1,.. .,s,}, with

m v n</<m+ n, and obtain a third representation

C = {we (w(qi ), , w(q,))E E}, EE.I(081).

By the same reasoning as before, we may assume that = 1 s j < n. Then E = A xLril-" and, by Case 2, Q,(A) = Qq(E) with q = (q1, qd. A dual argument showsQ1(B) = Qn(E)

The preceding method (adapted from Wentzell (1981)) shows that Q is well definedby (2.3). To prove finite additivity, let us notice that a finite collection {C.,}7=1 g_ if maybe represented, by enlargement of the index sequence if necessary, as

=_- {CO E REC"D); (0)(t1), ... , CO(t,)) E Al }, Ai .4(01")

for every 1 < j < in. The Al's are pairwise disjoint if the Cis are, and finite additivityfollows easily from (2.3), since

U c= fweffe°,-); (co(ti), , co(tnfle U A; }..i=1 .i=1

Finally, RE°'") = {COE till°'°°°; (0(t)e ER) for every t > 0, and so Q(081°.'")) = Q,(R) = 1.

2.4. Let 327 be the collection of all regular sets. We have OESc. To show .547 is closedunder complementation, suppose A E 37, so for each e > 0, we have a closed set Fand an open set G such that F g A g_ G and Q(G\F) < E. But then Fc is open,G` is closed, Gr c A' S F`, and Q(P\G`) = Q(G\F) < E; therefore, ACE F. Toshow ,F is closed under countable unions, let A = (.);,°=1 Ak, where Ak E3-4.- foreach k. For each E > 0, there is a sequence of closed sets {Fk} and a sequence ofopen sets {Gk} such that Fk g Ak g Gk and Q(Gk \Fk)< k = 1, 2, .... Let

G = Ur=1Gk and F = Uk1

Fk, where m is chosen so that Q(Ulf._, Fk\Ur=, <E/2. Then G is open, F is closed, F g A g G, and

E E

Q(G\F) Q (G\ U Fk) + Q(6° Fk\F) < E2

= E.k=1 k=1 k=1

Therefore, 3,7 is a a-field.Now choose a closed set F. Let Gk = {X E RI"; II Y < 11k, some y F}. Then

each Gk is open, G1 2 G2 2 , and nio_i Gk = F. Thus, for each e > 0, thereexists an m such that Q(G,\F) < E. It follows that F e ST; .

Since the smallest a-field containing all closed sets is .4(R"), we have= .4(R").

2.5. Fix t = t2, , t), and let s = (t,i, t,2,.. , t,n) be a permutation of t. We haveconstructed a distribution for the random vector (B,i,131, ..... BO under which

Q1(A1 x A2 x x A) = P[(13,, ..... x x An]

= ..... Ai, x x

= Qs(Ai, x x

Furthermore, for A E g(R"-1) and s' = (t1,t2,. ,t_,),

Qt(A x l) = P[(B,,, , B1_,)e 24] =

2.9. Again we take T = 1. It is a bit easier to visualize the proof if we use the maximumnorm ill(t, , t2, , td)ill A max , ,, I t,1 in Rd rather than the Euclidean norm.Since all norms are equivalent in Rd, it suffices to prove (2.14) with It - silreplaced by III t - s ill. Introduce the partial ordering in Rd by (s,, s2, sd)

(t,, t2,... , td) if and only if s, < t i = 1, ,d. Define the lattices

co

L = y,; k = 0, 1, , 2" - 1} ; n 1, L= U L,n=1

and for s E L, define N(s) = {t E L; s < t, IIt - sil = 2'}. Note that L has2nd elements, and for each s e L, N(s) has d elements. For s L and t o Nn(s),Cebygev's inequality applied to (2.13) gives (with 0 < y < GO),

P[I X, - XsI > 2-Yn] < C2-^(d+9 -2Y),

and consequently,

[P max IX, -X sl > 2-Y" < dC2-"(13-27) .

se Lt e N(s)

The Borel-Cantelli lemma gives an event Sr E 3'7. with P(S1*) = 1, and a positive,integer-valued random variable n*, such that

max I X,(0)) - Xs(co)I < 2-", n > n(a)), a) E ir.SE L

IE &(s)

Now let Rn(s) = {t e L"; s t, lilt - s ill = 2-11. For s E Ln and t E Rn(S), there isa sequence s s1, , s, = t with m < d and s, N(s,_,), i = 1, ..., m.Consequently,

(10.1) max IX,(co) - Xs(co)I 5 d2"", n n* (co), weir.re R,,(s)

We now fix w n > n*(w), and show that for every m > n, we have

171

(10.2) IX,(co) - Xs(co)I 5 2d E 2-7i; V t, s E L s < t, lilt - sill <j=n+1

For in = n + 1, we can only have tE R,(s), and (10.2) follows from (10.1). Suppose(10.2) is valid for m = n + 1, , M - 1. Take t, s E Lm, s t. There is a vectors' E L,_, n R m(s) and a vector ti E Lf_, with t E R f(t') such that s t1 t.

From (10.1) we have,

IX,,(co) - Xs(co)1 _5 dr'', PC, (co) - X,i(o.))1 5_ d2"m,

and from (10.2) with in = M - 1, we havem-i

PC,i(a)) - Xsi(co)1 5_ 2d E 2-i=n+1

We obtain (10.2) for in = M.For any vectors s, t E L with s t and 0 < lilt - sill < h(co) A 2 ""), we select

n > n*(w) such that 2-("+" 5_ lilt - sill < 2-". We have from (10.2)

I Xi(co) - Xs(co)I 5_ 2d 2 YJ 5_ Slllt -j=n+1

where (5 = 2d/(1 - 2-Y). We may now conclude as in the proof of Theorem 2.8.

4.2. The n-dimensional cylinder sets are generated by those among them which aren-fold intersections of one-dimensional cylinder sets; the latter are generated bysets of the form H = {we C[0, co); w(t,) G}, where G is open in R. But H isopen in C[0, co), because for each coc,eli, this set contains a ball B(coo, 8) A{we C[0, oo); p(co,coo) < El, for suitably small E > 0. It follows that W.4(C[0, co)). Because C[0, co) is separable, the open sets are countable unions ofopen balls of the form B(coo,E) as previously. Let Q be the set of rationals in[0, co). We have

EB(0)0,0=

{(.0: E - sup (lco(t) - wo(01 A 1) < El.n=1 2" 0<t<n

tEQ

The set on the right is W-measurable, so ,I(C[0, co)) cFor the second claim, notice that with any cylinder set C of the form (2.1)' we

have

cptI(C) = Ico E C[0, co); (0),(0)(t ,), ,(cp,w)(t))E A}

= Ico C[0, co); (o)(t h t1), co(t e Eic,

so cp,-1(f) g_ (et. On the other hand, for any Cegt we have t , to in [0, t] andA EM(6") so that

C = {we oo);(w(t,), co(t.))E A}

= C[0, co); (co(t co(t t,,)) E A = cp, 1 (C),

and thus Ce, (if). It follows that Ce, = (pt-1(W), which establishes the claim.

4.11. We have to show

(4.6)' lim sup P[IXr1 > A.] = 0, andA-.x mZ I

(4.7)' lim sup P I max I - n")I > E I = 0P10 m21 It-sl.p

0 .cr,s

for all positive numbers t, T. Relation (4.6)' is an immediate consequence of (i)and of the Ceby§ev inequality. It suffices to prove (4.7)' for T = 1. Let ri > 0 andE > 0 be given. We denote X(")) simply by X, and employ the notation of theproof of Theorem 2.8. With

ni g ñ { max I Xkir, X(k -1)/2.1 < 2-Y"n=1 1 ..ck .c 2^

we have from (2.9): PA) 5 C,E`:_,2-"(13-") < n, provided 1 is a large enoughinteger. Now for every CO e (21 and n > 1, we have from (2.12):

max I X,(a.)) - Xs(w)I < 62-7",

t,se D

where .5 2/(1 - 2-7). It follows that, given e > 0, n > 0, there exists an integern = n(e,n) such that for every to E Stn:

max I Xr(o.)) - n")(0))1 < e, V m > 1

O<I,S5 1

(we have used the continuity of the sample path ti-o.")(o.))), and consequently

sup P max IXr - .nn)I > e < P(SY) < ti.,,,1

[t-m<2-^()./,s51

4.16. Let (Q,,, .0)--; Fi.) denote the space on which X and Y are defined, and let E denoteexpectation with respect to P. Let X be defined on (II, 37;,P). We are given thatlim,, = Ef(X) for every bounded, continuous f: S -0 FI and thatp(X("),r")) .0 in probability. To prove Y(") 4 X, it suffices to show

lim E[f(X(")) - f(r"))] = 0n-ce

whenever f is bounded and continuous. Let such an f be given, and setM = suP..es If(x)1 < co. Since {X("1°'_, is relatively compact, it is tight; so foreach E > 0 there exists a compact set K c S, such that P [X(M)E K] > 1 - e/6M,V n > 1. Choose 0 < S < 1 so I f(x) - f(y)I < c/3 whenever XE K and p(x, y) < S.Finally, choose a positive integer N such that P[p(X("),Y(")) > (5] < e/6M,V n N We have

<3- P

n[X(") e K, p(X("), V")) < .5]-

+ 2M P[X(") IC]

+ 2M P[p(X("), r")) > (5] < E.

5.2. The collection of sets F E a(C. [0, 00 )d ) for which x 1- P'(F) is .4(08°)/,4([0, 1])-measurable forms a Dynkin system, so it suffices to prove this measurability for

all finite-dimensional cylinder sets F of the form

F = {aye CTO, oo)d; co(t 0) E FP, , co(t.)e F} ,

where 0 = t, < t, < < t, re.4)(Rd), i = 0, 1, , n. But

Jr,P'(F) = lro(x) I Pe(t 1; x, Yi) Pa(t. -4 -1; Y.-1, Y,OdY. dYi,

r, J rwhere pd(t; x, y) A (2nt)-da exp{ -(II x - y112 /2t) }. This is a Borel-measurablefunction of x.

5.7. If for each finite measure p on (S, AS)), there exists g,, as described, then for eachcc e R, {x ES;f (x) < a} 0 {x E S; g ,,(x) < a} has p-measure zero. But {go e.4(S), so {f e At(S)". Since this is true for every p, we have If eaW(S).

For the converse, suppose f is universally measurable and let a finite measurep be given. For r e Q, the set of rationals, let U(r) = {x e S; f(x) Thenf(x) = inf {r e Q; x e U (r)} . Since U(r) E AST', there exists B(r)E R(S) withp[B(r),a,U(r)] = 0, r e Q. Define

g m(x) A inf {r e Q; x e B(r)} = inf (1),(4reQ

where (p,.(x) = r if X E B(r) and (p,(x) = oo otherwise. Then gu: S IR is Borel-measurable, and {x ES; f(x) # g,(x)} U,e Q[B(r) 0 U(r)], which has p-measure zero.

5.9. We prove (5.5). Let us first show that for D E *Rd x Rd), we have

(10.3) P[(X, 11E1)0] = PUX, Y)EDIn

If D = B x C, where B, C E R(M), then

E[1{xee}1{YeC" = 1{yec}E{1{XeB" = i{yc}P[XE B].

For the same reasons, E[1{x E,3}1{y Ec} I = 1{yc}P[X en so (10.3) holds forthis special case. The sets D for which (10.3) holds form a Dynkin systemcontaining all measurable rectangles, so (10.3) holds for every D E .41(Rd x Rd).To prove (5.5), set D = {(x,y); x + ye r }.

A similar proof for (5.6) is possible.

6.9. (ii) By Corollary 1.2.4, S is a stopping time of {#;.,.}. Problem 1.2.17 (i) implies

E[Z2IFs+] = EV21,(5,,o+], a.s. on {S

This equation combined with (i) gives us the desired result.(iii) Suppose that S is an optional time of {A}, and that (e) holds for every

bounded optional time of {A}. Then for each s > 0,

Px[X(ss)+,e FI.Fas)+] = (U,1,)(Xs s), a.s.

But on {S < s}, we have X(s,,o+, = Xs+ so (ii) implies

Px[Xs+te rIFs+] = Px[x(s,,$),-,ErIF(s,,o+]

=(U,i,-)(xs,$)= (U,1,)(Xs), 13' a.s. on {S

Now let s T co to obtain (e) for the (possibly unbounded) optional time S.The argument for (e') is the same.

7.1. (1) = ns>tn.>03,sL= =(ii) The a-field Ax is generated by sets of the form F = {(X,... , 'COE r}, where

0 = t, < < t = t and I e AR"). Since X, = Xs_ for any sequenceg [0, t) satisfying smT t, we have F

(iii) Let X be the coordinate mapping process on C[0, co). Fix t > 0. Thenonempty set F A {w E C[0, co); w has a local maximum at t} is in sincefor each n > 0,

co

F =U nm=n reQ

lirst

{w; w(t) w(r)} E

On the other hand, a typical set in A has the form G = {we C[0, CO 00(t1),(0(t2), )eI' }, for {ti} g [0, t] and F e ( R x R x ). We claim thatF = G cannot hold for any G e Ax. Indeed, suppose we have F n Gfor some G e Ax. Then given any weFnG, the function

6(s)0 < s < t,

co(t) + s - t; s t,

is in G but not in F, so F cannot agree with any G E AX. This shows thatthe filtration 1,,x1 is not right-continuous. With ter, = A:+c, we have

= n 0-(u F(f+,),2)=e>0 s<t

so the failure of {V} to be right-continuous implies the failure of {..f;.'} tobe left-continuous.

7.3. Clearly, Ye is a a-field: QS E F e dr implies that there exists an event Gwith P A GC = FA GE Aim, and finally for any sequence {F}°°_, g_ le we havea companion sequence {G},;°_, g_ 1 such that

( U F) A (U G) s U (F A G ) E ,A°,

n=1

which yields FE 112.Further, the observation FAG = N =. F= GAN yields the characterization

Ye = g C2; 3 G el, N E.41" such that F = G A N}. It follows that ?° g 5".On the other hand, Jr contains both / and X°, and since it is a a-field:5" = cr( L.) ") g Jr.

For completeness, let us observe that the requirements F Pm(F) = 0 implythe existence of GE4 such that N = FA Ge X° and P"(G) = 0. Now F iscontained in N v G and hence F is in .AI", as is any subset of F.

7.4. Let {st; t > 0} be the coordinate mapping process on C[0, co). Let P" on(C[0, cc), ,4(C[0, co))) be the probability measure under which B = {B,, AB;

> 0} is a one-dimensional Brownian motion with initial distribution p. The setF = {w; w(1) = 0} has Po-measure zero, so FE .F4.. If F is also in thecompletion ,01' of under Pi', then there must be some G e,013 with F s Gand P"(G) = 0. Such a G must be of the form G = {w: w(0) e F} for some F E R(R),and the only way G can contain F is to have F = R. But then Pu(G) # 0. It followsthat F is not in Foi`.

2.10. Solutions to Selectee Problems 123

7.5. Clearly, AP g_ Fu holds for every 0 < t < o o , so ..f-sg, g_ F. For the oppositeinclusion, let us take any F ..F0; Problem 7.3 guarantees the existence of anevent G54...,ox such that N = PAGE Ac". But now .Foox .FX, (we have .97,x g

.97,,; for every 0 < t < co), and thus G E N e Ar" c ,9 ; implyF=G.LNE,Fog.

7.6. Repeat the argument employed in Solution 7.5, replacing ..F" by and ..FX by.5,'! and using the left-continuity of the filtration {V} (Problem 7.1).

7.18. Let {B t > 0 }, (0,1"), {P'}11 be a one-dimensional Brownian family. Forr E a(Q8), define the hitting time Hr(co) = inf It 0; B,(a)) According toProblem 1.2.6, H(O,) is optional, so {H(,,,s) = 0} is in .510, = A. Likewise,li,,mE37:0. Because of the symmetry of Brownian motion starting at theorigin, P°[H,0,, = 0] = = 0]. According to the Blumenthal zero-one law (Theorem 7.17), this common value is either zero or one. If it werezero, then P° [B, = 0, V 0 < t < s for some E> 0] = 1, but this contradictsProblem 7.14 (i). Therefore, P° [H(0a) = 0] = P°[H,_0,,o) = 0] = 1, and foreach we {H(0.0,) = 0} n {11(_..0)= 0}, there are sequences s 10, tn 10 withBs.(co) > 0, Bin(co) < 0 for every n 1.

7.19. For fixed wee, Tb(w) is a left-continuous function of b and Sb(w) is a right-continuous function of b. For fixed b E FR, "T is a stopping time and Sb is anoptional time (Problem 1.2.6), so both are 37"..w-measurable. According to Re-mark 1.1.14, the set A = {(b, co) E [0, CC) x 0; Tb(w) Sb(w)} is in .4([0, cc)) (3),F07. Furthermore, A g {(0 e f2; (b, 0)) E Al is included in the set

{we SI; BA(0) wrb(o+t(w) - Wrb(w)(0) < 0 for some E> 0 and all t E en,which has probability zero because {B F,B; 0 < t < co} is a standard Brownianmotion (Remark 6.20 and Theorem 6.16), and Problem 7.18 implies that B takespositive values in every interval of the form [0,E] with probability one. Thisestablishes (i). For (ii), it suffices to show

P[wES); (L(co),co)e A] = J P(Ab)P[L E db].OD

If A were a product set A = C x D; C E .4([0, co)), D e..Fw, this would followfrom the independence of L and The collection of sets A e .4([0, co)) (3).F,wfor which this identity holds forms a Dynkin system, so by the Dynkin systemTheorem 1.3, this identity holds for every set in 98([0, op)) ®

8.8. We have for s > 0, t > 0, b > a, b > 0;

P°[W,±s a, M,s < b1.5rJ = P°[W,±s < a, M, < b, max W,, < bou5s

= 1 { ,,,,,} P° [W.s < a, max W, < boso5s

The last expression is measurable with respect to the a-field generated by thepair of random variables (W M,), and so the process {(W,, M,); .F.,; 0 t < }

is Markov. Because V, is a function of (W Me), we have

P°[Y,A,EFIA] = P°[Yt+serlWr, rEge(R)-

In order to prove the Markov property for Y and (8.11), it suffices then to showthat

r[Yr-FsE Wr = W9 Mr = m] = P+(s; m - w,Y)dY

Indeed, for b > m > w,b > a, m > 0 we have

[Wr. da, Mr. W, = w, M, = m]

= P°[W,,,E da, max W,. E dbl W, = w, M, =13.5

Pw[WE da, Ms db] = [Wsda - w, Mse db - w]

=2(26 -a - w)

exp(26 -a - w)2}

da db,.12rz.s3 2s

thanks to (8.2), which also gives

r[Wr+s E da, Mt+s = ml W, = w,M, = m]

ll= r[W,+se da, max W +SmIW = w,Mr =mJ

= e da, Ms m] = [W da - w, Ms m -w]

(2m -a -= t -exp

2sexp

1

2its 2s

Therefore, P°[Y,+se dy1W, = w, M, = m] is equal to

P°[WE b - dy, 111,+ sE dbl 1N, Air = m] dbf(m, co)

+ P°[W,+sE m - dy, M,+s = mIW, = w, Mr = m] = p+(s; m - w, y)dy.

Since the finite-dimensional distributions of a Markov process are determinedby the initial distribution and the transition density, those of the processes I WIand Y coincide.

8.12. The optional sampling theorem gives

= Ex = Ex X, ,., T ATa = [exp W.,ATOAT- -1,12(t A To A TO)].

Since W(AMA T. is bounded, we may let t co to obtain

= Ex [exp 1:1.2(To 'TX]JI WT. A T -

= Ex[1{7-0<,-.}e- '111'0/2] + e"Ex[1{T.<,-.}e-'2T-12].

By choosing = ± 2a, we obtain two equations which can be solved simulta-neously and yield (8.27) and (8.28).

9.3. For every 0 < a < i, Doob's maximal inequality (Theorem 1.3.8 (iv)) gives

2

E[ sup (-) < -2 E( supW2) 4 4T

er<f<i t cf<fS 0-2

and with r = 2"+1 = 2a,

P[sup I > E] <--8 2-"

2^<(<2 ^. t

is seen to hold for every E > 0, n > 1. The result (9.2) follows now from the Borel-Cantelli lemma.

9.5. Use Problem 7.18 and Lemma 9.4 (ii).

9.8. If II = {to, t,} is a partition of [0, t], we write Vi(2)(11) - t =- (tk - 4_0} as a sum of independent, zero-mean random variables.

Therefore,

2

E(/,(2)(11) - = E - tk_02E[ 11 s tE(Z2 - 1)2111111,k=1 tk - tk-1

where Z is a standard normal random variable. The first assertion follows readily.For the second, we observe that

const.E P[iv,(2)01)- tl > 2 E 1111Il < co

n=1 E n=1

holds for every E > 0, and by the Borel-Cantelli lemma, P[IV,(2)(11) - t I > E,infinitely often] = 0. The conclusion follows.

9.11. (D. Freedman (1971))(i) Each point of the set

Mn = It E [0, cc); f(s) < f(t), V SERt --1),

is isolated, so Mn is countable. But U:3-1 Mn is the set of points of strict localmaximum for f.

(ii) It suffices to show that f has a local maximum in an arbitrary, nonemptyinterval (a, b) S [0, cc). Let us begin by assuming f(a) < f(b), and let t be thelargest number in [a, b) with f (t) = f (a). Because f is not monotone such a texists, and in (t, b) there are two numbers r and s with r < s and f(r) > f(t) vf(s). Being continuous, f must have a maximum over [t, s], which is a localmaximum in the sense of Definition 9.10 (iii).

If f(a) > f(b), we apply the preceding argument in reverse, defining t to bethe smallest number in (a, b] with f(t) = f(b). If f(a) = f(b), then f must havea maximum over [a, b] at some r e (a, b) and this r is also a point of localmaximum.

9.22. Use integration by parts (Chung (1974), p. 231; McKean (1969), p. 4).

9.26. It suffices to show that for every m > n No(co), we have

(10.4) I W(w) - Ws(co)I < (1 + [2 g(2-j) + g(t -=n+1

valid for every s, t E D, satisfying 0 < t - s < 2-11(1 -9). For m = n, (10.4) followsfrom (9.30). Let us assume that (10.4) holds for m = n, . . . , M - 1. With s, t E DMand 0 < t -s < 2-"("), we consider, as in the proof of Theorem 2.8, the num-bers t' = max {u E Dm_i; U < t} and sl = min{u E Dm_ ; u s} and observe the

relations t - tt < 2-1", s1 - s < 2-m, and 0 <11 - s1 < t - s < 2-"" -8) < 1/e,thanks to (9.29). We have

M -1

Wo(a)) - Ws.(01 5 (1 + E)[2 E g(2-j) + g(t' -j=n+I

by the induction assumption, and 1W, co) - W,,(01 5 (1 + e)g(2') as well as1Ws(w) - W.,((.0)1 < (1 + E)g(2') because of (9.30). Since g(t1 - s') 5 g(t - s),we conclude that (10.4) holds with m = M.

2.11. Notes

Section 2.1: The first quantitative work on Brownian motion is due toBachelier (1900), who was interested in stock price fluctuations. Einstein (1905)derived the transition density for Brownian motion from the molecular-kinetictheory of heat. A rigorous mathematical treatment of Brownian motion beganwith N. Wiener (1923, 1924a), who provided the first existence proof.

The most profound work in this early period is that of P. Levy (1939, 1948);he introduced the construction by interpolation expounded in Section 2.3,studied in detail the passage times and other related functionals (Section 2.8),described in detail the so-called fine structure of the typical sample path(Section 2.9), and discovered the notion and properties of the mesure duvoisinage or "local time" (Section 3.6 and Chapter 6). Most amazingly, hecarried out this program without the formal concepts and tools of filtrations,stopping times, or the strong Markov property.

Section 2.2: The construction of a probability measure from a consistentfamily of finite-dimensional distributions is clearly explained in Kolmogorov(1933); Daniell (1918/1919) had constructed earlier an integral on a space ofsequences. The existence of a continuous modification under the conditionsof Theorem 2.8 was established by Kolmogorov (published in Slutsky (1937));Loeve ((1978), p. 247) noticed that the same argument also provides localHolder-continuity with exponent y for any 0 < y < Pc. For related results,see also Centsov (1956a). The extension to random fields as in Problem 2.9was carried out by Centsov (1956b).

Section 2.3: The Haar function construction of Brownian motion wasoriginally carried out by P. Levy (1948) and later simplified by Ciesielski(1961). For a similar construction of Brownian motion indexed by directedsets, see Pyke (1983). Yor (1982) shows that the choice of the complete, ortho-normal basis in L2[0, 1] is not important for the construction of Brownianmotion.

Section 2.4 is adapted from Billingsley (1968). The original proof of Theorem4.20 is in Donsker (1951), but the one offered here is essentially due toProhorov (1956). It is also possible to construct a probability space on whichall the random walks are defined and converge to Brownian motion almostsurely, rather than merely in distribution (Knight (1961)).

Sections 2.5, 2.6: The Markov property derives its name from A. A. Markov,whose own work (1906) was in discrete time and state space; in that context,of course, the usual and the strong Markov properties coincide. It was notimmediately realized that the latter is actually stronger than the former; Ray((1956), pp. 463-464) provides an example of a continuous Markov processwhich is not strongly Markov. It seems rather amazing today that a completeand rigorous statement about the strongly Markovian character of Brownianmotion (Theorem 6.16) was proved only in 1956; see Hunt (1956).

A Markov family for which the function xl-- Ex f(X,) is continuous for anybounded, continuous f : II -4 ill and t e [0, co) is said to have the Feller prop-erty, and a right-continuous Markov family with the Feller property is stronglyMarkovian. Very readable introductions to Markov process theory can befound in Dynkin & Yushkevich (1969), Wentzell ((1981), Chapters 8-13), andChung (1982); more comprehensive treatments are those by Dynkin (1965),Blumenthal & Getoor (1968), and Ethier & Kurtz (1986). Markov processeswith continuous sample paths receive very detailed treatments in the mono-graphs by ItO & McKean (1974), Stroock & Varadhan (1979), and Knight(1981).

Section 2.7: The concept of enlargement of a filtration has become veryimportant in Markov process theory and in stochastic integration. There is asubstantial body of theory on this topic, which we do not take up here; weinstead send the interested reader to the articles in the volume edited by Jeulin& Yor (1985).

Sections 2.8, 2.9: The material here comes mostly from P. Levy (1939, 1948).Section 1.4 in D. Freedman (1971) was our source for Theorems 9.6, 9.9 and9.12, and can be consulted for further information on this subject matter. Ourdiscussion of the law of the iterated logarithm and of the Levy modulus followsMcKean (1969) and Williams (1979). Theorem 9.18 was strengthened byDvoretzky (1963), who showed that there exists a universal constant c > 0such that

P[co -,-1W+h((o) - 14/11041 > c, V t e [0, op)] = 1.al; limh40 Nfil

For every w e 0, S,,, -4 It E [0, CO); liMh10 (1 Wt+17(04 - Wt(0))01) < 091 hasbeen called by Kahane (1976) the set of slow points from the right for the pathW.(w). Fubini's theorem applied to (9.25) shows that meas(S,o) = 0 for P a.e.wee, but, for a typical path, S,,, is far from being empty; in fact, we have

P1 En; inf lim 1WH-h

(0) - W(01 = 11 = 1.cl.,<co it.i 0 112

This is proved in B. Davis (1983), where we send the interested reader for moreinformation and references on this subject.

Chung (1976) and Imhof (1984) offer excellent follow-up reading on thesubject matter of Section 2.8.

CHAPTER 3

Stochastic Integration

3.1. IntroductionA tremendous range of problems in the natural, social, and biological sciencescame under the dominion of the theory of functions of a real variable: whenNewton and Leibniz invented the calculus. The primary components of thisinvention were the use of differentiation to describe rates of change, the useof integration to pass to the limit in approximating sums, and the fundamentaltheorem of calculus, which relates the two concepts and thereby makes thelatter amenable to computation. All of this gave rise to the concept of ordinarydifferential equations, and it is the application of these equations to themodeling of real-world phenomena which reveals much of the power ofcalculus.

Stochastic calculus grew out of the need to assign meaning to ordinarydifferential equations involving continuous stochastic processes. Since themost important such process, Brownian motion, cannot be differentiated,stochastic calculus takes the tack opposite to that of classical calculus: thestochastic integral is defined first, in Section 2, and then the stochastic differ-ential is given meaning through the fundamental "theorem" of calculus. This"theorem" is really a definition in stochastic calculus, because the differentialhas no meaning apart from that assigned to it when it enters an integral.For this theory to achieve its full potential, it must have some simple rulesfor computation. These are contained in the change of variable formula(Ito's rule), which is the counterpart of the chain rule from classical calculus.We present it, together with some important applications, in Section 3.

Section 4 advances our recurrent theme that "Brownian motion is thefundamental martingale with continuous paths" by showing how to representcontinuous, local martingales in terms of it, either via stochastic integration


3.2. Construction of the Stochastic Integral 129

or via time-change. We also establish the representation of functionals ofthe Brownian path as stochastic integrals. The important Theorem 4.13 ofF. B. Knight is established as an application of these ideas.

Stochastic calculus has a fundamental additional feature not found in itsclassical counterpart, a feature based on the Girsanov change of measure(Theorem 5.1). This result provides a device for solving stochastic differentialequations driven by Brownian motion by changing the underlying probabilitymeasure, so that the process which was the driving Brownian motion becomes,under the new probability measure, the solution to the differential equation.This profound idea is first presented in Section 5, but does not reach itsculmination until the discussion of weak solutions of stochastic differentialequations in Chapter 5. In some cases, this device is merely a convenient wayof finding the distribution of a functional of an already existent stochasticprocess; such an example is provided by the computations related to Brownianmotion with drift in subsection 5.C. In other cases, the change of measureprovides us with a proof of the existence of a solution to a stochastic differ-ential equation, when the more standard methods fail. This is discussed insubsection 5.3.B. A particularly nice application of the Girsanov theoremappears in Section 5.8, where it is used in a model of security markets toremove the differences among the mean rates of return of the securities. Thisreduction of the model permits a complete analysis by martingale methods.Although "optional" in the sense that stochastic calculus can (and did for 20years) exist and be useful without it, the Girsanov theorem today plays sucha central role in further developments of the subject that the reader would beremiss not to come to grips with this admittedly difficult concept.

In Section 6 we employ the stochastic calculus in the study of P. Levy'smesure du voisinage or local time, a device for measuring the "amount of timespent by the Brownian path in the vicinity of a certain point." This concept hasbecome exceedingly important in both theory and applications; we examineits connections with reflected Brownian motion, extend with its help the Itorule to functions which are not necessarily twice continuously differentiable,and use it as a tool in the study of certain additive functionals of the Brownianpath. Finally, Section 7 extends the notion of local time to general, continuoussemimartingales.

3.2. Construction of the Stochastic Integral

Let us consider a continuous, square-integrable martingale M = {M0 < t < oo } on a probability space (K2, P) equipped with the filtration1.4r 1, which will be assumed throughout this chapter to satisfy the usualconditions of Definition 1.2.25. We have shown in Section 2.7 how to obtainsuch a filtration for standard Brownian motion. We assume Mo = 0 a.s. P.Such a process M e .i/4 is of unbounded variation on any finite interval [0, T]

130 3. Stochastic Integration

(cf. Problems 1.5.11, 1.5.12, and the discussion following them), and conse-quently integrals of the form

(2.1) /T(X) = I Xt(w)dM,(w)

cannot be defined "pathwise" (i.e., for each w e SI separately) as ordinaryLebesgue-Stieltjes integrals. Nevertheless, the martingale M has a finite second(or quadratic) variation, given by the continuous, increasing process <M>;cf. Theorem 1.5.8. It is precisely this fact that allows one to proceed, in a highlynontrivial yet straightforward manner, with the construction of the stochasticintegral (2.1) with respect to the continuous, square-integrable martingale M,for an appropriate class of integrands X. The construction is due to ItO (1942a),(1944) for the special case that M is a Brownian motion and to Kunita &Watanabe (1967) for general Me di2. We shall first confine ourselves toM e."2, and denote by <M> the unique (up to indistinguishability) adapted,continuous, and increasing process, such that IM,2 - <M> A; 0 < t < oolis a martingale (cf. Definition 1.5.3 and Theorem 1.5.13). The constructionwill then be extended to general continuous, local martingales M.

We now consider what kinds of integrands are appropriate for (2.1). Wefirst define a measure gm on ([0, oo) x S2, 8([0, co)) <91 by setting

(2.2) IM(A) = E J 1A(t,a) d<M>,(w).

We shall say that two measurable, adapted processes X = IX ,Fg; 0 < t < coland Y = { Y A; 0 < t < col are equivalent if

X,(w) = Y,(co); µM -a.e. (t, w).

This introduces an equivalence relation. For a measurable, 1,97,1-adaptedprocess X, we define

(2.3) [X]T °= E I X2 d<M>

provided that the right-hand side is finite. Then [X]j, is the L2-norm for X,regarded as a function of (t, w) restricted to the space [0, T] x CI, under themeasure We have [X - = 0 for all T > 0 if and only if X and Y areequivalent. The stochastic integral will be defined in such a manner thatwhenever X and Y are equivalent, then 1(X) and 1(Y) will be indistinguishable:

PUT(X) = /T(Y), V 0 T < oo] = 1.

2.1 Definition. Let 29 denote the set of equivalence classes of all measurable,IFJ-adapted processes X, for which [X],- < co for all T > 0. We define ametric on 2' by [X - Y], where

[X] -4-- 2-"(1 A [X].).n=i

3.2. Construction of the S:ochastic Integral 131

Let ..r* denote the set of equivalence classes of progressively measurableprocesses satisfying [X]T < oo for all T > 0, and define a metric on .2" in thesame way.

We shall follow the usual custom of not being very careful about the dis-tinction between equivalence classes and the processes which are members ofthose equivalence classes. For example, we will have no qualms about saying,".29* consists of those processes in .29 which are progressively measurable."

Note that .29 (respectively, .2") contains all bounded, measurable, {,F, }-adapted (respectively, bounded, progressively measurable) processes. Both2' and 2* depend on the martingale M = {M t > 0}. When we wish toindicate this dependence explicitly, we write 2'(M) and 2' *(M).

If the function ti- <M>,(co) is absolutely continuous for P-a.e. to, we shallbe able to construct P., X, dM, for all X e and all T > 0. In the absence ofthis condition on <M>, we shall construct the stochastic integral for X in theslightly smaller class .29*. In order to define the stochastic integral with respectto general martingales in A/2 (possibly discontinuous, such as the compensatedPoisson process), one has to select an even narrower class of integrands amongthe so-called predictable processes. This notion is a slight extension of left-continuity of the sample paths of the process; since we do not develop stochasticintegration with respect to discontinuous martingales, we shall forego furtherdiscussion and send the interested reader to the literature: Kunita & Watanabe(1967), Meyer (1976), Liptser & Shiryaev (1977), Ikeda & Watanabe (1981),Elliott (1982), Chung & Williams (1983).

Later in this section, we weaken the conditions that M e.ifz and [X]i < co,V T > 0, replacing them by M e Jr' and

T

P d<M>, < co]= 1, V T 0.

This is accomplished by localization.We pause in our development of the stochastic integral to prove a lemma

we will need in Section 4. For 0 < T < co, let .27 denote the class of processesX in ..r* for which Xt(to) = 0; V t > T, co en. For T = co, ..r; is defined as theclass of processes X e .2" for which E ft; X,2 d<M>, < co (a condition wealready have for T < co, by virtue of membership in _2"). A process X e 2,;!'can be identified with one defined only for (t, w) e [0, T] x n, and so we canregard 4` as a subspace of the Hilbert space

(2.4) YfT -4 L2([0, T] x a([0, T]) T Pm)

More precisely, we regard an equivalence class in A9T as a member of .27 ifit contains a progressively measurable representative. Here and later wereplace [0, T] by [0, co) when T = co.

2.2 Lemma. For 0 < T < co, .27 is a closed subspace of .09T. In particular, .,27is complete under the norm [X]i, of (2.3).

PROOF. Let {X(")},111 be a convergent sequence in .27 with limit X EST. Wemay extract a subsequence, also denoted by {X(")},,m11, for which

{ (t, co) e [0, T] x lim X: "°(co) 5 Xt(co)} = O.n-,co

By virtue of its membership in Yer, X is ,4([0, T]) .F-measurable, but maynot be progressively measurable. However, with

A -A {(t,co)e [0, T] x i2; lim X;"°(w) exists in 51),

the process

lim XP)(co); (t, w) e A)7, (w) _4_ {n-co

0; (t, CO) A

inherits progressive measurability from {X(")}c°_, and is equivalent to X.

A. Simple Processes and Approximations

0

2.3 Definition. A process X is called simple if there exists a strictly increasingsequence of real numbers {tn}n°3_0 with to = 0 and lim, t = oo, as well as asequence of random variables g1,110 with supn,oln(01 < C < co, for everywe 1, such that is Ft.-measurable for every n > 0 and

CO

X,(co) = 0(co) 1{0} (t) + i(co)10,,,,,i(t); 0 < t < co, co ei=o

The class of all simple processes will be denoted by Yo. Note that, becausemembers of ..F0 are progressively measurable and bounded, we have .290 g...?*(M) c .29(M).

Our program for the construction of the stochastic integral (2.1) can nowbe outlined as follows: the integral is defined in the obvious way for X eas a martingale transform:

n-1(2.5) it(X) A E - m) + um, - 4)

co

= E um, A ti+, - MIA t,), t <i=0

where n > 0 is the unique integer for which t < t < tn+1. The definition isthen extended to integrands X e 21* and X e .29, thanks to the crucial resultswhich show that elements of .21* and _V' can be approximated, in a suitablesense, by simple processes (Propositions 2.6 and 2.8).

2.4 Lemma. Let X be a bounded, measurable, {g;}-adapted process. Then thereexists a sequence {X('")}:=1 of simple processes such that

T

(2.6) sup lira E f RO'n) - X,12 dt = 0.T>0 m-.co 0

PROOF. We shall show how to construct, for each fixed T > 0, a sequence{x(n,T)} flop_ of simple processes so that

lim E f I XIn'T) - X,I2dt = 0.o

Thus, for each positive integer m, there is another integer n,,, such thatm

E 1,0"-m) -o

X,I2 dt < -1m

and the sequence {X("-m)},',., has the desired properties. Henceforth, T is afixed, positive number.

We proceed in three steps.

(a) Suppose that X is continuous; then the sequence of simple processes2.-1

)0 n)(C0) A X0(01{0}(t) + E X0-724(0) laT/2",(k+1)T/2.1(t); n _. 1,k=0

satisfies lim, E f o IX; ") - x,12 dt = 0 by the bounded convergencetheorem.

(b) Now suppose that X is progressively measurable; we consider the con-tinuous, progressively measurable processes

t T

(2.7) F,(w) J. XJw) ds; fes'n)(co) m[Ft(w) - Fo_(1 /m)".(co)]; m 1,

for t > 0, w e f2 (cf. Problem 1.2.19). By virtue of step (a), there exists, foreach m > 1, a sequence of simple processes {g(m")},T=1 such thatlimn E - -1:"1)12 dt = 0. Let us consider the A[0, n) .FT-measurable product set

A A {(t, w) e [0, T] x f2; lim it'n(co) = Xi(w)lc.

For each w e f2, the cross section A. {t e [0, T]; (t, e A} is ina([0, T]) and, according to the fundamental theorem of calculus, hasLebesgue measure zero. The bounded convergence theorem now giveslim,n_,. E fpfon)- X,I2 dt = 0, and so a sequence {g(m "-)}:=1 ofbounded, simple processes can be chosen, for which

T

lim E I i{"-) - X,I2 dt = 0.m- o

(c) Finally, let X be measurable and adapted. We cannot guarantee immedi-ately that the continuous process F = {Ft; 0 < t < co} in (2.7) is progres-sively measurable, because we do not know whether it is adapted. We do

know, however, that the process X has a progressively measurable modi-fication Y (Proposition 1.1.12), and we now show that the progressivelymeasurable process {G, g po^ T Ysds, A; 0 5 t 5 T} is a modification of F.For the measurable process ;Mu)) = 0 < t < T, w en, wehave from Fubini: E (T)ri,(w)dt = 11; P[x,(0)) Y,(w)]dt = 0. Therefore,14' ri,(w)dt = 0 for P-a.e. wen. Now {F, 0 G,1 is contained in the eventIw; f o ri,(w) dt > 0 }, G, is A- measurable, and, by assumption, A containsall subsets of P-null events. Therefore, F, is also A- measurable. Adaptivityand continuity imply progressive measurability, and we may now repeatverbatim the argument in (b).

25 Problem. This problem outlines a method by which the use of Proposition1.1.12, a result not proved in this text, can be avoided in part (c) of the proofof Lemma 2.4. Let X be a bounded, measurable, {. }- adapted process. Let0 < T < co be fixed. We wish to construct a sequence {X(k)},T_I of simpleprocesses so that

T

(2.8) lim E 1)0k) - X,I2 dt = 0.k co

To simplify notation, we set X, = 0 for t < 0. Let cp: Or"; j =0, + 1, + 2, ...1 be given by

(0 .(t)2"

for 1 < t <2"

(a) Fix s > 0. Show that t - (1/2") < cp(t - s) + s < t, and that

t > 0

is a simple, adapted process.(b) Show that lim,40 E 14. X, - X,_I2 dt = 0.(c) Use (a) and (b) to show that

T 1

lim E 1,O"'s) - X,I2 ds dt = 0.n- 0 0

(d) Show that for some choice of s > 0 and some increasing sequence Ink1,°_,of integers, (2.8) holds with X(k) = X("k's).

This argument is adapted from Liptser and Shiryaev (1977).

2.6 Proposition. If the function t <M>,(w) is absolutely continuous withrespect to Lebesgue measure for P-a.e. wen, then .290 is dense in .29 with respectto the metric of Definition 2.1.

PROOF.

(a) If X e .29 is bounded, then Lemma 2.4 guarantees the existence of abounded sequence {V")} of simple processes satisfying (2.6). From these

we extract a subsequence {X(mk)}, such that the set

{(t, w) e [0, co) x SI; lim Xrnkga)) = Xt(co)}`

has product measure zero. The absolute continuity of t <M>,(co) andthe bounded convergence theorem now imply [X(mk) - X] -+ 0 as k -+ co.

(b) If X e .2 is not necessarily bounded, we define

)(riga)) -4 Xi(a))1{1x,(,01.(n); 0 < t < co, a)E12,

and thereby obtain a sequence of bounded processes in 2. The dominatedconvergence theorem implies

T

[X(n) - E X,21{ nao 0

for every T > 0, whence lim, [X(n) - X] = 0. Each X(") can be ap-proximated by bounded, simple processes, so X can be as well.

When t H <M>, is not an absolutely continuous function of the timevariable t, we simply choose a more convenient clock. We show how to dothis in slightly greater generality than needed for the present application.

2.7 Lemma. Let {A1; 0 < t < co} be a continuous, increasing (Definition 1.4.4)process adapted to the filtration of the martingale M = 1111.F,,; 0 < t < col. IfX = {X..F,; 0 < t < oo} is a progressively measurable process satisfying

T

E X,2 dA, < oo

for each T > 0, then there exists a sequence {X(")}°11 of simple processes suchthat

sup lim E 1XP) - X,I2 dA, = 0.T>0 nom 0

PROOF. We may assume without loss of generality that X is bounded (cf. part(b) in the proof of Proposition 2.6), i.e.,

(2.9) IXt(o))) C < co; V t > 0, a) ED.

As in the proof of Lemma 2.4, it suffices to show how to construct, for eachfixed T > 0, a sequence {X(")},;°_, of simple processes for which

T

lim E !XI") - X,I2 dA, = O.n- 0

Henceforth T > 0 is fixed, and we assume without loss of generality that

(2.10) Xi(w) = 0; V t > 7', a) e

We now describe the time-change. Since A,(w) + t is strictly increasing int > 0 for P-a.e. w, there is a continuous, strictly increasing inverse functionT(w), defined for s > 0, such that

A ,;(,)(w) + Ts(w) = s; V s > 0.

In particular, Ts < s and {T, < t} = {A, + t > s} E. Thus, for each s > 0, Tsis a bounded stopping time for Taking s as our new time-variable, wedefine a new filtration IC by

= S>0,

and introduce the time-changed process

YS(w) = XT,(0,)(0)); s > 0, wen,

which is adapted to {Ws} because of the progressive measurability of X(Proposition 1.2.18). Lemma 2.4 implies that, given any e > 0 and R > 0, thereis a simple process { V, Ws; 0 s < col for which

(2.11) E - Y.,12 ds < 6/2.

But from (2.9), (2.10) it develops that

E J 1;2 ds = ECO CO

j'AT-1-7.= E X?..ds < C(EAT + T)< op,o

so by choosing R in (2.11) sufficiently large and setting V = 0 for s > R, wecan obtain

E J IYS`- Y1Zds <e.CO

Now V is simple, and because it vanishes for s > R, there is a finite partition0 = so < s, < < s < R with

n

Y:(0)) = o((.0)1{0}(5) + E 0 < s < 00,

where each s, is measurable with respect to Ws, = FT, and bounded inabsolute value by a constant, say K. Reverting to the original clock, we observethat

Xt A Yi+A, = 01{0}(t) + E , 1(Ts

T. 1(4 0 t < co,j =1 - J

is measurable and adapted, because restricted to { Ts, < t} is g,-measurable(Lemma 1.2.15). We have

E f IX: - X,I2 dA, < E J ixf - X,12(dA, + dt)

E f - 1;12 ds < E.0

CO

The proof is not yet complete because X' is not a simple process. To finishit off, we must show how to approximate

)1t(w) ,(0))1(7;,(.),7;(01(t); 0 < t < co, co ED,

by simple processes. Recall that Ts,_, < TS < s; and simplify notation bytaking sfri = 1, = 2. Set

1 + 2" kTi(m)(w) = kZ.1

1

2m((k -1)/2 P", k/2^)( T(W)), i = 1, 2

and define

q!'")((o) A (01(71-00o, 4m )(.)1(t)

2^'+I

= E IT <(k-1)/2'n T2)(W) I ((k -1)/2", k/2",1(t).k=1

Because {T1 < (k - 1)/2m < T2} e 37;(k_i vv. and 1 restricted to {T1 <(k - 1)12m} is (k -1)/2,measurable, ri(m) is simple. Furthermore,5"

E f In!'") - 71,12 dA, < K2[E(Ann) - AT2) + E(Avr - AT1)] .7-,03 O. 0

2.8 Proposition. The set Yo of simple processes is dense in Y* with respect tothe metric of Definition 2.1.

PROOF. Take A = <M> in Lemma 2.7.

B. Construction and Elementary Properties of the Integral

We have already defined the stochastic integral of a simple process X e ..To bythe recipe (2.5). Let us list certain properties of this integral: for X, Y e SP, and0 < s < t < co, we have

(2.12) 10(X) = 0, a.s. P

(2.13) EU,(X)IFJ = 4(X), a.s. P

(2.14) IE((X))2 = E ft d<M>.

(2.15) 111(X)II [X]

(2.16) E[(I,(X) - Is(X))21.F.J = E [ft d<M> .Fs], a.s. P

(2.17) 1(aX + 13Y) = al (X + 131(Y); a, /3e R.

Properties (2.12) and (2.17) are obvious. Property (2.13) follows from the factthat for any 0 < s < t < co and any integer i > 1, we have, in the notation of(2.5),

- Mt At,)Igs] = i(A4sAt,, - MsAt,), a.s. P;

this can be verified separately for each of the three cases s < ti, t1 < s < ti+1,and ti+, < s by using the gt,-measurability of Thus, we see that I(X) =11,(X),,,; 0 < t < co }is a continuous martingale. With 0 < s < t < oo and mand n chosen so that tm_i < s < tm and t,, < t < tn+1, we have (cf. the discus-sion preceding Lemma 1.5.9)

(2.18)

E[(I,(X) - 1,(X))21.F,s]

n-1= E[{,_1(Mtm - + -A41,)+ utii-min)}2

= E[a-i(Mtm - Ms)2 + E - mo2 + a(vit - Nit)2n-1

= -<M>) + M<M>t, <M)t,)i=m

+ (<M>, -

= E [I X,; d<M> gsl

This proves (2.16) and establishes the fact that the continuous martingale 1(X)is square-integrable: I(X)e,11, with quadratic variation

(2.19) </(X)>, = J t Xu d<M>.

Setting s = 0 and taking expectations in (2.16), we obtain (2.14), and (2.15)follows immediately, upon recalling Definition 1.5.22.

For X E 2' *, Proposition 2.8 implies the existence of a sequence {V)},;°_, g..ro such that [X(") - X] 0 as n co. It follows from (2.15) and (2.17) that

III(x("))-1(x(m))11 = III(x(n) - x(m))11 = [X(n) -x(m)].o

as n, m co. In other words, 11(X("))1;°_1 is a Cauchy sequence in .t.. ByProposition 1.5.23, there exists a process I(X) = {I,(X); 0 < t < co} in

defined modulo indistinguishability, such that II/(X(")) - /(X)I1 0 as n co.Because it belongs to Atl2, I(X) enjoys properties (2.12) and (2.13). For0 < s < t < co, {/s(V)));°-1 and {h(X("))),`,ci converge in mean-square toIS(X) and I,(X), respectively; so for A e gs, (2.16) applied to {V^)}:=1 gives

(2.20) E[lA(h(X) - 1s(X))2] = lim E[1A(I,(X(n)) - is(X(n)))2]n-.co

= lim E[1A ft (X,(,"))2 d<M>dn-.co s

= E[lA d<M>d,

where the last equality follows from [V) - X], = 0. This proves thatI(X) also satisfies (2.16) and, consequently, (2.14) and (2.15). Because X andM are progressively measurable, Its d<M>. is A- measurable for fixed0 < s < t < co, and so (2.16) gives us (2.19). The validity of (2.17) for X, Y ealso follows from its validity for processes in Yo, upon passage to the limit.

The process 1(X) for X e 2* is well defined; if we have two sequences{X(")},711 and { Pnl,°,11 in 2'0 with the property lim, [X" - X] = 0,

- X] = 0, we can construct a third sequence {Z(1:11 with thisproperty, by setting Z(2"-1) = X(n) and Z(2") = Y(n), for n > 1. The limit 1(X)of the sequence {/(Z("))},7°_, in dic2 has to agree with the limits of bothsequences, namely {/(X("))}:=, and 11(P"))1°11.

2.9 Definition. For X e...9", the stochastic integral of X with respect to themartingale M e .112 is the unique, square-integrable martingale I(X) ={Ii(X),g;; 0 < t < col which satisfies limn II/(X(")) - 1(X)II = 0, for everysequence {Vnl,;°_, g 210 with lim, [X(n) - X] = 0. We write

.1,(X) = .1' XsdMs; 0 <t < co.

2.10 Proposition. For M e,46 and X e.r*, the stochastic integral I(X) ={Ii(X),A; 0 5 t < 00} of X with respect to M satisfies (2.12)-(2.16), as well as(2.17) for every Ye 21*, and has quadratic variation process given by (2.19).Furthermore, for any two stopping times S < T of the filtration {A} and anynumber t > 0, we have

(2.21) ER T(X)j,,s] = s(X), a.s. P.

With X, Y e 21* we have, a.s. P:

(2.22) EUlt T(X) - A s(X))(I t A AY) - t A s(Y))1-Fs]

=

E[ft A T

XYd<M>t As

and in particular, for any number s in [0, t],

g.s],

(2.23) E[(I,(X) - 15(X))(1,(Y) - Is(Y))1.F5] = E[ ft X.Y.d<M>.s

Finally,

(2.24) 1,,,T(x) = It(g) a.s.,

where 51,(w) g Xt(a))1{tr( .)}

gd.

PROOF. We have already established (2.12)-(2.17) and (2.19). From (2.13) andthe optional sampling theorem (Problem 1.3.24 (ii)), we obtain (2.21). The sameresult applied to the martingale {/t2.(X) -Po XZ' d<M>., gr; t 0} providesthe identities

E[(I(AT(X) -I , A s(X))2I.Fs] = EMT(X) - It2As(X)1,0

= E[fAT

X! d<M>. .Fs], P-a.s.I As

t

Replacing X in this equation, first by X + Y and then by X -Y, and sub-tracting the resulting equations, we obtain (2.22).

It remains to prove (2.24). We write

/, A T(X) - Mg) = /tA T(X -JO -uf(S')_ it, Tan.Both {/tA T(X -fa A; t 0} and fli(.1) - 1,,,,,( ),gt; t 0} are in dic2; weshow that they both have quadratic variation zero, and then appeal toProblem 1.5.12. Now relation (2.22) gives, for the first process,

E[(I(AT(X - i?) -Is T(X - 2))2I-Fs]tA r

= ELI (X. - g.)2 d<M>. g;1= 0sAT

a.s. P, which gives the desired conclusion. As for the second process, we have,

E [(mg) -it A T(g))2] = E [f g! d<M>d = 0,tn.,'

and since this is the expectation of the quadratic variation of this process, weagain have the desired result.

2.11 Remark. If the sample paths t 1-- <M>,(co) of the quadratic variationprocess <M> are absolutely continuous functions of t for P-a.e. w, thenProposition 2.6 can be used instead of Proposition 2.8 to define 1(X) for everyX E Y. We have I(X)e.112 and all the properties of Proposition 2.10 in thiscase. The only sticking point in the preceding arguments under these condi-tions is the proof that the measurable process f; --4-- Ito Xs2 d<M>5 is {A}-adapted. To see that it is, we can choose Y, a progressively measurablemodification of X (Proposition 1.1.12), and define the progressively measur-

able process G, A Po Ys2 d<M>s. Following the proof of Lemma 2.4, step (c),we can then show that P[F, = = 1 holds for every t > 0. Because G, isA-measurable, and A contains all P-negligible events in F (the usual condi-tions), F is easily seen to be adapted to {A}.

In the important case that M is standard Brownian motion with <M>, = t,the use of the unproven Proposition 1.1.12 can again be avoided. For boundedX, Problem 2.5 shows how to construct a sequence {X(k) }f_, of bounded,simple processes so that (2.8) holds; in particular, there is a subsequence, alsocalled {Xm}f_, , such that for almost every to [0, T] we have

F, Xs' ds = lim f (X''))2 ds, a.s. P.0 k-.0z, 0

Since the right-hand side is A-measurable and A contains all null events inF, the left-hand side is also A- measurable for a.e. t e [0, T]. The continuityof the samples paths of {Ft; t > 0} leads to the conclusion that this process is.y,- measurable for every t. For unbounded X, we use the localization techniqueemployed in the proof of Proposition 2.6.

We shall not continue to deal explicitly with the case of absolutely con-tinuous <M> and X e .29, but all results obtained for X e .29* can be modifiedin the obvious way to account for this case. In later applications involvingstochastic integrals with respect to martingales whose quadratic variationsare absolutely continuous, we shall require only measurability and adaptivityrather than progressive measurability of integrands.

2.12 Problem. Let W = {W, A; 0 < t < co} be a standard, one-dimensionalBrownian motion, and let T be a stopping time of {A } with ET < co. Provethe Wald identities

E(WT) = 0, E(WT) = ET

(Warning: The optional sampling theorem cannot be applied directly becauseW does not have a last element and T may not be bounded. The stoppingtime t n T is bounded for fixed 0 < t < co, so E(W,,,T) = 0, E(W t2 T) =E(t n T), but it is not a priori evident that

(2.25) lim E(W( A T) = EWT, lim E(14 T) = E(WT2)-)t-co

2.13 Exercise. Let W be as in Problem 2.12, let b be a real number, and let Tbbe the passage time to b of (2.6.1). Use Problem 2.12 to show that for b 0 0,we have ET = co.

C. A Characterization of the Integral

Suppose M = {M 0 < t < } and N = {N.F.,; 0 < t < co} are in M,and take X e 2 *(M), Y e Y*(N). Then IM(X) .A SO Xs dMs, on A SO YS dNs

are also in .I12 and, according to (2.19),

</"(X)>, = f t XJ d<M>, <IN(Y)>, = f r Y2 d<N>.

We propose now to establish the cross-variation formula

(2.26) </"(X), /N(Y)>, = J r X Yd <M,N) ; t 0, P-a.s.0

If X and Y are simple, then it is straightforward to show by a computationsimilar to (2.18) that for 0 < s < t < co,

(2.27) E[e (X) - InX))(INY) -Is Y))1,9;]

= E[1, X d<M, N> gd; P-a.s.

This is equivalent to (2.26). It remains to extend this result from simpleprocesses to the case of X e -99*(M), Y e.29*(N). We carry out this extensionin several stages, culminating in Propositions 2.17 and 2.19 with a very usefulcharacterization of the stochastic integral.

2.14 Proposition (An Inequality of Kunita & Watanabe (1967)). If M, N e dr2,X e Y*(M), and Y e Y*(N), then a.s.

E Ysl d4, (fXs

d<M>s)1/21/2

(f d<N>s) ; 0 < t < co,

where 4, denotes the total variation of the process A <M, N> on [0,s].

PROOF. According to Problem 1.5.7 (iv), (w) is absolutely continuous withrespect to cp(co) A A<M> <N>] (co) for every co e n with P(C1) = 1, and forevery such w the Radon-Niko4m theorem implies the existence of functionsfie , w): [0, co) -> 111; i = 1, 2, 3, such that

<M> (w) = (s, cicos(0), <N> (co) = f2(s,(0)403(c0),

,(co) = <M, N>,(co) = f t f3 (s, co) dcps(w); 0 < t < co.

Consequently, for a, /3 elA and w e C satisfying P(Cap) = 1, we have

0 < <aM + 13N>,(co) - <aM + 131%1>(co)

= (cef,(s, w) + 2aflf3(s, w) + fl2f2(s, w)) chps(co); 0 < u < t < co.

This can happen only if, for every co en, there exists a set 7;3(co)e a([o, 00))with IT.ocodcpi(co) = 0 and such that

(2.28) c(2./1(t, co) + 20f3(t, co) + ig2f21t, (0) 0

holds for every t T(w). Now let a _4 n,,EQQ,, poi, A U243 7 (04 sothat P(C') = 1, $T(,,,) dpi(w) = 0; V wen. Fix wen; then (2.28) holds for everyt T(w) and every pair (a, /1) of rational numbers, and thus also for everyt T(w), (1,13)e R2; in particular,

a 2 I XtMl 2f1 (t, 2ocl X ,(a) Y(0)11 f 3(t, co)i + I Y*012 f 2(t, (0) 0;

V t T(w).

Integrating with respect to dp, we obtain

s12 d<m>s 20coz2 YI C14 + 1Y 12 d<N>5 0; 0 < t < CO,s s s

0 0 0

almost surely, and the desired result follows by a minimization over a.

2.15 Lemma. If M, N e diec2, X e .29*(M), and {X(")};'11 S .29*(M) is such thatfor some T > 0,

then

lim If Xi(in) - 2 d<M> = 0; a.s. P,n co o

lira <I(X(")),N>, = (1(X), N>,; 0 < t < T, a.s. P.n -0 co

PROOF. Problem 1.5.7 (iii) implies for 0 < t < T,

KI(X(")) - I(X),N>,12 <I(X" - X)>, <N>,

1XL° Xu12 d<M>.' <N>r.

2.16 Lemma. If M, N 6.41'2 and X e S9 *(M), then

(2.29) <Im (X), N>, = J t Xud<M,N>u; VOt< op, a.s.

PROOF. According to Lemma 2.7, there exists a sequence {X"} ,;°_, of simpleprocesses such that

sup limT>0 co

E JT IXL ")- XI2d <M>u =O.

Consequently, for each T > 0, a subsequence {;Y-(n)},;°_, can be extracted forwhich

T

lim .1 - X12 d<M>u = 0, a.s.n-:o

But (2.26) holds for simple processes, and so we have

, = f i g;,") d<M, N>; 0 S t< T0

almost surely; letting n -* co we obtain (2.29) from Lemma 2.15 and theKunita-Watanabe inequality (Proposition 2.14).

2.17 Proposition. If M, N e dll, X e_r*(M), and Y e 2*(N), then the equiva-lent formulas (2.26) and (2.27) hold.

PROOF. Lemma 2.16 states that d<M, IN(Y)>,, = y, d<M, NX. Replacing Nin (2.29) by /N(Y), we have

<Pm (X), IN (Y)>, = f t X,,d<M, IN(Y)>o

=ft X,,Y,,d<M, N>u; t _. 0, P-a.s.o

2.18 Problem. Let M = {M .9;; 0 S t < co} and N, = {Ng",; 0 < t < co} bein dr2 and suppose X e .9':,(M), Y e ..rt(N). Then the martingales /"'(X), /NY)are uniformly integrable and have last elements km(X), /NY), the cross-variation <IM(X), /N(Y)>, converges almost surely as t -) co, and

E[P0M,(X)IZ (Y)] = E<Im (X), INY)X0 = E c° X, Y, d <M , N X.o

In particular,

E(fX, dM,)2 = E X,2 d<MX.o o

2.19 Proposition. Consider a martingale M e ..e2 and a process X E ..r*(M). Thestochastic integral Pm (X) is the unique martingale 40 e.11z which satisfies

(2.30) <01), N>, = f t X d<M, N>; 0 < t < co, a.s. P,o

for every N e dic2.

PROOF. We already know from (2.29) that (1) = /m(X) satisfies (2.30). Foruniqueness, suppose (1) satisfies (2.30) for every N e ./11. Subtracting (2.29) from(2.30), we have

<0 - /M(X), NX = 0; 0 < t < oo, a.s. P.

Setting N = 41 - IM (X), we see that the continuous martingale (1) - /m(X)has quadratic variation zero, so (1:0 = IM(X).

Proposition 2.19 characterizes the stochastic integral IM(X) in terms of themore familiar Lebesgue-Stieltjes integral appearing on the right-hand side of(2.30). Such an idea is extremely useful, as the following corollaries illustrate.

In the shorthand "stochastic differential" notation, the first of these statesthat if dN = X dM, then Y dN = X Y dM.

2.20 Corollary. Suppose M e .ill, X E 2*(M), and N -1=-' IM(X). Suppose furtherthat Y e ..r*(N). Then X Y e ..?*(M) and IN(Y) = IM(X Y).

PROOF. Because <N>, = Po Xs2d<M>s, we have

E

fT T

X,2Y,2 d<M>, = E j l',2 d<Al>, < coo o

for all T > 0, so X Y e 1 "(M). For any F/ e .1,1, (2.26) gives d<N, IV>, =Xsd<M,R> and thus

i

</m(X Y), R>, = f XsYsd<M, R>,o

=ft Ysd<N,R>, = <IN (Y), R>,.o

According to Proposition 2.19, IM(X Y) = IN(Y).

2.21 Corollary. Suppose M, R e ./e2, X e Y*(M), and 2 e 2 *(1C1), and thereexists a stopping time T of the common filtration for these processes, such thatfor P-almost every co,

X, A T(04(W) = it A T(c))(W), Mt A T(u))(W) = lilt A T(u)/(W); 0 < t < CO.

Then

I7 T(,)(X)(co) = Ii 7-00(50(w); 0 5 t < 00, for P-a.e. w.

PROOF. For any N e ..11, we have <M - fa, N>t A T = 0; 0 < t < co, and so(2,29) implies </m(X) - Im (I), N>, A T = 0; 0 < t < co. Setting N = IM(X) -Im (1) and using Problem 1.5.12, we obtain the desired result. 111

D. Integration with Respect to Continuous, Local Martingales

Corollary 2.21 shows that stochastic integrals are determined locally by thelocal values of the integrator and integrand. This fact allows us to broadenthe classes of both integrators and integrands, a project which we nowundertake.

Let M = {M Ft; 0 5_ t < co} be a continuous, local martingale on aprobability space (a, ,, P) with Mo = 0 a.s., i.e., M e dr I" (Definition 1.5.15).Recall the standing assumption that {F,} satisfies the usual conditions. We

define an equivalence relation on the set of measurable, {g,}-adapted pro-cesses just as we did in the paragraph preceding Definition 2.1.

2.22 Definition. We denote by 91 the collection of equivalence classes of allmeasurable, adapted processes X = {X.Fi; 0 < t < col satisfying

(2.31) P[f d<M>, < co] = 1 for every T e [0, co).

We denote by 91* the collection of equivalence classes of all progressivelymeasurable processes satisfying this condition.

Again, we shall abuse terminology by speaking of 91 and g* as if they wereclasses of processes. As an example of such an abuse, we write g* g g, andif M belongs to lfZ (in which case both 2' and 2)* are defined) we write

g g and ..c," g".We shall continue our development only for integrands in g*. If a.e. path

t H <M>,(w) of the quadratic variation process <M> is an absolutely con-tinuous function, we can choose integrands from the wider class g. The readerwill see how to accomplish this with the aid of Remark 2.11, once we completethe development for g".

Because M is in .41"", there is a nondecreasing sequence {S},°=, of stoppingtimes of {A}, such that lim, S = oo a.s. P, and kmtAs,,,,Ft; 0 < t < col isin .41. For X e g", one constructs another sequence of bounded stoppingtimes by setting

R,,(w) n A inf {0 t < co; XR(o)d <M>s(co) n}.

This is also a nondecreasing sequence and, because of (2.31), R = co,a.s. P. For n > 1, co e Q, set

(2.32) T(w) = R(w) A Sn(w),

(2.33) M")(w) A Mt A "jai)/ )011)(W) = Xt(01 TWO> t} ; t <Then WI) e and X(")e ..9"(M(")), n > 1, so InX(")) is defined. Corollary2.21 implies that for 1 < n < m,

/r)(X(")) = /r""(X(''')), 0 < t < T,

so we may define the stochastic integral as

(2.34) /,(X) A /r)(X(")) on {0 < t < T}.

This definition is consistent, is independent of the choice of {S},fi and deter-mines a continuous process, which is also a local martingale.

2.23 Definition. For M e ../1"°e and X e 9", the stochastic integral of X withrespect to M is the process 1(X) {I,(X),,,; 0 < t < col in de.'" defined by(2.34). As before, we often write co Xs di I 1 instead of /,(X).

When M e di'mc and X e4)*, the integral I (X) will not in general satisfyconditions (2.13)-(2.16), (2.21)-(2.23), or (2.27), which involve expectations atfixed times or unrestricted stopping times. However, the sample path properties(2.12), (2.17), (2.19), (2.24), and (2.26) are still valid and can be easily proved bylocalization. We also have the following version of Proposition 2.19.

2.24 Proposition. Consider a local martingale M e dr' and a process X ego*(M). The stochastic integral IM(X) is the unique local martingale (De dr"which satisfies (2.30) for every N e de, (or equivalently, for every N e

2.25 Problem. Suppose M, N e d' and X e .9* (M) n .9* (N). Show that forevery pair (a, /3) of real numbers we have

rm+PN(X) = a/m(X) + 08(X).

2.26 Proposition. Let M e {X(")};,°_,, g .9* (M), X e g* (M) and supposethat for some stopping time T of {F,} we have II; I XI") -X, I2 d <M>, = 0,in probability. Then

sup0.1EST

X(s") dM, - J X,dMs11-.00 *

PROOF. The proof follows immediately from Problem 1.5.25 and Proposition2.24.

2.27 Problem. Let M e dic'" and choose X e g*. Show that there exists asequence of simple processes {X(")},;°_, such that, for every T > 0,

T

lim X: ")- X,I2 d<M>, =0n- o0 0

and

lim sup I /,(X(")) - 4(X)I = 0n- 051ST

hold a.s. P. If M is a standard, one-dimensional Brownian motion, then thepreceding also hold with X e .9.

2.28 Problem. Let M = W be standard Brownian motion and X e g. Wedefine for 0 <s<t<co

(2.35) Cf(X) A f t XdW -2-1

sdu; (,(X) CP(X)

The process lexp((,(X)).Ft; 0 < t < co} is a supermartingale; it is a martin-gale if X e.ro.

Can one characterize the class of processes X e Y*, for which the exponen-tial supermartingale {exp (MX)), .9;; 0 t < co) of Problem 2.28 is in fact a

martingale? This question is at the heart of the important result known as theGirsanov theorem (Theorem 5.1); we shall try to provide an answer in Section 5.

2.29 Problem. Let W be a standard Brownian motion, e a number in [0, 1],and ri = Ito,t..., tn, a partition of [0, t] with 0 = to < t, < < tm = t.Consider the approximating sum

(2.36)rn -1

um -4 E -1=0

for the stochastic integral Sto Ws dWs. Show that

(2.37) lim Sc(I1) =2-1 14/,' + (c - -2) t,

1

where the limit is in L2. The right-hand side of (2.37) is a martingale if andonly if e = 0, so that W is evaluated at the left-hand endpoint of each interval[t1, t 1] in the approximating sum (2.36); this corresponds to the Ito integral.With e = i we obtain the Fisk-Stratonovich integral, which obeys the usualrules of calculus such as ro Ws 0 dW, = 1147,2; we shall have more to say aboutthis in Problems 3.14, 3.15. Finally, e = 1 leads to the backward Ito integral(McKean (1969), p. 35). The sensitivity of the limit in (2.37) to the value of eis a consequence of the unbounded variation of the Brownian path.

We know all too well that it is one thing to develop a theory of integrationin some reasonable generality, and a completely different task to compute theintegral in any specific case of interest. Indeed, one cannot be expected torepeat the (sometimes arduous) process which fortunately led to an answer inthe preceding problem. Just as we develop a calculus for the Riemann integral,which provides us with tools necessary for more or less mechanical computa-tions, we need a stochastic calculus for the Ito integral and its extensions. Wetake up this task in the next section.

2.30 Exercise. For M a die' Inc, X ego *, and Z an .Fs-measurable random vari-able, show that

ZX.dM = Z ft X.dM.; s < t < co, a.s.S

3.3. The Change-of-Variable Formula

One of the most important tools in the study of stochastic processes of themartingale type is the change-of-variable formula, or Ito's rule, as it is betterknown. It provides an integral-differential calculus for the sample paths ofsuch processes.

3.3. The Change-of-Variable Formula 149

Let us consider again a basic probability space (0, P) with an associatedfiltration {,F,} which we always assume to satisfy the usual conditions.

3.1 Definition. A continuous semimartingale X = {X A; 0 < t < oo} is anadapted process which has the decomposition, P a.s.,

(3.1) X, = X0 + M, + Bt; 0 < t < co,

where M = {M 0 < t < co} e .11'''" (Definition 1.5.15) and B = {B.9",;0 < t < co} is the difference of continuous, nondecreasing, adapted processes{A,±,,,; 0 < t < co}:

(3.2) B, = A;F. -,c; 0 < t < GO,

with A,4-- = 0, P a.s. We shall always assume that (3.2) is the minimal de-composition of B; in other words, A,'" is the positive variation of B on [0, t]and 2=1- is the negative variation. The total variation of B on [0, t] is then

A,+ .

The following problem discusses the question of uniqueness for the de-composition (3.1) of a continuous semimartingale.

3.2 Problem. Let X = {X,,; 0 < t < co } be a continuous semimartingalewith decomposition (3.1). Suppose that X has another decomposition

X, = + +111,; 0 < t < co,

where M e../1'1" and is a continuous, adapted process which has finite totalvariation on each bounded interval [0, t]. Prove that P-a.s.,

M, = -M B, = 0<t<co.

A. The ItO Rule

Ito's formula states that a "smooth function" of a continuous semimartingaleis a continuous semimartingale, and provides its decomposition.

3.3 Theorem. (Ito (1944), Kunita & Watanabe (1967)). Let f:[18-[18 be afunction of class C2 and let X = {X A; 0 < t < co} be a continuous semi-martingale with decomposition (3.1). Then, P-a.s.,

(3.3) f(X,) = f(X0)+ f'(Xs)dM, + f f'(Xs)dBs

f+2 0

f "(Xs)d<M> s, 0< t< co .

3.4 Remark. For fixed co and t> 0, the function Xs(w) is bounded for 0 < s 5 t,so f ' (X s(co)) is bounded on this interval. It follows that ft° f ' (X s) dM s is defined

as in the last section, and this stochastic integral is a continuous, localmartingale. The other two integrals in (3.3) are to be understood in theLebesgue-Stieltjes sense (Remark 1.4.6 (i)), and so, as functions of the upperlimit of integration, are of bounded variation. Thus, {f (X,),,; 0 < t < col isa continuous semimartingale.

3.5 Remark. Equation (3.3) is often written in differential notation:

(3.3)' df(X,) = f '(X,)dM, + f ' (X,) dB, +2-1 f "(X,) d Of> ,

= f(Xt) dXt + -2 f "PO d 00,, 0 < t < oo.

This is the "chain-rule" for stochastic calculus.

PROOF OF THEOREM 3.3. The proof will be accomplished in several steps.

Step 1: Localization. In the notation of Definition 3.1 we introduce, for eachn _>_ 1, the stopping time

0; if I X0 I .. n,

T = {inf t 0; I{ Mt 1 n or 4 _. n or <M>, n}; if I Xo I < n,

co; if I Xo I < n and { ... } = 0.

The resulting sequence is nondecreasing with lim, T. = co, P-a.s. Thus, if--i T ,we can establish (3.3) for the stopped process X:0 .4--, X : t > 0, then we

obtain the desired result upon letting n co. We may assume, therefore, thatX0(w) and the random functions M ,(co), 13,(w), and <M>,(w) on [0, co) x ("/ areall bounded by a common constant K; in particular, M is then a boundedmartingale. Under this assumption, we have I X*0)1 < 3K; 0 < t < co, wee,so the values of f outside [ - 3K, 3K] are irrelevant. We assume without lossof generality that f has compact support, and so f, f', and f" are bounded.

Step 2: Taylor expansion. Let us fix t > 0 and a partition II = {to,ti,...,t,,,} of [0, t], with 0 = to < t, < < t, = t. A Taylor expansion yields

../(Xt) -f(x0)= E {f(xt,)- fix,_,)}

k=1

. l -= kE f'(x,_,)(x,- x,_,)+

k7, E f "(nk)(xt, - x(,_,)2,

=1 =1

where 1h(w) = X,k_t(w) + 0(co)(X,k(w) - X,k_i(w)) for some appropriate Ok (CO)satisfying 0 < Ok(w) < 1, co E n. We conclude that

(3.4) ffX() - fiXo) = -11 (II) + J2(1-1) + 4J3(H),

where

mJ1(1-1) s E f-

- Btk ,),E'

.12(11) A Ek=1

J3(11) E flnk)(xt, - x( 1)2.k=1

It is easily seen that .11(11) converges to the Lebesgue-Stieltjes integralfof '(Xs)dBs, a.s. P, as the mesh 111111 = maxi <,n 1 tk - tk_11 of the partitiondecreases to zero. On the other hand, the process

Ys(w) 9 f'(Xs(co)); 0< s< t, (yen,

is in ..r* (adapted, continuous, and bounded); we intend to approximate it bythe simple process

Kn(w) f'(x0(0)1{0}(S)

Indeed, we have E1,2(Y" - Y) = EP° 1 Ys' - K12 d<M>, 0 as 111111 - 0, bythe bounded convergence theorem, and so

J2(11) = YsndMs KdMsJo

in quadratic mean.

Step 3: The quadratic variation term. J3(II) can be written as

J3(11) = 4(11) + J5(11) + J6(11),

where

m

4(11) E flIk)(13-k=1

J5(r1) 'A 2m

flnk)(13,- Aki)(111,- m(k_,),

J6(11) A E f"(N)(Altk-k=1

Because B has total variation bounded by K, we have

14(11)1 + 1J5(11)1

-21(11f "11 co ( max I Bfk - Bik_ii + max I Mtk II)15k5nt 1<k5m

and, thanks to the continuity of the processes B and M, this last term convergesto zero almost surely as uno 0 (as well as in L (II, "", P), because of thebounded convergence theorem). As for J6(II), we define

.'A E flx,_,)(mtk- mik 1)2

k=1

and observe

14(11) - J6(n)I < vi(2)(n)- max If"(h)- f"(x,_1)I,1 Sk <rn

where V,(2)(11) is the quadratic variation of M over the partition n (c.f.Theorem 1.5.8 and the discussion preceding it). According to Lemma 1.5.9and the Cauchy-Schwarz inequality,

EIJ:(11) - J6(II)I .,/48K4.1E ( max1 'icSm

2

If"(nk) - f."(Xti,_ i )1 )

and this is seen to converge to zero as II nil -+ 0 because of the continuity ofthe process X and the bounded convergence theorem. Thus, in order toestablish the convergence of the quadratic variation term J3(11) to the integralSI, f "(Xs) d<M>, in 1.1 (SI, .97, P) as 111111 - 0, it suffices to compare J:(11) tothe approximating sum

.17(11) A i f"(x,_,)(<m),k- <m>,_,)k=i

Recalling the discussion just before Lemma 1.5.9, we obtain

WWI) -.17(1)12m 2

= EL f"(xi, )1(14,,- 14,_,)2 -(04),- <M>tk_,)}

k=1

= E[X___1 [f "(Xtk)]2 - Alti,_,)2 - (<m)t - <m>tk_1)}2]

. .211f"ii2 'E[I, (M -M )4 + E Km\ - <m>, i )21

k=1 k=1

max (Oki/Xi, - <M>tk id'211.i" II 2.D'E [ Vt(4(1-1) + OW >t1<k <m

and Lemma 1.5.10 together with the bounded convergence theorem showsthat the last term in the preceding equations goes to zero as 1111 11 -, 0. Sinceconvergence in L2 implies convergence in L1, we conclude that

in LI (11,,, P).J3(11) 11-=-+.0 Jo f"(Xs)d<M>,

Step 4: Final Touches. If Inool-_, is a sequence of partitions of [0, t] with1111 °'l _-77- o, then for some subsequence In(^01;10_1 we have, P-a.s.,

ilim J10^0) = f f'(Xs)d135,k-kco 0

lim .12(-100) = f f'(Xs)dMs,k-oo 0

lim J3(fl(k)) = ft f"(Xs)d<M>sk-.0o 0

Thus, passing to the limit in (3.4), we see that (3.3) holds P-a.s. for each0 < t < co. In other words, the processes on the two sides of equality (3.3) aremodifications of one another. Since both of them are continuous, they areindistinguishable (Problem 1.1.5). I=1

We have the following, multidimensional version of Ito's rule.

3.6 Theorem. Let {M, A (Mr), , Mr)), ,F,; 0 < t < col be a vector of localmartingales in .11`''", {B, A (BP), ,Br)), g;; 0 5 t < col a vector of adaptedprocesses of bounded variation with Bo = 0, and set X, = X0 + M, + B,;0 < t < co, where X0 is an go- measurable random vector in Rd. Let f(t,x):[0, co) x Rd R be of class C"2. Then, a.s. P,

t d

(3.5) f(t, X,) = f(0, X0) + f - f(s, Xs)ds + E f(s, Xs)dBli)o at ox,

d a+ E xodmv

1=1 Oxi

1d d 32

+5EE f(s, X s) d <M"), Mul>s, 0 < t < co.- i=1 j=1

0

3.7 Problem. Prove Theorem 3.6.

3.8 Example. With M = W = Brownian motion, X0 = 0, B, = 0 and f(x) =x2, we deduce from (3.3):

Wi2 = 2 I WsdW, t.

Compare this with Problem 2.29.

3.9 Example. Again with M = W = Brownian motion, let us consider X EYand recall the exponential supermartingale of Problem 2.28:

Z, exp(C,); 0 t < co

where = C(X) of (2.35). We now check by application of ItO's rule that thisprocess satisfies the stochastic integral equation

(3.6) Z, = 1 + ZsX,dWs; 0 < t < co.

Indeed, {(t; .9;; 0 < t < co} is a semimartingale, with local martingale partYo X, dW, and bounded variation part B, -z f o X,2 ds. With f(x) = ex,

we have

Z` = f(Cr) = f(C0) + f t f '(C)dN, + ft f'gs)dB, +1 ft f"((s)d<N>,o o 2 0

ft ! f= 1 + f Z,X,d4 + Zs + Z,X2 dso o 2 2 0

= 1 + f' Zs Xs d W.0

The replacement of dN, by X,dW, in this equation is justified by Corollary2.20 (actually, the extension of Corollary 2.20 to allow for the present case ofM = W and X e1). It is usually more convenient to perform computationslike this using differential notation. We write

= X, dW, - IX,2 dt,

and, to reflect the fact that the martingale part of ( has quadratic variationwith differential X,2 dt, we let (c/(,)2 = X,2 dt. One may obtain this from theformal computation

(c/(,)2 = (X, dW, - 1-X,2 dt)2

= X,2(dW,)2 - X,3 dW, dt + 4V(dt)2

= X,2 dt,

using the conventional "multiplication table"

dt dW,

dt 0 0 0dW,dW

00

dt0

0dt

where W, W are independent Brownian motions (recall Problem 2.5.5). Withthis formalism, Ito's rule can be written as

df((t) = f ' (Cdcg, + -if " (C,) (dCt)2,

and with f(x) = ex, we obtain

dZ, = Z,X, dW, - 4Z,X,2 dt + dt

= Z,X,dW,.

Taking into account the initial condition Z0 = 1, we can then recover (3.6).

3.10 Problem. With {Z1; 0 5 t < co} as in Example 3.9, set Y = 1/4 0 5t < oo, which is well defined because P[info ,, < T 4 > 0] = P[info T Ct >

-OA = 1. Show that Y satisfies the stochastic differential equation

dy = Y, X,2 dt -Y, X, dW Yo = 1.

3.11 Example. One of the motivating forces behind the ItO calculus was adesire to understand the effects of additive noise on ordinary differentialequations. Suppose, for example, that we add a noise term to the linear,ordinary differential equation

4(t)= a(t)(t)

to obtain the stochastic differential equation

4 = a(t)e, dt + b(t)dW

where a(t) and b(t) are measurable, nonrandom functions satisfying

JTT

la(t)Idt + f b2(t) dt < oo; 0 < T < co,1o o

and W is a Brownian motion. Applying the HO rule to XP)XP) with )01) 4exp[Sto a(s) ds] and X:2) = 4 + Po b(s) exp[ -sso a(u) du] dWs, we see that , =XP'XP) solves the stochastic equation. Note that is well defined because,for 0 < T < co:

s T T

ob (s)exp [ -2 f

oa(u)du]ds exp[2 f ja(u)I did .1 b2(s)ds < co.

o o

A full treatment of linear stochastic differential equations appears in Sec. 5.6.

3.12 Problem. Suppose we have two continuous semimartingales

(3.7) X, = Xo + M, + B Y, = Yo + N, + c; 0 < t < oo ,

where M and N are in dr.'" and B and C are adapted, continuous processesof bounded variation with Bo = Co = 0 a.s. Prove the integration by partsformula

(3.8) t Xsdy= X, Y, -X 0 Yo -f t ydX - <M, N>,.fo o

The Ito calculus differs from ordinary calculus in that familiar formulas,such as the one for integration by parts, now have correction terms such as<M, N>, in (3.8). One way to avoid these corrections terms is to absorb theminto the definition of the integral, thereby obtaining the Fisk-Stratonovichintegral of Definition 3.13. Because it obeys the ordinary rules of calculus(Problem 3.14), the Fisk-Stratonovich integral is notationally more conve-nient than the HO integral in situations where ordinary and stochastic calculusinteract; the primary example of such a situation is the theory of diffusions on

differentiable manifolds. The Fisk-Stratonovich integral is also more robustunder perturbations of the integrating semimartingale (see subsection 5.2.D),and thus a useful tool in modeling. We note, however, that this integral isdefined for a narrower class of integrands than the HO integral (see Definition3.13) and requires more smoothness in its chain rule (Problem 3.14). Wheneverthe Fisk-Stratonovich integral is defined, the Ito integral is also, and the twoare related by (3.9).

3.13 Definition. Let X and Y be continuous semimartingales with decom-positions given by (3.7). The Fisk-Stratonovich integral of Y with respect toXis

1(3.9) 01 lc 0 dX, -A Ysdill, + YsdB, + 2 <M, N>,; 0 t < co,o o

where the first integral on the right-hand side of (3.9) is an Ito integral.

3.14 Problem. Let X = , V)) be a vector of continuous semi-martingales with decompositions

= Xg) + Ati)+ BP; i = d,

where each Mo) e I" and each B(1) is of the form (3.2). If f: l -R is ofclass C3, then

(3.10) f(x,)= fixo) + ft ,a fajodxr.=i 0 ox;

3.15 Problem. Let X and Y be continuous semimartingales and n ={t0,t1,...,tm} a partition of [0, t] with 0 = t0 < t1 < < = t. Show thatthe sum

m-i

Y'+` + 2 IC' (X`'` X`')

converges in probability to Po Yso dX, as 111111 0.

B. Martingale Characterization of Brownian Motion

In the hands of Kunita and Watanabe (1967), the change-of-variable formula(3.5) was shown to be the right tool for providing an elegant proof of P. Levy'scelebrated martingale characterization of Brownian motion in Rd. Let us recall

Ai); jr); F; 0here that if {B, < t < col is a d-dimensional Brownianmotion on (0, P) with P[B0 = 0] = 1, then <01°, Bul>, = bkit; 1 < k, j < d,0 < t < cc (Problem 2.5.5). It turns out that this property characterizesBrownian motion among continuous local martingales. The compensatedPoisson process with intensity A = 1 provides an example of a discontinuous,

square-integrable martingale with <M>, = t (c.f. Example 1.5.4 and Exercise1.5.20), so the assumption of continuity in the following theorem is essential.

3.16 Theorem (P. Levy (1948)). Let X = {X, = (V), . , V)), 0 < t < co}be a continuous, adapted process in Rd such that, for every component 1 < k < d,the process

Ar) iylk) 4k); 0 < t < 00,

is a continuous local martingale relative to {,F,}, and the cross-variations aregiven by

(3.11) <AP), 0-5>t = Skit; 1 < k, j < d.

Then {X1,.97,; 0 < t < oo} is a d-dimensional Brownian motion.

PROOF. We must show that for 0 < s < t, the random vector X, -Xs isindependent of and has the d-variate normal distribution with mean zeroand covariance matrix equal to (t - s) times the (d x d) identity. In light ofLemma 2.6.13, it suffices to prove that for each u e Rd, with i = I-1,

(3.12) E[ei( = e-(1/2)iiuli2o-s), a.s. P.

For fixed u = (u , u) e Rd, the function f(x) = ei("'') satisfies

a a2f(x) = f(x) =

ax. ax. axJ k

Applying Theorem 3.6 to the real and imaginary parts off, we obtain

d(3.13) ei( xt) E _1 E uj el("'x ) dv.

j=1 s 2.1=1 s

Now If(x)1 < 1 for all xe Rd and, because <Mo>, = t, we have Mti)Thus, the real and imaginary parts of {r, eio"-)divt-h, A; 0 < t < co} are notonly in .1r1, but also in .1'. Consequently,

E[ f t eR.,xv) dmy)

For A E., we may multiply (3.13) by e-1".)1A and take expectations toobtain

= 0, P-a.s.

E[ein,,xt-xoiA] = P(A) - -2 E[ei(". x '4) 1A] dv.'

This integral equation for the deterministic function t E[er( ",x,-X.)1A] is

readily solved:E[eu,xi-xd.A, = P(A )e-(1124.112(i-s), V e

and (3.12) follows.

3.17 Exercise. Let W = (W('), W,(2), Wi(3)) be a three-dimensional Brownianmotion starting at the origin, and define

3

X = n sgn(WP),

Mu) wi(i), m:2) pvt(2) and M:3)= X W,(3).

Show that each of the pairs (M(1), M(2)), (M(1), M(3)) and (M(2), M(3)) is atwo-dimensional Brownian motion, but (Mu), m(2), m(3)) is not a three-dimensional Brownian motion. Explain why this does not provide a counter-example to Theorem 3.16, i.e., a three-dimensional process which is not aBrownian motion but which has components in d)" and satisfies (3.11).

3.18 Problem. Let W = 1W, = (Wu), ... , W(d)), g,; 0 < t < cdo I be a d-dimensional Brownian motion starting at the origin, and let Q be a d x dorthogonal matrix (QT = 12-1). Show that 11/, A QW, is also a d-dimensionalBrownian motion. We express this property by saying that "d-dimensionalBrownian motion starting at the origin is rotationally invariant."

C. Bessel Processes, Questions of Recurrence

Another use of the P. Levy Theorem 3.16 is to obtain an integral representa-tion for the so-called Bessel process. For an integer d _. 2, let W = { W, =(IV,(1), ... , Wt(d)), ,i; 0 < t < col, {Px1,.. Rd be a d-dimensional Brownian familyon some measurable space (SI, F). Consider the distance from the origin

(3.14) R, A II Hitli = Wt(1)/2 + + (Wi(d))2; 0 < t < oo,

[R = Ilxli]so Px o = 1. If x, y e Rd and Ilxil = OA, then there is an orthogonalmatrix Q such that y = Qx. Under Px, (V = 111/, A QW ..F,; 0 < t < ool is ad-dimensional Brownian motion starting at y, but II afili = II Will, so for anyF EM(C[0, co)), we have

(3.15) Px[R.e F] = Px[110<11 e n = PY[R.e F].

In other words, the distribution of the process R under P depends on x onlythrough Ilx11-

3.19 Definition. Fix an integer d > 2, and let W = { W, .57;,; 0 < t < co},{ Px}x.R. be a d-dimensional Brownian family on (SI, ,). The process R ={R, = 11W,II, ..F,; 0 < t < co} together with the family of measures {P' },>0 .

{Po .o.....o»>0i (S/,.F) is called a Bessel family with dimension d. For fixedr > 0, we say that R on (SI, g", Pr) is a Bessel process with dimension d startingat r.

3.20 Problem. Show that for each d > 2, the Bessel family with dimension dis a strong Markov family (where we modify Definition 2.6.3 to account forthe state space [0, co)).

3.21 Proposition. Let d > 2 be an integer and choose r > 0. The Bessel processR with dimension d starting at r satisfies the integral equation

' -(3.16) R, ---- r + fo

d

2R,1

ds + Bt; 0 < t < co,

where B = {B..; 0 5 t < co} is the standard, one-dimensional Brownianmotion

(3.17) B Q > 13(1) Iwith BY) A' We dWP); 1 < i < d.0 R,

PROOF. We use the notation of Definition 3.19, except we write P instead ofP. Note first of all that R, can be at the origin only when Ws") is, and so theLebesgue measure of the set {0 < s < t; Rs= 0} is zero, a.s. P (Theorem 2.9.6).Consequently, the integrand (d - 1)/2R3 in (3.16) is defined for Lebesguealmost every s, a.s. P.

Each of the processes B(i) in (3.17) belongs to .At e2, because

' 2

E f (-1W(i)) ds < t; 0 < t < co.

0 Rs

Moreover,

J

1 . .r 1

<Bo) Bo >, = wscowsco d<wo, Ivo> = 45_____ wo Ivo) dso R52

5 u D2s s ,0 ls

which implies

, =-- E <Bo), = t,

and we conclude from Theorem 3.16 that B is a standard, one-dimensionalBrownian motion.

It remains to prove (3.16). A heuristic derivation is to apply Ito's rule(Theorem 3.6) to the function f(x) -A 11x11 = xi + + ald [0, co, forwhich

,i

c5 xx=

4-1oxif(x)

axax;f(x)

Ilx,11 11391 5 i, j 5_ d,

hold on le \ {0}. Then R, = f(W,) and (3.16) follows from (3.5). The difficultyhere is that f is not differentiable at the origin, and so Theorem 3.6 cannot beapplied directly to f. This problem is related to our uneasiness about whetherthe integral in (3.16) is finite. Here is a resolution. Define

Yr 11147,112 =

and use Ito's rule to show thatd

`

= r2 + 2E Wswd147,(i)+ td.J.1=1

Let g(y) = .67, and for e > 0, define

3 3 1/-+ ,y y2;

gjy) = 4,./e 8e,fi

fi; y 6,

SO g, is of class C2 and lime4.0 g,(y) = g(y) for all y > 0. Now apply Ito's ruleto obtain

(3.18) ge(Y,) = ge(r2) + e(e) + .1,(e) + K,(e),1=1

where

1

44(0 [1{r >0 -+ 1{r.<02

1

Ve(3 - rs)] Wsoldwp,

Rs

d 1

jt(g) g 2Rs ds'

Kt(e) A f 1 ,[3d - (d + 2) ]ds.0 4/s 8

We now show that, as el 0, (3.18) yields (3.16). From the monotone conver-gence theorem, we see that

t -lim .1,(e) =

1

{y.,0}d2 -Rs

1

ds =d

0 2R51 ds, a.s.

We also have 0 < EK,(e) < (3d/4,./i)PoP[Y, < el ds. The probability in theintegrand is bounded above by

(yvs(2))2 < e] = -2itse-P212spdpd0,2ir .1;

Jo 0

1

and so the integral becomes, upon using Fubini's theorem and the change ofvariable = p/fs:

t

P[Y < &]ds 5 p -e-P212s ds dpo o s

=2 f p(f -e-4212 d)dp.o if

But now it is easy to see that this expression is o(ft) as t 10, using the ruleof l'HOpital. Therefore, lime,1,0 EK,(E) = 0. Finally

r 1 1 (3 - Ys )i2(,(0,2 dsE[BP) - IP)(En2 = E ifr.<0ks

= E it __ _171(3 _ 114(ws"))2 dsJo E /J /

P[Ys < E] ds = 0(10 asel O.

This establishes (3.16).

Let {R .9;; 0 < t < co} be a Bessel process with dimension d 2 startingat r > 0. Then, for each fixed t > 0, it is clear from (3.14) that P[R, > 0] = 1.A more interesting question is whether the origin is nonattainable:

(3.19) P[R, > 0; V 0 < t < oo] = 1.

The next proposition shows that this is indeed the case. Of course the situationis drastically different in one dimension, since P[I W:(1)1 > 0; V 0 < t < cc] = 0(Remark 2.9.7).

3.22 Proposition (Nonattainability of the Origin by the Brownian Path inDimension d > 2). Let d > 2 be an integer and r > 0. The Bessel process R withdimension d starting at r satisfies (3.19).

PROOF. It is sufficient to treat the case d = 2, since, for larger d,(w(1))2 (wt(d),

)2 can reach zero only if (W;"))2 + (147,(2))2 does.

We consider first the case r > 0. For positive integers k satisfying (1/k)k <r < k and n > 1, define stopping times

7', = inf T 0; Ri = (k-1y}

kS = inf {t 0; R, = k}, tk= TkASknn.

Because P-almost every Brownian path is unbounded (Theorem 2.9.23), wehave

(3.20)P[n {sk < cc} n thin

k=1 k-co

CO

Sk (X)}] = 1.

Using (3.16), apply Ito's rule to log(Rt) to obtain

tk 1

log R, = log r+ -dB,.o Rs

This step is permissible because log is of class C2 in an open interval contain-ing [(1/k)k, k]. For 0 < s < Tk, 11/Rsi is bounded, and since Tk is also bounded,we have E frel/Rs)dB, = 0. Therefore,

(3.21) log r = E [log Rrk] = - k(log k) P[Tk < Sk A n]

- tn<Si, A TO].+ (log k) P[Sk < Tk A n] + E[(log R)1

For every n 1, log R on {n < Sk A TO is bounded between - k(log k) andlog k. According to (3.20), as n co we have P[n < Sk A Ti] 0. Thus, lettingn -co in (3.21), we obtain

log r = - k(log k) P[Tk < Sk] + (log k) P[Sk < Tk].

If we divide by k(log k) and let k -* cc, we see that

(3.22) lim P[Tk < Sk] = 0.k -. co

Now set T = inf{t > 0; R, = 0 }, so that Tk < T for every k > 1. From (3.20)and (3.22), we have

(3.23) P[T < co] = lim PET < Sk] < lim P[Tk < Sk] = 0.k -.cok - co

It follows that P[R, > 0,V0 < t < co] = 1.Finally, we consider the case r = 0. Recalling the indexing of probability

measures in Definition 3.19, we have from Problem 3.20:

PqR, > 0; V s < t < co] = oE { pR,,R,i > 0; V 0 < t < co]} = 1

for any fixed s > 0, by what was just proved and the fact that P° [R, > 0] = 1.Letting a 4 0, we obtain the desired result.

3.23 Problem. Let R = {R,F,; 0 t < col be a Bessel process with dimen-sion d > 2 starting at r > 0, and define

m = inf R,.:)1<0,

(i) Show that if d = 2, then m = 0 a.s. P.(ii) Show that if d > 3, then m has the beta distribution

cy- 2P[m < c] = G 0 < c < r.

(Hint: Adapt the proof of Proposition 3.22. For (ii), an appropriate substitutefor the function f(r) = log r must be used.)

Proposition 3.22 says that, with probability one, a two-dimensionalBrownian motion never reaches the origin. Problem 3.23 (i) shows, however,that it comes arbitrarily close. By translation, we can conclude that for anygiven point ze R2, a two-dimensional Brownian path, with any starting posi-tion different from z, never reaches the point z, but does reach every disc ofpositive radius centered at z. In the parlance of Markov chains, one says that"every singleton is nonrecurrent," but that "every disc of positive radius isrecurrent." For a Brownian motion of dimension 3 or greater, Problem 3.23

(ii) shows that, once it gets away from the origin, almost every path of theprocess remains bounded away from the origin; this lower bound depends, ofcourse, on the particular path. Thus, d-dimensional spheres are nonrecurrentfor d-dimensional Brownian motion when d > 3.

3.24 Problem. Let R be a Bessel process with dimension d > 3 starting atr > 0. Show that P[Iim,-, R, = oc] = 1.

D. Martingale Moment Inequalities

As a final application of Ito's rule in this section, we derive some usefulbounds on the moments of martingales. The following exercise illustrates thetechnique.

3.25 Exercise. With W = W, .97,; 0 < t < oo a standard, one-dimensionalBrownian motion and X a measurable, adapted process satisfying

(3.24) E J T IX,12mdt < co

for some real numbers T > 0 and m > 1, show that

(3.25) f: X tdW,2m

(m(2m - 1))mrn-lE 1X,12'n dt.

(Hint: Consider the martingale {Mt = f o Xs dW, 0 < t < T }, and applyIto's rule to the submartingale IMt12m .)

Actually, with a bit of extra effort, we can obtain much stronger results. Weshall show, in effect, that for any M E be the increasing functions E(1 12m)

and E(<M>r), with the convention

(3.26) itit* -A max IMsl,o<st

have the same growth rate on the entire of [0, co), for every m > 0. This isthe subject of the Burkholder-Davis-Gundy inequalities (Theorem 3.28). Wepresent first some preliminary results.

3.26 Proposition (Martingale Moment Inequalities [Millar (1968), Novikov(1971)]). Consider a continuous martingale M which, along with its quadraticvariation process <M>, is bounded. For every stopping time T, we have then

(3.27) E(1/147-12"1) < C;E(<M)7); m > 0

(3.28) BmE(<M)7) < E(IMr12m); m > 1/2

(3.29) B,E(<M>7) _5 EUM1)21 5 CmE(<M>7); m > 1/2

for suitable positive constants B C., C,, which are universal (i.e., depend onlyon the number m, not on the martingale M nor the stopping time T).

PROOF. We consider the process

Y, -A 6 + t<M>, + M? = 6 + (1 + t)<M>, + 2 f Ms c/M 0 t < oo,o

where 6 > 0 and E z 0 are constants to be chosen later. Applying the change-of-variable formula to f(x) = xm, we obtain

(3.30)

Y,'" = 6m + m(1 + E) il Ysm-1 d<M>s + 2m(m - 1).1 Yr2M: d<M>o o

+ 2m f i YriMsdMs; 0 t < co.o

Because M, Y, and <M> are bounded and Y is bounded away from zero,the last integral is a uniformly integrable martingale (Problem 1.5.24). TheOptional Sampling Theorem 1.3.22 implies that E 11; Yri Ms dM, = 0, sotaking expectations in (3.30), we obtain our basic identity

T

(3.31) EYE = 6m + m(1 + OE 1 Ysm-1 d<M>so

T

+ 2m(m - 1)E f yr2ms2 d<m>s.o

Case 1: 0 < m < 1, upper bound: The last term on the right-hand side of(3.31) is nonpositive; so, letting 610, we obtain

T

(3.32) E[E<M>T + Mflm < m(1 + E)E J (E<M>s + M:)"1-1 d<M>,0

MO + c)cm -1 E I <M>;' d <M>so

= (1 + E)E'E(<M>7).

The second inequality uses the fact 0 < m < 1. But for such m, the functionf(x) = xm; x > 0 is concave, so

(3.33) 2"1-1(xm + ym) < (x + y)m; x > 0, y > 0,

E

and (3.32) yields: Em EVM >7) + E(IMTI2m) (1 + E)G E(< M >7), whence

(3.34) E(I MTI2m) [(1 + E) (-21 m - em1E(<M>7).

Case 2: m > 1, lower bound: Now the last term in (3.31) is nonnegative, andthe direction of all inequalities (3.32)-(3.34) is reversed:

E(IMT121 [(1 + E) (c/m-1

- EmiE(01)7).

Here, c has to be chosen in (0,(2'1 -

Case 3:1 < m < 1, lower bound: Let us evaluate (3.31) with E = 0 and thenlet 810. We obtain

(3.35) E(IMTI2m) = 2m(m - f IM,12(m-1)d<M>s.

On the other hand, we have from (3.33), (3.31):

2m-1[emE(<M>7) + E(5 + Mi)m]

E[E<M>T + + MD]mT

6" + ± (6 + Ms2)m-1 d<M>s-o

Letting S 4 0, we see that

(3.36) 2' CemE(01>7) + I MT 12m)] < m(1 + f I Ms12("1-1) d<M>s.

Relations (3.35) and (3.36) provide us with the lower bound

-1E(1/1/7.12m)

cm (12+:211 m 1) E1<m>71

valid for all c > 0.

Case 4: m > 1, upper bound: In this case, the inequality (3.36) is reversed, andwe obtain

E(IMTI2m) < Ern((1+ c)21-m

1)-1 E(<M>7!),2m - 1

where now c has to satisfy E > (2m - 1)2' - 1.This analysis establishes (3.27) and (3.28). From them, and from the Doob

maximal inequality (Theorem 1.3.8) applied to the martingale IMTAt, -Ft;0 < t < oo}, we obtain for m > 1/2:

B,E(<M>TA E(IMT A il2m) < EUM1 A t/21( 2m )

1"IMTAtI2m)

2 - 1)2171

< C; (2m 2m_ 1)21"E(<M>7,);

which is (3.29) with T replaced by T n t. Now let t co in this version of (3.29)and use the monotone convergence theorem. 11:1

3.27 Remark. A straightforward localization argument shows that (3.27),(3.29) are valid for any M edrI". The same is true for (3.28), provided thatthe additional condition E( <M >) < co holds.

We can state now the principal result of this subsection.

3.28 Theorem (The Burkholder-Davis-Gundy Inequalities). Let M e dric'cand recall the convention (3.26). For every m > 0 there exist universal positiveconstants km, Km (depending only on m), such that

(3.37) kmE(<M >) < E[(M1)21 < KmE(<M>7)

holds for every stopping time T

PROOF. From Proposition 3.26 and Remark 3.27, we have the validity of (3.37)for m > 1/2. It remains to deal with the case 0 < m < 1/2; we assume withoutloss of generality that M, <M> are bounded.

Let us recall now Problem 1.4.15 and its consequence (1.4.17). The right-hand side of (3.29) permits the choice X = (M*)2, A = C, <M> in the former,and we obtain from the latter

EL(MP)21 <2 -m1 -m

CrE(<M>7)

for every 0 < m < 1. Similarly, the left-hand side of (3.29) allows us to takeX = B, <M >, A = (M*)2 in Problem 1.4.15, and then (1.4.17) gives for0 < m < 1:

1 -mBTE(<M>7) <

2 -m 0

3.29 Problem. Let M = (M"),..., M(d)) be a vector of continuous, local mar-tingales, i.e., M") e dr I°c, and denote

d

II M ii ;'` '''' max IlMs11, At -- E or>,; 0 t < cc.0,s,, -'

,=,

Show that for any m > 0, there exist (universal) positive constants ,I.m, Am suchthat

(3.38) AmE(AT) < E(11M111)2m < AmE(AT)

holds for every stopping time T

3.30 Remark. In particular, if the AO in Problem 3.29 are given by

t

MP n) = E ,Pwschd,tr=1 jo

where { W = (W,(1), ... , W,(')), .Ft; 0 < t < co} is standard, r-dimensionalBrownian motion, {X, = (VP); 1 __ i < d, 1 < j 5 r,0 < t < co} is a matrix

of measurable processes adapted to and 11Xt112 A Ef.-.1 (XP.J.))2, then(3.38) holds with

(3.39) AT =o

E. Supplementary Exercises

331 Exercise. Define polynomials H(x, y); n = 0, 1, 2, ... by

H(x, y) =ace

exp ax -2-1 (x2 y)

0"x, y e 11;t

a=0

(e.g., Ho(x, y) = 1, H, (x, y) = x, 112(x, y) = x2 - y, H3(x, y) = x3 - 3xy,H4(x, y) = x4 - 6x2y + 3y2, etc.). These polynomials satisfy the recursiverelations

(3.40) -ax = y); n = 1, 2, ...

as well as the backward heat equation

a az(3.41) H,,(x, v) + axe H.(x, = 0; n = 0, 1, ...

For any M e.A,,,k,c, verify

(i) the multiple Ito integral computation

1

t1 41 dM . . dM dM = H,,(M <M>,),in t2 t, n!o o

(ii) and the expansionot.

exp (xM, -2

<M>,,,

) = E -H(M<M>,).=0 n!

(The polynomials H(x, y) are related to the Hermite polynomials

(-1r dnh ^(x) ex2/2 e-x2a

p-ir dx"

by the formula H(x, y) = .111! y"/2 h(x/fy).)

3.32 Exercise. Consider a function a: (0, co) which is of class C1 and suchthat 1/a is not integrable at either +co. Let c, p be two real constants, andintroduce the (strictly increasing, in x) function f(t, x) = e" f o dy /a(y);0 < t < co, x e II and the continuous, adapted process = 4 + p roe" ds +c'oe" dW, A; 0 < t < co. Let g(t, -) denote the inverse of f(t, -). Show that the

process X, = g(t,i) satisfies the stochastic integral equation

(3.42) X, = X0 + 1 t b(Xs) ds + ft 01)0 dWs; 0 < t < ooo o

for an appropriate continuous function b: gl -> DI, which you should determine.

333 Exercise. Consider two real numbers .5, it; a standard, one-dimensionalBrownian motion W; and let W,(") = W, + pt; 0 < t < co. Show that theprocess

X, = f t exp[ol W,(P) - Ws(P)} - +62(t - s)]ds; 0 < t < coo

satisfies the Shiryaev-Roberts stochastic integral equation

X, = it (1 + .3,1Xs)ds + .3 f XsdWs.o o

334 Exercise. Let W be a standard, one-dimensional Brownian motion and0 < T < co. Show that

lim supp-00 05t5T e-st

fefts diV,o

= 0, a.s.

335 Exercise. In the context of Problem 2.12 but now under the conditionE.,/T < co, establish the Wald identities

E(WT) = 0, E(W7) = ET.

336 Exercise (M. Yor). Let R be a Bessel process with dimension d z 3,starting at r = 0. Show that {M, A (1/Rd,-2); 1 < t < co}

(i) is a local martingale,(ii) satisfies sups ,,<, E(Mr) < co for every 0 < p < d/(d - 2) (and is thus

uniformly integrable),(iii) is not a martingale.

3.37 Exercise (M. Yor). Let R be a Bessel process with dimension d = 2starting at r = 0. Show that {X, = - log Rt; 1 < t < co} is a local martingalewith Ee'x, < co for -co < a < 2, t > 1, but X is not a martingale.

338 Exercise (Yor, Stricker). Let X be a continuous process and A a con-tinuous, increasing process with X0 = A0 = 0, a.s.

(i) Suppose that for every OE 01, the process

V) A exp(OX, - 102A,); 0 < t < co

is a local martingale. Prove that X E.i1°c and <X> = A.

3.4. Representations of Ccntinuous Martingales in Terms of Brownian Motion 169

(ii) Suppose that both X and Z") = exp(X - IA) are local martingales.Then again <X> = A.

3.39 Exercise (Wong & Zakai (1965b)). Let X be a continuous semimar-tingale of the form (3.1), and {/34")}Th a sequence of processes of boundedvariation, such that P Dim .1r) = X,] = 1 holds for every finite t > 0. If thefunction f: R is of class 0(R), show that

lim f(g.))dgio. f(Xs)dX, + - ft f'(Xs)d<M>,-..7) 0 0 2 0

holds a.s. P, for every fixed t > 0.

3.4. Representations of Continuous Martingalesin Terms of Brownian Motion

In this section we expound on the theme that Brownian motion is the fun-damental continuous martingale, by showing how to represent other con-tinuous martingales in terms of it. We give conditions under which a vectorof d continuous local martingales can be represented as stochastic integralswith respect to an r-dimensional Brownian motion on a possibly extendedprobability space. Here we have r < d. We also discuss how a continuous localmartingale can be transformed into a Brownian motion by a random time-change. In contrast to these representation results, in which one begins witha continuous local martingale, we will also prove a result in which one beginswith a Brownian motion W = { W Ft; 0 < t < co} and shows that everycontinuous local martingale with respect to the Brownian filtration {..F,} is astochastic integral with respect to W. A related result is that for fixed 0 < T <co, every g-r-measurable random variable can be represented as a stochasticintegral with respect to W.

We recall our standing assumption that every filtration satisfies the usualconditions, i.e., is right-continuous, and ..F0 contains all P-negligible events.

4.1 Remark. Our first representation theorem involves the notion of theextension of a probability space. Let X = {X.Ft; 0 < t < co} be an adaptedprocess on some (0, P). We may need a d-dimensional Brownian motionindependent of X, but because (0,,,P) may not be rich enough to supportthis Brownian motion, we must extend the probability space to construct this.Let (0, 327, P) be another probability space, on which we consider a d-dimensional Brownian motion B----{B A; 0 < t < co }, set nAo.n,g "ost-,15.4p x P, and define a new filtration by gt g; ®A. Thelatter may not satisfy the usual conditions, so we augment it and make itright-continuous by defining

A A n aosus>,

where Ai is the collection of P-null sets in g. We also complete g by defining= cr(g u .4'). We may extend X and B to {A}-adapted processes on

P) by defining for (w, E

gi(0), = x ,(0)), A(co = ).

Then 11 = {R A; 0 < t < col is a d-dimensional Brownian motion, inde-pendent of 5C-, A; 0 < t < col. Indeed, B is independent of the exten-sion to a of any ,-measurable random variable on U. To simplify notation,we henceforth write X and B instead of fe and 11 in the context of extensions.

A. Continuous Local Martingales as Stochastic Integralswith Respect to Brownian Motion

Let us recall (Definition 2.23 and the discussion following it) that if W =ig .Ft; 0 < t < col is a standard Brownian motion and X is a measurable,

adapted process with P[yo Xs ds < oo] = 1 for every 0 < t < co, then thestochastic integral /,(X) = Po X sc11415 is a continuous local martingale withquadratic variation process <1(X)>1 ro X s2 ds, which is an absolutely con-tinuous function of t, P a.s. Our first representation result provides theconverse to this statement; its one-dimensional version is due to Doob (1953).

4.2 Theorem. Suppose M = {M, = (MP), , AV)), A; 0 < t < col is definedon (a, , P) with Mt') E e' I", 1 ,(w) is an absolutely continuous function of t forP-almost every w. Then there is an extension P) of (S2, P) on which isdefined a d-dimensional Brownian motion W = {W, = (W,(1), , W(°)), A;0 < t < col, and a matrix X = {(X;'" )°,k=1, A; 0 < t < co} of measurable,adapted processes with

(4.1) p [f (xli,k))2 ds col= 1 < k < d; 0 < t < oo,

such that we have, P-a.s., the representations

d t

(4.2) E x , k) d ws(k); 1 ,=Ji

E x!")XP'k) ds ; 1 < j < d, 0 t < oo.d 'k =1 0

PROOF. We prove this theorem by a random, time-dependent rotation ofcoordinates which reduces it to d separate, one-dimensional cases. We beginby defining

3.4. Representations of Continuous Martingales in Terms of Brownian Motion 171

(4.4). . d= =

dt

= lim nUM(`), Mtn >, - <M(i), Mcn>0_0 f)).],

so that the matrix-valued process Z = {4 = (4.1;=,,A; 0 < t < col issymmetric and progressively measurable. For a = (a,,..., ota)e Rd, we have

Ed d dE ccizIja = - E aior) 0,dt i=i

so 4 is positive-semidefinite for Lebesgue-almost every t, P-a.s.Any symmetric, positive-semidefinite matrix Z can be diagonalized by an

orthogonal matrix Q, i.e., = QT, so that ZQ = A and A is diagonalwith the (nonnegative) eigenvalues of Z as its diagonal elements. There areseveral algorithms which compute Q and A from Z, and one can easily verifythat these algorithms typically obtain Q and A as Borel-measurable functionsof Z. In our case, we start with a progressively measurable, symmetric,positive-semidefinite matrix process Z, and so there exist progressively mea-surable, matrix-valued processes {Qt(co) = (q1-1(co))1.;_,; gf,; 0 < t < co} and{A,(w) = (.5u4(a)))1,;=, , 0 < t < co} such that for Lebesgue-almost everyt, we have

d d

(4.5) Egc.i1k.J

E Si;1:5_

k=1 k=1

d d

(4.6) E E (5v1.1 > 0 ; 1 j < d,k=1 1=1

a.s. P. From (4.5) with

'i

= j we see that (e1)2 <1,s0

ft (e.`)2 d<M(k)>s < <Mm>, < co,

and we can define continuous local martingales by the prescription

(4.7) Nio) A dIVIlk); 1 , = E Ek=1 1=1

We see, in particular, that

kqs

.iqs j d<M(k), Al")).s

Jo

d d .t

= E E q! i 4 I 11.! i dsk=1 1=1

= Si; Aids.

(4.9) Asi ds = <N(')>, < o o .

We now represent the vector of local martingales N = MP), - , Nt(d)), <9;;0 < t < col as a vector of stochastic integrals on an extended probabilityspace 0, g, F), which supports a d-dimensional Brownian motion B ={13, = (BP), . . . , MI)), .A; 0 < t < co 1 independent of N (cf. Remark 4.1). Since

1 t

1{,1,0} d <N(I)>s = 1 {,1).0) ds t,o

we can define continuous, local martingales

Jo

1N(4.10)Wto g'-- 1{4,0} d li) + 1{4.0} di3.11);

o .1.1 Jo1 , = but, 1 < i, j < d; 0 < t < cc,

so, according to Theorem 3.16, W = {W = (W,(1), ..., Wi(d)), gi; 0 < t < col isa d-dimensional Brownian motion. Moreover,

Jt

(4.11) .1,17, dWso) = ft

11A1,.01dNr = NP, 1 , = 1 {,11.0} As ds = 0,

o

is itself identically zero.Having thus obtained the stochastic integral representation (4.11) for N in

terms of the d-dimensional Brownian motion W, we invert the rotation ofcoordinates (4.7) to obtain a representation for M. Let us first observe thatfor 1 < i, k < d,

j. f0(qi;k12 4 ds < Ask ds < co; 0 t < co0

by (4.9), so with XP'k) A ql'k , condition (4.1) holds. Furthermore, (4.11),(4.7), and (4.5) imply

d

t(4.12) j'X!") dWs(k) = qs kdNsk)k=1 p k=1 0

d

= E Ed

j=1 k=1 0

q!, k tips

k d

t= ditt) = MP),J=1 0

which establishes (4.2). Equation (4.3) is an immediate consequence of (4.2). LI

4.3 Remark. If for P-a.e. w E SI, the matrix-valued process Zt(co) = (4'i(co))1,;,,has constant rank r,1 5 r 5 d, for Lebesgue-almost every t, then the Brownianmotion W used in the representation (4.2) can be chosen to be r-dimensional,and there is no need to introduce the extended probability space (n,,, P).Indeed, we may take At 5... , to be the r strictly positive eigenvalues of Zand replace (4.10) by

Jo "(4.10)' = 1

dN(i)- 1 i r.

Since /VP = 0; r + 1 5 i 5 d, 0 5 t < co (witness (4.9)), (4.12) becomes

r t d

i(4.12)' E xv,k)dws(k) = fg" = 1 < i d.k 0 k =1 o

Because (4.10)' defines W without reference to the Brownian motionB, there is no need to extend the original probability space.

The following exercise shows that any vector of continuous local martin-gales can be transformed by a random time-change into a vector of continuouslocal martingales satisfying the hypotheses of Theorem 4.2.

4.4 Exercise. Let {M = (MP . , gd)), .5r.t; 0 < t < co} be a vector of con-tinuous local martingales on some (SI, P), and define

d d

A" <M(i),M(i)>, Ai(w) E E AV(0),1=1 ;=,

where AP) denotes total variation of A(Li) on [0, t]. Let 7;(0)) be the inverseof the function At(w) + t, i.e., il,s(0(co) + Ts(co) = s; 0 5 s < co.

(i) Show that for each s, T., is a stopping time of(ii) Define Ws A ..f.T.; 0 5 s < co. Show that if {.Fi} satisfies the usual condi-

tions, then {Ws} does also.(iii) Define

Nr MP, 1 5 i d; 0 < s < oo.

Show that for each 1 5 i 5 d: N(i) drloc, and the cross-variation<MI), /NM>, is an absolutely continuous function of s, a.s. P.

B. Continuous Local Martingales as Time-ChangedBrownian Motions

The time-change in Exercise 4.4 is straightforward because the function A, + tis strictly increasing and continuous in t, and so has a strictly increasing,continuous inverse T. Our next representation result requires us to considerthe inverse of the quadratic variation of a continuous local martingale; be-

cause such a quadratic variation may not be strictly increasing, we begin witha problem describing this situation in some detail.

4.5 Problem. Let A = IA(t); 0 < t < co } be a continuous, nondecreasingfunction with A(0) = 0, S -4a', A(co) 5 co, and define for 0 < s < co:

T(s) ={in* > 0; A(t) > s };

oo;

0 5 s < Ss > S.

The function T = { T(s); 0 5 s < co } has the following properties:

(i) T is nondecreasing and right-continuous on [0, S), with values in [0, co).If A(t) < S; V t .- 0, then limsts T(s) = co.

(ii) A(T(s)) = s A S; 0 _< s < op.(iii) T(A(t)) = supIr t: A(r) = A(t)}; 0 5 t < co.(iv) Suppose cp: [0, co) -> R is continuous and has the property

A(t1) = A(t) for some 0 < t1 < t cp(ti)= p(t).

Then p(T(s)) is continuous for 0 < s < S, and

(4.13) p(T(A(t))) = cp(t); 0 < t < c0.

(v) For 0 5 t, s < co: s < A(t).. T(s) < t and T(s) S t s < A(t).(vi) If G is a bounded, measurable, real-valued function or a nonnegative,

measurable, extended real-valued function defined on [a, b] c [0, co),then

A(b)

(4.14) G(t) dA(t) = 1 G(T(s))ds.A(a)

4.6 Theorem (Time-Change for Martingales [Dambis (1965), Dubins & Sch-warz (1965)]). Let M = {M.9;;0 5 t < co } e../r1" satisfy lim,o, <M>, =co, a.s. P. Define, for each 0 < s < co, the stopping time

(4.15) T(s) = inf {t > 0; <M>, > s }.

Then the time-changed process

(4.16) B, '' MT(,), -41.. LFT(s); 0 ''. S < 00

is a standard one-dimensional Brownian motion. In particular, the filtration {W,}satisfies the usual conditions and we have, a.s. P:

(4.17) M, = B<m>,; 0 5 t < co.

PROOF. Each T(s) is optional because, by Problem 4.5 (v), { T(s) < t} =I<M>, > sl e .5c and {..F,} satisfies the usual conditions; these are also satis-fied by Al just as in Problem 4.4 (ii). Furthermore, for each t, <M>, is astopping time for the filtration Is} because, again by Problem 4.5 (v),

{ <M>, 5 s} = { T(s) t} egT(s) .--- Is; 0 ..' S < 0 0 .

Let us choose 0 5 s, < s2 and consider the martingale {R, = M T(s2), .57t;0 < t < co}, for which we have

<R>, = <M>, A T(S2) <M>T(s2) = s2; 0 5- t < GO

by Problem 4.5 (ii). It follows from Problem 1.5.24 that both Cf and SI2 - <Al>are uniformly integrable. The Optional Sampling Theorem 1.3.22 implies, a.s. P:

E[Bs2 - = - KIT(si).LFT(s,)] = 0,

E[(13,2 - Bs,)20si] = EURT(s2)- MT 100 I .-Fro,)]

= EC<R>ps2)- <R>T401,-FT(so] = S2 -S I

Consequently, B = {Bs, Ws; 0 s < co} is a square-integrable martingale withquadratic variation , = s. We shall know that B is a standard Brownianmotion as soon as we establish its continuity (Theorem 3.16). For this we shalluse Problem 4.5 (iv).

We must show that for all co in some Sr f2 with P(S2*) = 1, we have:

(4.18) <M>ti(w) = <M>,(w) for some 0 < t, < t mii(w) = mt(c0).

If the implication (4.18) is valid under the additional assumption that t, isrational, then, because of the continuity of <M> and M, it is valid even withoutthis assumption. For rational t > 0, define

o- = inf t > t,: <M>, > <M>,, I,

N, = M(,,), - M,,, 0 < s < co,

so {N,, ..F,1; 0 < s < co} is in dr'1" and

<N>, = - <M>,, = 0, a.s. P.

It follows from Problem 1.5.12 that there is an event f2(t, ) s 12 with P(C1(t,))=1 such that for all co e S1(t,),

<M>,,(co) = <M>,(co), for some t > t, M,,(w) = Mi(co).

The union of all such events fl(t, ) as t, ranges over the nonnegative rationalswill serve as fr, so that implication (4.18) is valid for each co E sr. Continuityof B and equality (4.17) now follow from Problem 4.5 (iv). 11:1

4.7 Problem. Show that if P[S Q <M> co] > 0, it is still possible todefine a Brownian motion B for which (4.17) holds. (Hint: The time-changeT(s) is now given as in Problem 4.5; assume, as you may, that the probabilityspace has been suitably extended to support an independent Brownian motion(Remark 4.1).)

The proof of the following ramification of Theorem 4.6 is surprisinglytechnical; the result itself is easily believed. The reader may wish to omit thisproof on first reading.

4.8 Proposition. With the assumptions and the notation of Theorem 4.6, we havethe following time-change formula for stochastic integrals. If X = {X.;0 < t < co} is progressively measurable and satisfies

(4.19)

then the process

(4.20)

.X; d<M>t < co a.s.,

Ys X2(,), W's; 0 <s<oo

is adapted and satisfies, almost surely:

(4.21) J0Y52ds <co

<m>,(4.22) X

v

dMv

= - 17d13.; 0 < t < co,o

f'sj.T(s)(4.23) X dm, = d13.; < s < cc.

PROOF. The process Y is adapted to {WO because of Proposition 1.2.18.Relation (4.21) follows from (4.19) and (4.14).

Consider the continuous local martingale 1.4 pc, XvdMv,..F,; 0 < t < col.If

(4.24)

then

<M>t,(0)) = <M>t(w) for some 0 < t, < t,

<J>,,(co) =o

XR(D)d<M>,#0) = io XR(0)d<M>v(w) = <J>t(co)

Applying to J the argument used to obtain (4.18), we conclude that for all win some n* g SI with F(K-e)- 1, (4.24) implies the identity Jti(w)= Ji(co).According to Problem 4.5 (iv), we have that

(4.25) JT(s) X dMv; 0 < s < co

is continuous, and

(4.26) .7<m>t = 17.(of)t) = ft; 0 < t < cc,

almost surely. Let ; = inf {0 < t < co; <J>, > n}, so {J, Ft; 0 < t < co} isa martingale. For 0 < s, < s2 and n > 1, we have from the optional samplingtheorem:

E[Js2 A <m),,,,0 s i] = EUT(s2)A n rI'T(s,)]

= JT(si)A n = si a.s.

Each is a stopping time of the filtration {A} because t<M>. A s}

= {n A t Lc. T(s)} = {n 5 T(s), <J>m) E 'T(s). By assumption,<M>n = oo a.s.; it follows that .7 is in Jr' (relative to {C).

Using this fact, we may repeat the preceding argument to show that Joy>, =J, is a continuous local martingale relative to the filtration Mm>,}, whichcontains and may actually be strictly larger.

In fact, we can choose an arbitrary continuous local martingale N relativeto {Ws} and construct Art = Rof>,, a continuous local martingale relative to{W00,}. If we take N = B, then N = M from (4.17) and so M is in dr'"relative to {<,%}. We now establish (4.23) by choosing an arbitrary N ={fs?5s; 0 < s < co} e.4/"c and showing that

rs(4.27) <J, N>s Yud<B,1%7>; 0 < s <

holds a.s. (see Proposition 2.24). Let N, = Si<m> fix t1, and set

Mti = M1+1, - 41, Nil = +, - Nit; 0 < t < co.

Both M', N' (respectively; M, N) are local martingales relative to {Wow>,,,}(resp. {500,}). We may compute cross-variations thus:

KM, N>t+ - <M, N>t, I = I <M1,Ni>,1 N/<M1>t<N1>,

= NA<M>1.4-1, - <M>ii)(<N>:+1, - <N>1),

where Problem 1.5.7 (iii) has been invoked. We see that if <M> is constant onan interval, so is <M, N>. From Problem 4.5 (iv) we conclude that <M, N>T(s)is almost surely continuous. Because <M, N>, has finite total variation for tin compact intervals, the composition <M, N>T(,) has finite total variationfor s in compact intervals. Finally,

Bs Ns - <M,N>Tco= MT(S)NT(S) - <M N> T(s)

is in dr' relative to {Ws}, so

(4.28) <B, R>, = <M,N>T(s); 0 < s < co, a.s. P.

Setting s = <M>, in (4.28), we obtain

(4.29) <B,R>olf>, <M, N>T(<m>) = <M,N>,; 0 < t < co

almost surely; we have used Problem 4.5 (iv) with cp = <M, N>. Choose w e CIfor which (4.29) holds. Then

(v)d<M N> ((o) - <M5 bN> (w) <M,N>.(w)c. 1 fo,b) v

= <B,N>00,(.#0)- <B,A7><m>.(.)(0))

= f 1(<m).(.),<m>b(w )(u)d<B, Mco)

= J 1(a,b)(T(u,w))d<B,N>(co),

by virtue of Problem 4.5 (v) in the last step. Thus we have for step functions,and consequently for all Borel-measurable functions G: [0, oo) -> [0, co], that

(4.30) G(v)d <M, N> = G(T(u)) d<B, R>.

holds a.s. Returning to (4.27), we can use the optional sampling theorem and(4.30) to write

>s = <JT()1NT()>s = <J1 N>T(s)T(s)

= Xvd<M,N> = Yud<B,R>; 0 s < co, a.s.

This concludes the proof of (4.23). Replacing s by <M>, in (4.23) and using(4.26), we obtain (4.22).

4.9 Remark. If, in the context of Theorem 4.6, we have P[S A <M>co< co] > 0, we may take a Brownian motion B on an extended probabilityspace for which (4.17) holds (Problem 4.7), and the conclusions of Proposition4.8 are still valid, except now we must define Y by

(4.20)' Yf XT(s); 0 s < S,10; S s < co.

The proof is straightforward but tedious, and it is omitted.

4.10 Remark. Levy's characterization of Brownian motion (Theorem 3.16)permits a bit more generality than expressed in Theorem 4.6. If X = {X..F;: 0 < t < co} is an adapted process with M A X - Xoe Jr' and

<M>, = co a.s., then the time-changed process

X0 + MT(s), Ws A. LFT(s); 0 s < co

is a one-dimensional Brownian motion with initial distribution PXV. Inparticular, X0 is independent of

Bs = MT(s) , oksB ; 0 < s < co

(Definition 2.5.1). Similar assertions hold in the context of Problem 4.7,Proposition 4.8, and Remark 4.9.

4.11 Problem. We cannot expect to be able to define the stochastic integralJo XsdW, with respect to Brownian motion W for measurable adapted pro-cesses X which do not satisfy 14 Xs ds < co a.s. Indeed, show that if

P[f Xs ds oo]= 1, for 0 < t < 1 and E A lc ds = co},

then

tlim f X, dW, -lim I Xs dW, = +a), a.s. on E.tti o tt o

4.12 Problem. Consider the semimartingale X, = x + M, + C, with xeMe.11`.1", C a continuous process of bounded variation, and assume thatthere exists a constant p > 0 such that IC,C + <M>, < pt, V t 0 is validalmost surely. Show that for fixed T > 0 and sufficiently large n > 1, we have

F'[max IX,I > n] < exp { -n2 1.0 18p T

C. A Theorem of F. B. Knight

Let us state and discuss the multivariate extension of Theorem 4.6. The proofwill be given in subsection E.

4.13 Theorem (F. B. Knight (1971)). Let M = {M, = (MP), , Md)),F,; 0 <t < co} be a continuous, adapted process with M") diecmc, = 00;

a.s. P, and

(4.31)

Define

<M"),110>, = 0; 1 , > s}; 0 < s < co, I < i < d,

so that for each i and s, the random time Ti(s) is a stopping time for the(right-continuous) filtration {.9;;}. Then the processes

Br M)(s); 0 < s < co, 1 < i d,

are independent, standard, one-dimensional Brownian motions.

Discussion of Theorem 4.13. The only assertion in Theorem 4.13 which is notalready contained in Theorem 4.6 is the independence of the Brownianmotions B(i); 1 < i < d. Theorem 4.6 states, in fact, that B(1) is a Brownianmotion relative to the filtration {(e) -4- ,T4 ),,o, but, of course, these filtra-tions are not independent for different values of i because g!;) = Fes; 1 < i < d.The additional claim is that the a-fields Fr, , For are independent,where {.F.,B1 is the filtration generated by B(i). This would follow easily if theassumption (4.31) were sufficient to guarantee the independence of M(I), Mu)for i j; in general, however, this is not the case. Indeed, if W = {0 < t < co} is a standard Brownian motion, then with

AV) 1{w,> o} dWs,

we have M(I), M(2) e ..fiq and

0 t < co,

<M(1), M(2)>, = f 1 {w,} 1 {w.<0} ds = 0; 0 5 t< co.o

But M") and M(2) are not independent, for if they were, <M")> and <M(2)>would also be independent. On the contrary, we have

<M")>, + <M(2)>, = 1{,.> ds + 1{w.<0} ds = t 0 < t < aD.

F. B. Knight's remarkable theorem states that when we apply the propertime-changes to these two intricately connected martingales, and then forgetthe time-changes, independent Brownian motions are obtained. Forgetting thetime-changes is accomplished by passing from the filtrations 1101 to the lessinformative filtrations {.FsIr')}.

We shall use this example in Section 6.3 to prove the independence of thepositive and negative excursion processes associated with a one-dimensionalBrownian motion.

D. Brownian Martingales as Stochastic Integrals

In preparation for the proof of Theorem 4.13, we consider a different class ofrepresentation results, those for which we begin with a Brownian motionrather than constructing it. We take as the integrator martingale a standard,one-dimensional Brownian motion W = W ";; 0 < t < ao on a probabilityspace (SI, .F, P), and we assume satisfies the usual conditions. For 0 <T < oo, we recall from Lemma 2.2 that 4 is a closed subspace of the Hilbertspace YfT. The mapping X 1- IT(X) from .r.P to L2 (n, P) preserves innerproducts (see (2.23)):

E f X, Y, dt = E[IT(X)IT(Y)]

Since any convergent sequence in

(4.32) MT -4 {IT(X); X E

is also Cauchy, its preimage sequence in ..r? must have a limit in .r.P. It followsthat MT is closed in L2(0., ,T,P), a fact we shall need shortly.

Let us denote by 111 the subset of .11`2. which consists of stochastic integrals

/,(X) = Xs dWs; 0 < t < a),0

of processes X e

(4.33) llZ {/(X); X e ..T*} di`j g 2.

Recall from Definition 1.5.5 the concept of orthogonality in .112. We have thefollowing fundamental decomposition result.

4.14 Proposition. For every M e .112, we have the decomposition M = N + Z,where N Z e .112, and Z is orthogonal to every element of

PROOF. We have to show the existence of a process Y e 2 " such that MI( Y) + Z, where Z e A2 has the property

(4.34) <Z,I(X)> = 0; V X e

Such a decomposition is unique (up to indistinguishability); indeed, if we haveM = /( Y') + Z' = /(Y") + Z" with Y', Y" e .29* and both Z' and Z" satisfy(4.34), then

Z Z" - Z' = I( Y' Y")

is in and <Z> = <Z, I( Y' - Y")> = 0. It follows from Problem 1.5.12 thatP[Z, = 0, V 0 < t < oo] 1.

It suffices, therefore, to establish the decomposition for every finite time-interval [0, T]; by uniqueness, we can then extend it to the entire half-line[0, co). Let us fix T > 0, let gt, be the closed subspace of P) definedby (4.32), and let giq- denote its orthogonal complement. The random variableMT is in /2(52, 9T, P), so it admits the decomposition

(4.35) MT = IT(Y) ZT,

where Y e 2' and ZT e L2 (SI, .97T, P) satisfies

(4.36) E[ZTIT(X)] = 0; V X e

Let us denote by Z = {Z Ft; 0 < t < col a right-continuous version of themartingale E(ZTigc)(Theorem 1.3.13). Note that Z, = ZT for t > T ObviouslyZ e.42 and, conditioning (4.35) on .F we obtain

(4.37) Ait = 4(Y) + Zt; 0 < t < T, a.s. P.

It remains to show that Z is orthogonal to every square-integrable mar-tingale of the form I(X); X e-C1, or equivalently, that {Z,/,(X), A; 0 < t < T}is a martingale. But we know from Problem 1.3.26 that this amounts to havingE[ZsIs(X)] = 0 for every stopping time S of the filtration {A}, with S Lc. T.From (2.24) we have Is(X) = IT(X), where Xi(w) = Xi(o.01{, <s(,)} is a processin Therefore,

E[ZsIs(X)] = E[E(ZTI,$)ls(X)] = E[ZTIT(X)] = 0

by virtue of (4.36).

It is useful to have sufficient conditions under which the classes llZ and./#1 actually coincide; in other words, the component Z in the decompositionof Proposition 4.14 is actually the trivial martingale Z = 0. One such condi-tion is that the filtration {A} be the augmentation under P of the filtration{.F,W} generated by the Brownian motion W. (Recall from Problem 2.7.6 andProposition 2.7.7 that this augmented filtration is continuous.) We state and

prove this result in several dimensions. A martingale relative to this aug-mented filtration will be called Brownian.

4.15 Theorem (Representation of Brownian, Square-Integrable Martingalesas Stochastic Integrals). Let W = tW, = (W,"), W,(d)),..F,; 0 < t < ool be ad-dimensional Brownian motion on (S/,..F, P), and let {..F,} be the augmentationunder P of the filtration {..F,w } generated by W. Then, for any square-integrablemartingale M = {M, 0 < t < co) with Mo = 0 and RCLL paths, a.s., thereexist progressively measurable processes Y = {Y,(1), g. 0 < t < col such that

(4.38) E j

Yoe

* (Y,i)2 dt < oo; 1 j< d

for every 0 < T < co, and

t

(4.39) M, = E ysodwp; 0 < t < CO .J=1 0

In particular, M is a.s. continuous. Furthermore, if fth; 1 < j < d, are any otherprogressively measurable processes satisfying (4.38), (4.39), then

f I - fru) 12 dt = 0, a.s..i=1

PROOF. We first prove by induction on m, where m = d, that there areprocesses Y(1),..., Y(m) in .29* such that

m

(4.40) Z, A M, - E ysti)dwp; o < t < co.i=1

is orthogonal to every martingale of the form ET_1 f o XS dW,,u), where X(J e2*; 1 _< j m. If m = 1, this is a direct consequence of Proposition 4.14.Suppose such processes exist for m - 1, i.e.,

m-iZ A M, - E ypdwp; o < t < cc,

J=1 0

is orthogonal to ET_-11 Ito xpdwso for all Xul e 2*; 1 sjsm - 1. ApplyProposition 4.14 to write

Z =fYs(m) d Ws(m) + 4; 0 < t < oo,

0

for some 11(" e 2*, where Z is orthogonal to 1.0 gn) d Ws(m) for all X(m) e2*. For 1 s j< m- 1 and X01 e 2*, we have

<Z,/w(-"(X0)> = <2, I') (X(i))> - <1'"(Y(m)), lw())(X0)> = 0.

Thus, we have the decomposition (4.40) for M. In particular, with m = d,

(4.41) <M, Hi(h>, = I Yu) ds; 0 5 t < co, 1 < j < d.0

Following Liptser & Shiryaev (1977), pp. 162-163, we now show that, inthe notation of (4.40) with m = d, we have P-a.s. that

Z, 0; 0 < t < co.

First, we show by induction on n that if 0 = so < s, < < s < t, and if thefunctions fk: C, 0 < k < n are bounded and measurable, then

(4.42) E[Z, fl fk(Wsk)] = 0.k=0

When n = 0, (4.42) can be verified by conditioning on .90 and using the factZo = 0 a.s. Suppose now that (4.42) holds for some n, and choose s < t. For0 = (0,, . . ,0)E Rd fixed and s. < s < t, define with i = 1:

co(s) -4 E[Z, n fk(Wsdeimwd= E[Z n fk(wsdeimwd.k=0 k=0

Using Ito's rule to justify the identity

d

el("'w.) =s

+ E ie. ei(e'w-i dly(i) 110112 s e"'w-) du,.i=--1 2

s"

we may write

(4.43) E[Zse"-woi,s.] = zs.ei(t 3 'W..)

But (4.40) and (4.41) imply

E[Z, f et(9'w-) dW,P)s

+ E le i E[Zs f ei(e'w-) dWu)i=i

d s

3,,

110112 ErZs f eimw.)d2 L j u,

grd = E [(Z., - Zs)Js

eimw-) dWli) .FS.1 = 0.

Multiplying (4.43) by FP,:_o f(Wsk) and taking expectations, we obtain

110112 f s(4.44) co(s) = Os.) E[Zs- fl f5(14/5,)e"w-)]du

2 k =1s110112

= co(s.) - 4910 du; s s t.2

By our induction hypothesis, cp(s) = 0, and the only solution to the integralequation (4.44) satisfying this initial condition is 9(s) = 0; s < s < t. Thus

(4.45) E = 0; v e Rd.k=o

With D A Zt irk'=o fk(Wsk), we define two measures on (Old, M(Rd)) by

11±(F) = ELD± ir(Ws)i; Fe M(fre).

Equation (4.45) implies

c."'x) (dx) e'(°xl (dx); V 0fo;sa

and by the uniqueness theorem for Fourier transforms, we see that it+ --=Thus

E[D+ f (Ws)] = E[D f(Ws)]; s < 4+1 g s < t,

for any bounded, measurable f:01" -> C. This proves (4.42) for n + 1 andcompletes the induction step.

A standard argument using the Dynkin System Theorem 2.1.3 now showsthat we have

(4.46) E[Z,fl = 0

for every A w-measurable indicator and thus, for every 547,w-measurable,bounded Since A differs from Aw only by P-null sets, (4.46) also holdsfor every A-measurable, bounded Setting = sgn(Z,), we conclude thatZ, = 0 a.s. P for every fixed t, and by right-continuity of Z, for all t e [0, oo)simultaneously.

The uniqueness assertion in the last sentence of the theorem is proved byobserving that the martingale El=, f o (Yso - )dHis(i) is identically zero, andso is its quadratic variation.

4.16 Problem. Let W = {W, = , Wi(d)), A; 0 < t < a)} be a d-dimensional Brownian motion as in Theorem 4.15. Let M = {M A;0 < t < oo } be a local martingale with M, = 0 and RCLL paths, a.s. Thenthere exist progressively measurable processes Y(i) {Y,(j), A; 0 < t < co}such that

and

('T(Y, dt < co; < j d, 0 T < op ,

a

lye = E yso) dwsu);J=1 J'io

In particular, M is a.s. continuous.

0 < t < co.

4.17 Problem. Under the hypotheses of Theorem 4.15 and with 0 < T < co,let c be an .97T-measurable random variable with EV < co. Prove that there

are progressively measurable processes Y"),... , I'm satisfying (4.38), and suchthat

d

(0(4.47) = E(0 + E Ystb d wsch ; a.s. P.J=1 o

E. Brownian Functionals as Stochastic Integrals

We extend Problem 4.17 to include the case T = co. Recall that for M ewe denote by ..C1(M) the class of processes X which are progressively measur-able with respect to the filtration of M and which satisfy E f o X,2 d<M>, < oo.According to Problem 1.5.24, when X e Yt(M), we have .1`6° X dM, defined a.s.P. If W is a d-dimensional Brownian motion, we denote by ..r:(W) the set ofprocesses X which are progressively measurable with respect to the (aug-mented) filtration of W and which satisfy E roc' X,2 dt < co.

4.18 Proposition. Under the hypotheses of Theorem 4.15, let be anmeasurable random variable with EV < oo. Then there are processes Y(1), ...,Y(') in ..TVW) such that

d CO

= E() + E I yso d Ws° ; a.s. P..i=i o

PROOF. Assume without loss of generality that E() = 0, and let M, be aright-continuous modification of E(ig;). According to Theorem 4.15, thereexist progressively measurable Y"),..., Y(d) satisfying (4.38) and (4.39). Jensen'sinequality implies MI S E(215";), so

E i't (Ysti))2 ds = E<M>, = E(M,2) E(V) < co; 0 < t < co.i=1 o

Hence, Yti)E .29,(W) and M. f;°, Ysth MP.) is defined for 1 < j < d. Problem1.3.20 shows that M. = = c.

We leave the proof of uniqueness of the representation in Proposition 4.18as Exercise 4.22 for the reader.

In one dimension, there is a representation result similar to that of Proposi-tion 4.18 in which the Brownian motion is replaced by a continuous localmartingale M. This result is instrumental in our eventual proof of Theorem4.13. Uniqueness is again addressed in Exercise 4.22.

4.19 Proposition. Let M = {M; 0 < t < co} be in Jr' and assume that= co, a.s. P. Define T(s) by (4.15) and let B be the one-dimensional

Brownian motion

Mr(s), es; 0 s < co

as in Theorem 4.6, except now we take the filtration AI to be the augmentationwith respect to P of the filtration {AB} generated by B. Then, for everygo,-measurable random variable satisfying E2 < oo, there is a process X e43.(4) for which

(4.48) = E() J. X, dM,; a.s. P.CO

PROOF. Let Y = { Ys, gs; 0 < s < co} be the progressively measurable processof Proposition 4.18, for which we have

(4.49)

(4.50)

E f ys2 ds < co,

= E() + J Ys dBs.

Define X,= Yol>,; 0 t <We show how to obtain an {.97,}-progressively measurable process X which

is equivalent to X. Note that because {A} {,T(01 contains {,23} andsatisfies the usual conditions (Theorem 4.6), we have es A; 0 5 s < oo.Consequently, Y is progressively measurable relative to If Y is a simpleprocess, it is left-continuous (c.f. Definition 2.3), and it is straightforward toshow, using Problem 4.5, that { Yoki>,; 0 < t < cc} is a left-continuous processadapted to {.F(}, and hence progressively measurable (Proposition 1.1.13). Inthe general case, let { Y('')}c°_, be a sequence of progressively measurable(relative to {6'0), simple processes for which

lim E I Ys(") -Ys 12 ds = O.n- co o

(Use Proposition 2.8 and (4.49)). A change of variables (Problem 4.5 (vi)) yields

(4.51) lim E J IX:") - ,1 2 d<M>, = 0,o

where X:n) A Vn>,. In particular, the sequence {X(")},13_, is Cauchy in 2 t(M),and so, by Lemma 2.2, converges to a limit X e _KAM). From (4.51) we musthave

E 12, - X,12 d<M>, = 0,

which establishes the desired equivalence of X and X.It remains to prove (4.48), which, in light of (4.50), will follow from

foc° Ys dB, = f X, dM,; a.s. P.

This equality is a consequence of Proposition 4.8.

PROOF OF F. B. KNIGHT'S THEOREM 4.13. Our proof is based on that of Meyer(1971). Under the hypotheses of Theorem 4.13, let {dl')} be the augmentationof the filtration {..fn generated by B"); 1 5 i 5 d. All we need to show isthat el'2), ... , dr are independent.

For each i, let V') be a bounded, (IV-measurable random variable. Accord-ing to Proposition 4.19, there is, for each i, a progressively measurable processX") = {X1'),.; 0 < t < co} which satisfies

and for which

Ef(r)2 d<M(')>, < co; 1 < i < d,

o

00

foo(i) = E(V)) + XP) dMP; 1 5i < d.o

Let us assume for the moment that

(4.52) E(V)) = 0; 1 5 i < d,

and define the (5,}-martingale

) -4- ft Xli) dM11); 0 < t < co , 1 s< co; 1 < i, d.o Jot

Repeated application of Holder's inequality yields:

E

f1 2

( n ix)) d<m(0>s0 Joi

,E{n[ sup (VI)2] - <e>,}Joi :)..s..t

1/2

[E sup (V))4] IE sup (V))80 <s <, 0 .5.r

2-,[E sup (V))21 [E<e>712; 0 < t t < oo.

ossst

For m > 1, Doob's maximal inequality (Theorem 1.3.8 (iv)), and Jensen'sinequality (Proposition 1.3.6) give

E[ sup Ill)12"11 ( 2m 2m El ai)12m11:),s..t 2m - 1

(2m - 1)2m 2m

El Vi)I2m < a).

We have from Proposition 3.26 (see also Remark 3.27):

B E<V)Va I I

for some positive constant B which does not depend on t. Thus (4.54) holds,and letting t oo in (4.53) we obtain the representation

d d m= E ,v)x,i)dm).

1=1 1=1 0

The right-hand side, being a sum of martingale last elements (Problem2.18), has expectation zero. Thus, under the assumption (4.52), we haveErr, = 0. Equivalently, we have shown that for any set of boundedrandom variables V1),..., V), where each V) is 61-measurable, the equality

(4.55) E fl [Vi) - E(V))] = 0i=1

holds. Using (4.55), one can show by a simple argument of induction on d that

d d

E ji = Evi).i=1 i=1

Taking Vi) = 1,; Ai e 8,;;), 1 < i < d, we conclude that the a-fields 6V, ), , crt')

are independent.

What happens if the random variable in Problem 4.17 is not square-integrable, but merely a.s. finite? It is reasonable to guess that there is still arepresentation of the form (4.47), where now the integrands Y(1), , Y(d) can

only be expected to satisfy

T

In fact, an even stronger result is true. We send the interested reader to Dudley(1977) for the ingenious proof, which uses the representation of stochasticintegrals as time-changed Brownian motions. There is, however, no unique-ness in Dudley's theorem; see Exercise 4:22 (iii).

4.20 Theorem (Dudley (1977)). Let W = 0 < t < co} be a standard,one-dimensional Brownian motion, where, in addition to satisfying the usualconditions, {9,} is left-continuous. If 0 < T < co and is an 9,-measurable,a.s. finite random variable, then there exists a progressively measurable process

Y = { -.Ft; 0 < t 5 T} satisfying

17,2 dt < co; a.s. P,o

such that

=T

Yt dllit; a.s. P.

4.21 Remark. Note in Theorem 4.20 that the filtration 1.97,1 might be generatedby a d-dimensional Brownian motion of which W is only one component. Foran amplification of this point, see Emery, Stricker & Yan (1983).

4.22 Exercise.

(i) In the setting of Proposition 4.18, show that the processes y"), Y(d)

are unique in the sense that any other processes P"), f(d) in .29t(W)which permit the representation

d CO

= E(0 + > f cc(i)dw,u), a.s.J---1 0

must also satisfy

CO d

E yi(j) - i7;012 dt = 0, a.s.Jo ;=-1

(ii) In the setting of Proposition 4.19, show that X is unique in the sense thatany other process X E .99*(M) which permits the representation

= E() + J 17, dM a.s.,0

must also satisfy

ro° IX: - g:12 d<M>: = 0, a.s.

(iii) Find a progressively measurable process Y such that1 1

0 < Y,2 dt < oo, a.s., but Y, dWi = 0, a.s.j.o o

In particular, there can be no assertion of uniqueness of Y in Theorem4.20.

4.23 Exercise. Is the following assertion true or false? "If M = {M .Ft;0 5 t < co} and N = {N Ft; 0 < t < co} are in ../1"" and <M> <N>, thenM and N have the same law."

4.24 Exercise (Hajek (1985)). Consider the semimartingales

X, = X0 + j't ji ds + j't as dWs, Y = Yo + m(s)ds + p(s)dVS

05t<oofor 0 < t < cc, where W and V are Brownian motions; the progressivelymeasurable processes ft, a and the Borel-measurable functions m: [0, co) [R,

p: [0, cc) [0, co) are assumed to satisfy

thisltit mW, atl + m(s) + p2(s)} ds < co, V0<t<co

almost surely. If X0 < Y0 also holds a.s. and f is a nondecreasing, convexfunction on show that

P(X, > c) < 2P(Y, > c); V c e

Ef(X,) < Ef(Y)

hold for every t > 0. (Hint: By extending the probability space if necessary,take W to be a Brownian motion independent of W, and consider the con-tinuous semimartingales

Y(i) A Y0 + J m(s)ds + J as dl/K + (-1)i 2 (s) 0.s2 dws; i = 1, 2.)

3.5. The Girsanov Theorem

In order to motivate the results of this section, let us consider independentnormal random variables Z1, Z,, on (52,,F, P) with EZi = 0, EZ? = 1.Given a vector (pi,. .., un)egr, we consider the new probability measure Pon (s2,9%) given by

1 nP(dw) = exp piZi(w) - - E id P(dw).

2 t =1

Then 15[Z le dz1,.. .,Zne dzn] is given by

exp [tE - 1- E id P [Z1 e dz 1, , Zn e dzn]

1 n= (2n)-"12 exp [ - - E (zi - pi)2]dz dzn.

2 i=t

Therefore, under P the random variables Z1, Zn are independent andnormal with EZi = pi and £[(Z1 - yi)2] = 1. In other words, {Z = Zi -1 < i < n} are independent, standard normal random variables on (s2, y, P).The Girsanov Theorem 5.1 extends this idea of invariance of Gaussian finite-dimensional distributions under appropriate translations and changes of the

3.5. The Girsanov Theorem 191

underlying probability measure, from the discrete to the continuous setting.Rather than beginning with an n-dimensional vector (Z1,...,4) of indepen-dent, standard normal random variables, we begin with a d-dimensionalBrownian motion under P, and then construct a new measure P under whicha "translated" process is a d-dimensional Brownian motion.

A. The Basic Result

Throughout this section, we shall have a probability space (n, P) and ad-dimensional Brownian motion W = {W = Wm), A; 0 < t < co}defined on it, with P[Wo = 0] = 1. We assume that the filtration {F,} satis-fies the usual conditions. Let X = {X, = (V), , V)), A; 0 < t < ool bea vector of measurable, adapted processes satisfying

(5.1) P dt < co]= 1; 1 < i < d, 0 5 T < co.

Then, for each *, the stochastic integral /"(X(')) is defined and is a memberof eirioc. We set

(5.2)1

Zi(X) exp [ it dWs(i) ft=1 a 2 o

11Xsil 2 dd.

Just as in Example 3.9, we have

(5.3) WZ,(X)= 1 d (1),

which shows that Z(X) is a continuous, local martingale with Zo(X) = 1.Under certain conditions on X, to be discussed later, Z(X) will in fact be a

martingale, and so EZ,(X) = 1; 0 < t < co. In this case we can define, for each0 < T < co, a probability measure PT on FT by

(5.4) PT(A) E[lAZT(X)]; AE FT T.

The martingale property shows that the family of probability measures{PT; 0 < T < col satisfies the consistency condition

(5.5) PT(A) A(A); AE 0 < t < T.

5.1 Theorem (Girsanov (1960), Cameron and Martin (1944)). Assume thatZ(X) defined by (5.2) is a martingale. Define a process W = {W, =Ito)), A; 0 < t < col by

(5.6) It(l) Wt(i) - ds; 1 < i < d, 0 < t < co.

For each fixed T e [0, co), the process {It, A; 0 < t < T} is a d-dimensionalBrownian motion on (CI, FT, PT).

The preparation for the proof of this result starts with Lemma 5.3; the readermay proceed there directly, skipping the remainder of this subsection on firstreading.

Discussion. Occasionally, one wants 1V, as a process defined for all t e [0, op),to be a Brownian motion, and for this purpose the measures {PT; 0 < T < co}are inadequate. We would like to have a single measure P defined on sothat P restricted to any ..FT agrees with PT; however, such a measure does notexist in general. We thus content ourselves with a measure P defined only onFcw, the o--field generated by W, such that P restricted to any ..*-TW agrees with

PT, i.e.,

(5.7) P(A) = E[1AZT(X)]; AE , 0 5_ T < co.

If such a P exists, it is clearly unique. The existence of P follows from theDaniell-Kolmogorov consistency Theorem 2.2.2. We show this when d = 1;only notational changes are required for the multidimensional case.

Let t = (t . . . , t) be a finite sequence of distinct, nonnegative numbers, asin Definition 2.2.1, and let t = max {t,,... , to. Define

Qt(r) = [WE CI; Mit i((0), Wtn(a))) r]; re.6$(R ").

Then {$21} is a consistent family of finite-dimensional distributions, so thereis a probability measure Q on (RE°''), gi(Rtc"'))) such that

Q,(1") = Q[w e (w(t ,), , w (t )) e 11; e .4(01").

But the typical set in ..ny has the form {co e W(w)e B}, where B e Mi(Rt"')).Consequently, Q induces a probability measure P on F' defined by

Flu) e n; WOO e B] A Q(B); B e .4(011°'')),

and this measure satisfies (5.7).

The process 1;17 in Theorem 5.1 is adapted to the filtration {",}, and so isthe process {ft, Xr ds; 0 < t < co}; this can be seen as in part (c) of the proofof Lemma 2.4, which uses the completeness of .97,. However, when workingwith the measure P which is defined only on Fes, we wish 1,r to be adaptedto Ign. This filtration does not satisfy the usual conditions, and so we mustimpose the stronger condition of progressive measurability on X. We havethe following corollary to Theorem 5.1.

5.2 Corollary. Let W = {W, gft; 0 < t < co} be a d-dimensional Brownianmotion on (12, P) with P[Wo = 0] = 1, and assume that the filtration {.F,}satisfies the usual conditions. Let X = {X ; 0 < t < co} be a d-dimensional,progressively measurable process satisfying (5.1). If Z(X) of (5.2) is a martingale,then 1,V = {14-7 ; 0 < t < co) defined by (5.6) is a d-dimensional Brownianmotion on (11," cow , P).

PROOF. For 0 5 ti < < t S t, we have

PC(1171,..., G = Pt[(147/t. ai)e r EAR").The result now follows from Theorem 5.1.

Under the assumptions of Corollary 5.2, the probability measures P andP are mutually absolutely continuous when restricted to Fr ; 0 < T < cc.However, viewed as probability measures on P and 15 may not bemutually absolutely continuous. For example, when d = 1 and X, = it, anonzero constant, then

Z,(X) = exp[i2147, - t]; 0 < t < cc

is easily seen to be a martingale. Corollary 5.2 and the law of large numbersimply

1 1 - 1

P lim -W, = pl = P lim -W, = 01= 1, P[lim -W, = pi= O., t ,-00 t.."D t

In particular, the P-null event flim, (1/t) W, = /II is in FT for every 0 _5T < co so 15 and PT cannot agree on T. This is the reason we require (5.7)to hold only for A E ..FTw.

B. Proof and Ramifications

We now proceed with the proof of Theorem 5.1. We denote by ET (E.) theexpectation operator with respect to PT (/-5).

5.3 Lemma. Fix 0 < T < co and assume that Z(X) is a martingale. If 0 < s _5t < T and Y is an .5(7,-measurable random variable satisfying ETjYI < co, thenwe have the Bayes' rule:

G[YI T's] = (ma Yzi(x)IFJ, a.s. P and PT.

PROOF. Using the definition of E T, the definition of conditional expectation,and the martingale property, we have for any A e

ETI1A Zs(X)EEYZt(X)IFJ} = EflAE[YZt(X)1Fs]}

= E[IA YZ,(X)] = ET[lA

We denote by J1;4' the class of continuous local martingales M = IM0 5 t < T1 on (12, P) satisfying P[MO = 0], and define similarly,with P replaced by PT.

5.4 Proposition. Fix 0 S T< co and assume that Z(X) is a martingale. IfM e..16:1" then the process

(5.8) A A Mt - i't XV d<M, W(II>s, .97,; 0 5 t< T1=1 o

is in ,thl". If N e .11,:l" and

IC I, A N, -i ft XV d <N ,14/IiI> s; 0 5 t 5 T,i=1 o

then

<1171,R>, = <M, N>,; 0 < t < T, a.s. P and PT,

where the cross-variations are computed under the appropriate measures.

PROOF. We consider only the case where M and N are bounded martin-gales with bounded quadratic variations, and assume also that Z,(X) and11=i ro (XS )2 ds are bounded in t and w; the general case can be reduced tothis one by localization. From Proposition 2.14

ft XV d<M,14/IiI>,2

< <M>, 1 t (X.1`))2 ds,o

and thus SI is also bounded. The integration-by-parts formula of Problem 3.12gives

Z,(X)S1, = 1 Zu(X)d111,, + > ft 1171.XLIZAX)d14;V,o i=1 o

which is a martingale under P. Therefore, for 0 < s < t < T, we have fromLemma 5.3:

EAR tl.F.,1 = Z,(X)[E Z,(X)1171,1,j = las, a.s. P and 15T.

It follows that AI e 2'1".The change-of-variable formula also implies:

AR, - <M, N>, = ft M dN, + ft IV. dM, -i [ ft 111XLII d<N,14/ItI>.o o i=1 o

+ft I1 XV d<M, Ww>.1o

as well as

Z,(X)[1171,1C, - <M, N> J = ft Zu(X)111dN. + ft Zu(X)R. dM,o o

+ f t [M.N. - <M, N >] XV 4,(X) dW,V .

i=1 0

This last process is consequently a martingale under P, and so Lemma 5.3implies that for 0 .s5t5T

EAR iNt - <IV N>t1.97s] = - N> s; a.s. P and PT.

This proves that <M, N>, = <M, N),; 0 5 t < T, a.s. PT and P.

PROOF OF THEOREM 5.1. We show that the continuous process Won (II, ,FT, PT)satisfies the hypotheses of P. Levy's Theorem 3.16. Setting M = Wth in Prop-osition 5.4 we obtain SI = IP) from (5.8), so Wu) e i/V". Setting N = HM),we obtain

<1170 VV(k)> <W-1),W(``)>, = Sj.kt; 0 < t < T, a.s. PT and P.

Let {M 0 < t < T} be a continuous local martingale under P. Withthe hypotheses of Theorem 5.1, Proposition 5.4 shows that M is a continuoussemimartingale under PT. The converse is also true; if {114 .Ft; 0 < t <is a continuous martingale under PT, then Lemma 5.3 implies that for0 < s < t < T:

E[Z,(X)1171,IF] = Zs(X)ET[1171,13U = Zs(X)M- s; a.s. P and PT,

so Z(X)/171 is a martingale under P. If SI EMI", a localization argumentshows that Z(X)M edie;21". But Z(X) e dr, and so Ito's rule implies that Al =[z(X)iii]/Z(X) is a continuous semimartingale under P (cf. Remark 3.4).Thus, given gi eA:'°c, we have a decomposition

git = Mt + Bt; 0 < t < 'T,

where M e .11P° and B is the difference of two continuous, nondecreasingadapted processes with Bc, = 0, P-a.s. According to Proposition 5.4, the process

d- (M, - E d<M, Wffi> )i=1 o

d=A+if t d<M,W")>s; 0 t 5 T,i=1 o

is in A:1", and being of bounded variation this process must be indistinguish-able from the identically zero process (Problem 3.2). We have proved thefollowing result.

5.5 Proposition. Under the hypotheses of Theorem 5.1, every M e 22;1'°` hasthe representation (5.8) for some M e

We note now that integrals with respect to dlit(1) have two possible inter-pretations. On the one hand, we may interpret them by replacing dl,Pi(l) byd147,") - XY)dt so as to obtain the sum of an Ito integral (under P) and aLebesgue-Stieltjes integral. On the other hand, 1,V(`) is a Brownian motion

under PT, so we may regard integrals with respect to dii as Ito integralsunder PT. Fortunately, these two interpretations coincide, as the next problemshows.

5.6 Problem. Assume the hypotheses of Theorem 5.1 and suppose Y = { Yt, "t;0 < t < co} is a measurable adapted process satisfying P[11; Y2 dt < co] = 1;0 < T < co. Under P we may define the Ito integral Sto I's dWs"), whereasunder PT we may define the Ito integral Po Ys dlik,(1), 0 < t < T Show that for1 < i < d, we have

= Ys dW(i) - Ut I

Y dW(i) V ds; 0 < t < T, a.s. P and PT.jot s --0 o

(Hint: Use Proposition 2.24.)

C. Brownian Motion with Drift

Let us discuss a rather simple, but interesting, application of the Girsanovtheorem: the distribution of passage times for Brownian motion with drift.We consider a Brownian motion W = { 14/ ,F,; 0 < t < co} and recall fromRemark 2.8.3 that the passage time Tb to the level b 0 has density andmoment generating function, respectively:

(5.9) P[Tb e dt] = exp [--b2] dt; t > 0,/ 27tt3 2t

(5.10) Ee-2Tb = e-lbl; a > 0.

For any real numbers 0, the process W = { IT - pt, gir; 0 < t < colis a Brownian motion under the unique measure P(") which satisfies

P(n(A) = EDAM; A e

where Z, A exp(gW, - iii2t) (Corollary 5.2). We say that, under 1341), 14/, =

pt + i1 is a Brownian motion with drift p. On the set { Tb < t} e n =3'7,w Tb we have Z, A 71, = ZTb, so the optional sampling Theorem 1.3.22 andProblem 1.3.23 (i) imply

(5.11) P(u)[Tb < t] = {Tb,,}Z,] = E[1{4.,,}E(Z,Ig7,,,,b)]

= E[1{Tb<,}4,rb] = E[l{rbst}ZTb]

= (7.b <06,0-012w2Tb]

= J t exp (µb -2-1 µ2s) P[T, e ds].

The relation (5.11) has several consequences. First, together with (5.9) ityields the density of Tb under P"):

(5.12) P(*)[Tbedt] = I

2b I

t3exp [ (b

2tPt)] dt, t> 0.

Second, letting t co in (5.11), we see that

P(")[T < co] = e"E[exp( -114270],

and so we obtain from (5.10):

(5.13) /34")[Tb < co] = exp[ab -

In particular, a Brownian motion with drift p # 0 reaches the level b 0 0 withprobability one if and only if p and b have the same sign. If It and b have oppositesigns, the density in (5.12) is "defective," in the sense that P(")[Tb < co] < 1.

5.7 Problem. Let T be a stopping time of the filtration {..fr} with P[T < co] =1. A necessary and sufficient condition for the validity of the Wald identity

(5.14) E[exp(pWr - 1/42 T)] = 1,

whereµ is a given real number, is that

(5.15) P(PIT < co] = I.

In particular, if be fR and pb < 0, then this condition holds for the stoppingtime

(5.16) S, inff t 0; W, - pt =- 131.

5.8 Problem. Denote by

h(t;b, p) ,Ibl

exp(b - 2

2ttit)];t > 0, b 0, p e 1,

7-ct3

the (possibly defective) density on the right-hand side of (5.12). Use Theorem2.6.16 to show that

h( ; b, + b2, p) = h( ;hi, * h( h2, P); bi b2 > 0, eR,

where * denotes convolution.

5.9 Exercise. With kt > 0 and W* inft>0 Wt, under P(") the random variable-W* is exponentially distributed with parameter 2a, i.e.,

P(*)[- W* e db] = 2pe-2*b db, b > 0.

5.10 Exercise. Show that

E(u)e-ar, = exp(ab - Ibl,//22 + 2a), a > 0.

5.11 Exercise (Robbins & Siegmund (1973)). Consider, for v > 0 and c > 1,the stopping time of {77,"1:

Show that

R, = inf {t > 0; exp(v W, =

1P[R, < co] E(Y)R, -

2log c.

c v2

D. The Novikov Condition

In order to use the Girsanov theorem effectively, we need some fairly generalconditions under which the process Z(X) defined by (5.2) is a martingale. Thisprocess is a local martingale because of (5.3). Indeed, with

T. A inf It > 0; max f (Z,(X)Xli))2 ds = n}i<i<d 0

the "stopped" processes Z(") A {Z:") A 4,., T.(x), .Ft; 0 < t < co} are martin-gales. Consequently, we have

E[ZtATI's] = A ; 0 < S < t, n > 1,

and using Fatou's lemma as n co, we obtain E[Z,(X)I.Fs] < 4; 0 s s s_ t.In other words, Z(X) is always a supermartingale and is a martingale if andonly if

(5.17) EZ1(X) = 1; 0 < t < co

(Problem 1.3.25). We provide now sufficient conditions for (5.17).

5.12 Proposition. Let M = {M gr,; 0 < t < oo} be in Jr' and define

4 = exp[M, - i<M>1]; 0 < t < co.

If

(5.18) E[exp{ <M>,}] < oo; 0 < t <

then EZ, = 1; 0 < t <

PROOF. Let T(s) = inf {t > 0; <M>, > s }, so the time-changed process B of(4.16) is a Brownian motion (Theorem 4.6 and Problem 4.7). For b < 0, wedefine the stopping time for IC as in (5.16):

S, = infls > 0; Bs - s = bl

Problem 5.7 yields the Wald identity E[exp(Bsb - ISO] = 1, whenceE[exp(lS,)] = e-b. Consider the exponential martingale lc A exp(B, -(s /2)), 0 < s < ool and define {N, A Y-3Asb, 4s; 0 < s < co}. According toProblem 1.3.24 (i), N is a martingale, and because P[Sb < co] = 1 we have

N. = lim Ns = exp(Bsb - iSb).CO

It follows easily from Fatou's lemma that N = {Ns, '.,; 0 < s < co) is a super-martingale with a last element. However, ENS = 1 = EN0, so N = IN 's;0 < s < co) has constant expectation; thus N is actually a martingale witha last element (Problem 1.3.25). This allows us to use the optional samplingTheorem 1.3.22 to conclude that for any stopping time R of the filtration {/s}:

E[exp{Bsb - l(R A Sz, ) } ] = 1.

Now let us fix t e [0, 00) and recall, from the proof of Theorem 4.6, that <M>,is a stopping time of {/s}. It follows that for b < 0:

(5.19) E[1(sb <,,,,),) exp(b + ISM + E[1(0,0,<sbi exp(M, - i<M>,)] = 1.

The first expectation in (5.19) is bounded above by ebE[exp(Z <M>,)], whichconverges to zero as b1 -co, thanks to assumption (5.18). As b. -co, thesecond expectation in (5.19) converges to EZ, because of the monotoneconvergence theorem. Therefore, EZ, = 1; 0 < t < co.

5.13 Corollary (Novikov (1972)). Let W = {W, = (W,(1),..., W,(d)), A; 0t < co) be a d-dimensional Brownian motion, and let X = pc, = (Xi r)),A; 0 < t < co) be a vector of measurable, adapted processes satisfying (5.1). If

(5.20) E[exp (-21 f To 11Xs 112 ds)]< co; 0 < T < co,

then Z(X) defined by (5.2) is a martingale.

5.14 Corollary. Corollary 5.13 still holds if (5.20) is replaced by the followingassumption: there exists a sequence {t},T=0 of real numbers with 0 = to <t1 < < tni co, such that

(5.21) E [exp (-1 f in IlXs112ds)1 < oo; do 1.2 tn -1

PROOF. Let X,(n) = (X)1)1i,n_1.,)(t),..., Xrlitn_i.t.)(0), so that Z(X(n)) is amartingale by Corollary 5.13. In particular,

E[Z,.(X (n))1,,._i] = Z,,,_i(X(n)) = 1; n > 1.

But then,

E[Z,(X)] = E[Z,_1(X)EIZi(X(n))1A_,}i = E[4_1(X)],

and by induction on n we can show that E[Z,.(X)] = 1 holds for all n > 1.Since E[Z,(X)] is nonincreasing in t and limn_o, t = co, we obtain (5.17).

CI

5.15 Definition. Let C[0, cod be the space of continuous functions x: [0, oo) ->Rd. For 0 < t < co, define % A r(x(s); 0 < s < t), and set / = /co (cf. Prob-lems 2.4.1 and 2.4.2). A progressively measurable functional on C[0, ce)d is a

mapping /2: [0, oo) x C[0, oar -> IR which has the property that for each fixed0 5 t < co, ti restricted to [0, t] x C[0, oor is .4([0, t]) %/.4([11)-measurable.

Ifµ = tion) is a vector of progressively measurable functionals onC[0, °or and W = = (W,(1), Wt(d)), 37".,; 0 < t < co} is a d-dimensionalBrownian motion on some P), then the processes

(5.22) ,qi)(u)) A p(l)(t, W(oo)); 0 < t < co, 1 0 depending on T, the condition

(5.23) ti(t, x)II < K T(1 + x*(t)); 0 < t

where x*(t) g maxo Ilx(s)11. Then with X, = , Xr) defined by (5.22),Z(X) of (5.2) is a martingale.

PROOF. If, for arbitrary T > 0, we can find {to, , tnal} such that 0 = to <t1 < < = T and (5.21) holds for 1 n < n(T), then we can constructa sequence ItnI,T.0 satisfying the hypotheses of Corollary 5.14. Thus, fix T > 0.We have from (5.22), (5.23) that whenever 0 < ti_, < t,, < T, then

Itt,_,

where W; A maxor II NI. According to Problem 1.3.7, the process Y, Aexp[(t. - )K + II Wt )2/4] is a submartingale, and Doob's maximalinequality (Theorem 1.3.8(iv)) yields

t11)(3112 ds < (t,, - 4_1)1(1(1 + WT*)2,

E exp[f(t - )1q(1 + WT*)2] = EI max Yi2 < 4EY,4,0<t<T

which is finite provided that t,, - t_, _< 1/Tiq. This allows us to construct{to, , tn(T)} as described previously. El

5.17 Remark. Liptser & Shiryaev (1977), p. 222, show that with d = 1 and0 < e < 1, one can construct a process X satisfying the hypotheses of Corol-lary 5.13 but with (5.20) replaced by the weaker condition

E [exp IG - e) fo dt}1 < co; 0 < T < co,

such that Z(X) is not a martingale.

The next exercise, taken from Liptser & Shiryaev (1977), p. 224, provides asimple example in which Z(X) is not a martingale. In particular, it shows thata local martingale (cf. (5.3)) need not be a martingale.

3.6. Local Time and a Generalized ItO Rule for Brownian Motion 201

5.18 Exercise. With W = { 14/,, .; 0<t<1} a Brownian motion, we define

T = inf {0 < t < 1; t + WW2= 1 },

2

(1- t)20 < t < 1,

0; t = 1.

(i) Prove that P[T < 1] = 1, and therefore .1,`, X,2 dt < cc a.s.(ii) Apply Ito's rule to the process {(41/1(1 - t))2; 0 < t < 1} to conclude that

X, d147, dt0

1 ft

T1

o (1 - 031Wt2

(iii) The exponential supermartingale {Z,(X), ,F,; 0 5 t II is not a mar-tingale; however, for each n > 1 and an = 1 - ( 1 /.\Ai), {Z, (X), .??7,;0 < t < 1} is a martingale.

5.19 Exercise. Let W = {W, g",; 0 < t < co} be a Brownian motion on(0,F, P) with P[Wo = 0] = 1, and assume that {Ft} is the augmentationunder P of the Brownian filtration {,,W}. Suppose that, for each 0 < T < oo,there is a probability measure PT on .FT which is mutually absolutelycontinuous with respect to P, and that the family of probability measures{PT; 0 < T < oo} satisfies the consistency condition (5.5). Show that thereexists a measurable, adapted process X = {X 317,; 0 < t < co satisfying (5.1),such that Z(X) defined by (5.2) is a martingale and (5.4) holds for 0 < T < co.(Hint: Apply Problem 4.16 to the Radon-Nikodj/m derivative process dP, /dP.)

5.20 Exercise. Suppose that {L .Ft; 0 < t < op} e Jr' is such that Z, Aexp[L, -1<L>,] is a martingale under P, and define the new probabilitymeasure PT(A) E(1,4); A e FT. Establish the following generalization ofProposition 5.4 and of the Girsanov theorem: if M e ../r1", then

/17f, A Mt - <L, M>, = M - d<Z M> , ;007", ; 0 t To Zs "is in di'. (Hint: Imitate the proof of Proposition 5.4.)

3.6. Local Time and a Generalized ItO Rulefor Brownian Motion

In this section we devise a method for measuring the amount of time spentby the Brownian path in the vicinity of a point x e R. We saw in Section 2.9that the Lebesgue measure of the level set .1°.(x)= {0 5 t < oo; 14/,(co) =

turns out to be zero, i.e.,

(6.1) meas 2',,,(x) = 0, for P-a.e. co ER

yielding no information whatsoever about the amount of time spent in thevicinity of the point x (Theorem 2.9.6 and Remark 2.9.7). In search of anontrivial measure for this amount of time, P. Levy introduced the two-parameter random field

(6.2) Lt(x) = lim -1

meas {0 < s < t; I Ws - xi < e}; t e [0, oo), x e gla 0 4a

and showed that this limit exists and is finite, but not identically zero. Weshall show how Lt(x) can be chosen to be jointly continuous in (t, x) and, forfixed x, nondecreasing in t and constant on each interval in the complementof the closed set 2',,,(x). Therefore, (d/dt)L,(x) exists and is zero for Lebesguealmost every t; i.e., the function t i- Lt(x) is singularly continuous. P. Levycalled Lt(x) the mesure du voisinage, or "measure of the time spent by theBrownian path in the vicinity of the point x." We shall refer to Lt(x) as localtime.

This new concept provides a very powerful tool for the study of Browniansample paths. In this section, we show how it allows us to generalize It6'schange-of-variable rule to convex but not necessarily differentiable functions,and we use it to study certain additive functionals of the Brownian path. Thesefunctionals will be employed in Chapter 5 to provide solutions of stochasticdifferential equations by the method of random time-change. Local time willbe further developed in Chapter 6, where we shall use it to prove that theBrownian path has no point of increase (Theorem 2.9.13). In this section, thereader can appreciate the application of local time to the study of Brown-ian sample paths by providing a simple proof of their nondifferentiability(Exercise 6.6). This exercise shows that jointly continuous local time cannotexist for processes whose sample paths are of bounded variation on boundedintervals.

Throughout this section, 1147 A; 0 < t < col, (f2,F), {Pz }Z Ft denotes theone-dimensional Brownian family on the canonical space f2 = C[0, co). Thisassumption entails no loss of generality, because every standard Brownianmotion induces Wiener measure on C[0, co) (Remark 2.4.22), and resultsproved for the latter can be carried back to the original probability space. Wetake the filtration {..F, } to be {A} defined by (2.7.3), and we set .F = .7'03. Thisfiltration satisfies the usual conditions. In this situation, PZ is just a translateof P°, i.e.,

(6.3) P(F) = P°(F - z); F e g ,

(cf. (2.5.1)). We also have at our disposal the shift operators {0,},,,, definedby (2.5.15).

6.1 Definition. A measurable, adapted, real-valued process A = { A .; 0 .-

t < co} is called an additive functional if, for every z e 18 and Pz-a.e. co e f2,

3.6. Local Time and a Generalized Ito Rule for Brownian Motion 203

we have

(6.4) At+s(w) = As(co) + At(Osto); 0 < s, t < co.

6.2 Example. For every fixed Borel set Be .41([11), we define the occupation timeof B by the Brownian path up to time t as

(6.5) co) -4 J t 1B( Ws)ds = meas {0 Lc. s Lc t; Ws e B}; 0 ...5_ t < co,o

where meas denotes Lebesgue measure. The resulting process F(B) --- {F1(B),,F;; 0 < t < co} is adapted and continuous, and is easily seen to be an additivefunctional.

A. Definition of Local Time and the Tanaka Formula

Equation (6.2) indicates that doubled local time 24(x) should serve as a densitywith respect to Lebesgue measure for occupation time. In other words, weshould have

(6.6) TAB, co) = i 24(x, w) dx; 0 < t < co, B e .4(01).B

We take this property as part of the definition of local time.

63 Definition. Let L = {L,(x, w); (t, x) e [0, oo) x R, to ell} be a random fieldwith values in [0, co), such that for each fixed value of the parameter pair (t, x)the random variable Lt(x) is .- measurable. Suppose that there is an eventll* e g" with Pz(11*) = 1 for every z e 0:8 and such that, for each w e fe, the func-tion (t, x)i-- Lt(x, co) is continuous and (6.6) holds. Then we call L Brownianlocal time.

6.4 Remark. There is no universal agreement in the literature as to whetherL in Definition 6.3 or 2L is to be called local time. We follow the normalization(6.6), used by Ikeda & Watanabe (1981).

6.5 Remark. With L as in Definition 6.3 and w e 11*, one can immediatelyderive (6.2) from (6.6) and the continuity of x 1-- Lt(x, co). Further, L(a) ={L1(a), g;; 0 < t < co} is easily seen to inherit the additive functional property(6.4) from its progenitor, the occupation time I" (Example 6.2).

6.6 Exercise. Assume that Brownian local time exists and show that for eachw ell* of Definition 6.3, the sample path t F-4 Wt(o)) cannot be differentiableanywhere on (0, co). (Hint: If t'-' Wt(w) is differentiable at t, then for somesufficiently large C and sufficiently small S > 0 we must have iWt+h(w) -W,(w)1 _._ Ch; 0 5 h 5 b.)

6.7 Problem.

(i) Show that the validity of (6.6) is equivalent to

(6.7) fo f(147,(co)) ds = 2 ff (x) Lt (x, co) dx; 0 t < co,

for every Borel-measurable function f : l -4 [0, co).(ii) Let ft' be the class of continuous functions h: [11 - [0, 1] of th,:. form

(6.8) .112h(x) =

0;

x -q i

x

q1

q2

q3

x

q1 ,

x

x

x

q4,

q2,

q3,

q4,

- q1

1;

q4 -xq4 - q3

0;

where q1 < q2 < q, < q4 are rational numbers.

1

q2 (13 q4

Show that if (6.7) holds for all h E .°, then it holds for every Borel-measurable function f : [18 -, [0, co).

Our plan is the following: we shall assume in the present subsection thatBrownian local time exists, and we shall derive a convenient representationfor it, the Tanaka formula (6.11). In the next subsection it will be shownthat the right-hand side of (6.11) leads to a random field which satisfies therequirements of Definition 6.3, thus establishing existence.

Let us fix a number a e R, and take f(x) in (6.7) to be the Dirac deltaevaluated at x - a, and derive formally the representation

1 f '(6.9) Lt(a, co) = 6 (117 s(co) - a) ds.

But the integral on the right-hand side is only formal. In order to give itmeaning, we consider the nondecreasing, convex function u(x) = (x - a) +,which is continuously differentiable on R \Ial and whose second derivative inthe distributional sense is u"(x) = 5(x - a). Bravely assuming that Ito's rulecan be applied in this highly irregular situation, we write

(6.10) (Wt - a)+ - (z - a)+ = f0

1(,,00)(14/3)dW, +2f (5(Ws - a) ds,

0

and in conjunction with (6.9), we have

(6.11) L,(a) = (W, - a)+ - (z - a)+ -f I 1(..00(4 )dWs; 0 ._ t < co0

Pz-a.s. for every z E R. Despite the heuristic nature of both (6.9) and (6.10), therepresentation (6.11) for local time is valid and will be established rigorously.

6.8 Proposition. Let us assume that Brownian local time exists, and fix a numbera e R. Then the process L(a) = {L,(a), ,F,; 0 _<_ t < co} is a nonnegative, continu-ous additive functional which satisfies (6.11) and the companion representations

(6.12) L,(a) = (W, - a)- - (z - a)- + ft 1(,,,l(Ws)dWs; 0 < t < co,0

(6.13) 24(a) = I W, - al - lz - al - ft sgn(Ws - a) digs; 0 t < ooo

a.s. Pz, for every z E R.

6.9 Remark. Any of the formulas (6.11), (6.12), or (6.13) is referred to as theTanaka formula for Brownian local time. We need establish only (6.11); then(6.12) follows by symmetry and (6.13) by addition, since

Pz[i. 1{.}(Ws) dW, = 0; V0 < t < oo]=-- 1; V z e R.o

In particular, it does not matter how we define sgn(0) in (6.13); we shall definesgn so as to make it left-continuous, i.e.,

(6.14)1;

sgn(x) =x > 0x _. 0.

6.10 Remark. The process { (W, - a) +, ,; 0 < t < col is a continuous, non-negative submartingale (Proposition 1.3.6); it admits, therefore, a uniqueDoob-Meyer decomposition (under 1', for any z e R):

(6.15) (W, - a)+ = (z - a)+ + M,(a) + A,(a); 0 < t < co,

where A(a) is a continuous, increasing process and M(a) is a martingale(see Section 1.4). The Tanaka formula (6.11) identifies both parts of thisdecomposition, as A,(a) = L,(a) and

t

(6.16) M,(a) = 1. 1(,,o)(Ws) dWs; 0 5 t < co.o

Similar remarks apply to the representations (6.12) and (6.13).

PROOF OF PROPOSITION 6.8. In order to make rigorous the heuristic discussionwhich led to (6.11), we must approximate the Dirac delta 5(x) by a sequenceof probability densities with increasing concentration at the origin. Morespecifically, let us start with the C' function

(6.17) p(x)cexp[(x - 1)2 - 1];

1

0 < x < 2,

0; otherwise

which satisfies f°20, p(x) dx = 1 by appropriate choice of the constant c, anduse it to define the probability density functions (called mollifiers)

(6.18)

as well as

un(x)

/Mx) -4- np(nx)

Pn(z - a) dz dy; xeIll, n > 1.

We observe that un(x) = f X p(z - a) dz , and so we have the limiting relations

lim u'(x) = 1(,,,n)(x), lim u(x) = (x - a) +, x e141.

We now choose an arbitrary z e R. According to Ito's rule,

(6.19) un(W,) - u,,(z) = un(14/s)dW, + ft pn(W, - a) ds; 0 t < co,2 ,

a.s. Pz. But now from (6.7) and the continuity of local time,t

p(14!, - a) ds = 2 pn(x - a)Li(x)dx 2L,(a);

On the other hand,

Ez j; un(Ws)dW 0,- 1o,)(Ws) dW,2

a.s. P2.

= Ez j . lun(Ws) - 10,,)(Ws)12 ds Pz[IW, - al -2nids,

which converges to zero as n co. Therefore, for each fixed t, the stochasticintegral in (6.19) converges in quadratic mean to the one in (6.16), and (6.11)for each fixed t follows by letting n co in (6.19). Because of the continuityof the processes in (6.11), we obtain that, except on a Pz-null event, (6.11) holdsfor 0 < t < co.

B. The Trotter Existence Theorem

We can employ now the Tanaka representation (6.11) to settle the questionof existence of Brownian local time. This is in fact the only result proved inthis subsection.

6.11 Theorem (Trotter (1958)). Brownian local time exists.

PROOF. We start by showing that the two-parameter random field, obtainedby setting z = 0 on the right-hand side of (6.11), admits a continuous modifi-cation under P°. The term (W, - - ( -a)+ is obviously jointly continuousin the pair (t, a). For the random field {M,(a); 0 5 t < oo, a E RI in (6.16) wehave, with a < b, 0 <s<t5Tand any integer n > 1:

E° I M1(a) - Ms(b)I2"

< 4" {E°2n

+ E° 1(..b)(W.)dW.s f

< 4"Cn[E° (ft 1.(,,,,)(Wn)du)n + E° (cthanks to (3.37). The first expectation is bounded above by (t - s)", whereasthe second is dominated by

e[f 1,0.6,(W,)dt0

T T

= E° [1 (a , b]( Wt 1, 1 (a, bl(W tn)] dt. dt,

= n!r

[1(a, 11( WI, ) 10 Wt2). . . 10,,bi(Win)]dtn...dt2dt1.ti

With 0 < t < 0 < T, we have for every y E R,

P° [a < Wo < blW, = y] < P° [a < Wo b

(6-- 012.18 -t b -a= - e-Z212 dZ <fo / - tand so

a + blW= 2J

E°[T 1(,,,,,i(W)dto

(b -a n T T T

Cti(t2 - ti)...(tn - )]-1/2 dt . dt2 dti

en (b - a)",

where en, T is a constant depending on n and T but not on a or b. Therefore,witha<band 0 <s < t 5 T, wehave

(6.20) E°14(a) - 1115(b),' 5 Cn, - sr + (b - 47]5 C., T a) - (s, b)11"

for some constant C,,.. By the version of the Kolmogorov-Centsov theoremfor random fields (Problem 2.2.9), there exists a two-parameter random fieldIlt(a); (t, a) e [0, GO) x RI such that for P°-a.e. wee, the mapping (t, a)1-.I t(a, w) is locally Holder-continuous with exponent y, for any y e (0,1), and foreach fixed pair (t, a) we have

(6.21)

Now we define

P° [It(a) = Mt(a)] = 1.

Lt(a) A (W - a)+ - (- a)+ - I t(a); 0 <_ t < GO, a e R.

For fixed (t, a), Lt(a) is an ,t-measurable random variable, and the randomfield L is P°-a.s. continuous in the pair (t, a). Indeed, because 14/, and I t(a) areboth locally HOlder-continuous for any exponent y e (0,1), the local time Lalso has this property: for every y e (0,1), T > 0, K > 0, there exists a P°-a.s.positive random variable h(w) and a constant 6 > 0 such that

[ I Lt(a, co) -L s(b, (0)1(6.22) P° co e SI; sup < 6]= 1.

o<11060-(s,b)11<h(o) Il(t, a) - (s, b)IlY0 <s,t< T

-K<a,b<K

Our next task is to show that the random field Lt(a) satisf es the identity(6.6), or equivalently (6.7). For every function h in the class Yt of Problem 6.7,define

'D x YH(x) A I h(u) - u)+ du = Ih(u) du dy; x e R

and observe the identities

H' (x) = 1 h(u)1 (,,,c0(x) du = 1x

h(u) du, H"(x) = h(x).

By virtue of It6's rule and Problem 6.12, we have P°-a.s. for fixed t > 0:

01 1 t2

h(Ws)ds

= H(W) - H(0) - ft H' (Ws) dWs0

=

1.0 t .0

h(a) {(Wt - a)+ - (- a)+ 1 da -f (J h(a)1(a,c0)(Ws)da)dWs-.0 o

=fx' h(a){(Wt - a)+ - (- a)+ - ft 1(a,c0)(Ws)dWs} da-.0 o

= fm + fm - Mt(a)} da.

But E° ITco (I t(a) - Mt(a))2 da = Icf,, E° (L(a) - Mt(a))2 da = 0 by (6.21).

3.6. Local Time and a Generalized It6 Rule for Brownian Motion 209

Thus, for each fixed t > 0, we have for P°-a.e. w e S2

(6.23) co h(Ws(co))ds = 2 i h(x)L,(x, to) dx .- -co

Since both sides of (6.23) are continuous in t and Jr is countable, it is possibleto find an event M e.F with P°(123) = 1 such that for each w e 03, (6.23) holdsfor every h e Ye' and every t > 0. Problem 6.7 now implies that for each w e 03,(6.7) holds for every Borel function f: fll -) [0, co).

Recall finally that LI = C[0, co) and that P assigns probability one to theevent S2z -A {w e SI; w(0) = z }. We may assume that C23 g 00, and redefineLt(x, w) for w 0 Slo by setting

Li(x, w) A Lt(x - w(0), w - w(0)).

We set SP = {w ell; w - w(0) e M}, so that MCP) = 1 for every z e DI(c.f. (6.3)). It is easily verified that L and Q* have all the properties set forthin Definition 6.3. 1:1

6.12 Problem. For a continuous function h: R -> [0, co) with compact support,the following interchange of Lebesgue and Ito integrals is permissible:

(6.24) f 1 h(a)( f : 1 (,,,,,,)(Ws) d Ws) da

=ft (..ix' h(a)1()(Ws)da)dWs, a.s. P°.o -co

6.13 Problem. We may cast (6.13) in the form

(6.25) I Hit - al = I z - al - 13,(a) + 24(a); 0 < t < co,

where B,(a) A -Po sgn( Ws - a) digs, for fixed a e R.

(i) Show that for any z e R, the process B(a) = {B,(a), .97i; 0 .< t < col is aBrownian motion under P, with Pz [B0(a) = 0] = 1.

(ii) Using (6.25) and the representation (6.2), show that L(a) = {L,(a), ,-,;0 < t < co} is a continuous, increasing process (Definition 1.4.4) whichsatisfies

(6.26) J 101\{.}(NdLi(a) = 0; a.s. P=.1o

In other words, the path t'--> L,(a, w) is "flat" off the level set Y,,,(a) ={0 t < co; W(w) =a }.

(iii) Show that for P°-a.e. w, we have L,(0, w) > 0 for all t > 0.(iv) Show that for every z e R and Pz-a.e. w, every point of Yo,(a) is a point of

strict increase of ti--+ L ,(a , w).

C. Reflected Brownian Motion and the Skorohod Equation

Our goal in this subsection is to provide a new proof of the celebrated resultof P. Levy (1948) already discussed in Problem 2.8.8, according to which theprocesses

(6.27) {4w - w, A max W- W; 0 -t <co} and {I W; 0 5t <co}4:..st

have the same laws under P°. In particular, we shall present the ingeniousmethod of A. V. Skorohod (1961), which provides as a by-product the factthat the processes

(6.28) {Mr A max Ws; 0 < t < a)} and {2L,(0); 0 < t < op}ossst

also have the same laws under P°.

6.14 Lemma (The Skorohod equation (1961)). Let z > 0 be a given numberand y() = {y(t); 0 t < co} a continuous function with y(0) = 0. There existsa unique continuous function k() = {k(t); 0 5 t < co}, such that

(i) x(t) A z + y(t) + k(t), 0; 0 t < aD,(ii) k(0) = 0, k() is nondecreasing, and(iii) k() is flat off {t 0; x(t) = 0 }; i.e., l'o° 1{xop.0}dk(s) = 0.

This function is given by

(6.29) k(t) = max [0, max { -(z + y(s)) }], 0 5 t < co.1:..,..1

PROOF. To prove uniqueness, let k() and ic() be continuous functions withproperties (i)-(iii), where x() and Si) correspond to k() and k(-), respec-tively. Suppose there exists a number T > 0 with x(T) > i(T), and letT A- max {0 < t < T; x(t) - Z(t) = 0} so that x(t) > 5e(t) > 0, V t e(T, T]. Butk() is flat on {u 0; x(u) > 0 }, so k(t) = k(T). Therefore,

0 < x(T) - z(T) = k(T) - k(T) < k(t) - kW = x(t) - 54-r),

a contradiction. It follows that x(T) < Z(T) for all T > 0, so k 5 k. Similarly,k > k.

We now take k( ) to be defined by (6.29). Conditions (i) and (ii) are obviouslysatisfied. In order to verify (iii), it suffices to show Jo 1{x(s),,} dk(s) = 0 forevery E > 0. Let (t t2) be a component of the open set {s > 0; x(s) > el andnote that

-(z + y(s)) = k(s) - x(s) _-_ k(t2) - E; t, .5 s 5 t2.But then

k(t2) = max [k(t,), max {-(z + y(s)) }] 5 max[k(ti), k(t2) - a],t,...st,

which shows that k(t2) = k(t,).

6.15 Remark. For every z > 0 and y() e C[0, co) with y(0) = 0, we denote byif the class of functions k e C[0, co) which satisfy conditions (i) and (ii) ofLemma 6.14 and introduce the mappings

(6.30) T,(z; y) A max [0, max 1 -(z + y(s)) }1; 0 < t < coo<s<t

(6.31) R,(z; y) A z + y(t) + T(z; y); 0 5 t < co.

In terms of these, the solution to the Skorohod equation is given by

(6.32) k(t) = 7;(z; y), x(t) = R,(z;y),

and T(z; y) is the minimal element of , as can be seen from the first part ofthe proof of Lemma 6.14.

6.16 Proposition. Let z > 0 be a given number, and B = {B 1,; 0 < t < 00}a Brownian motion on some probability space (0,1,Q) with Q[B0 = 0] = 1.We suppose there exists a continuous process k = {k1,; 0 < t < co} such that,for Q-a.e. 0 e 0, we have

(i) X JO) z - + k,(0) 0; 0 _5 t < co ,(ii) ko(0) = 0, t k,(0) is nondecreasing, and(iii) 1,`;) 10.00)(X 5(0)) dk,(0) = 0.

Then X = {Xt; 0 < t < 00} under Q has the same law as 1141 = {I H1,1;0 s t < col under Pz.

PROOF. The law of the pair (k, X) is uniquely determined, since by Lemma 6.14k,(0) = T(z; - B.(0)), X,(8) = R,(z; - B.(0)); 0 < t < co, for Q-a.e. B e O. Itsuffices, therefore, on our given measurable space (S1,") equipped with theBrownian family I/V .9,; 0 < t < co}, {Px}),14, to exhibit a standard Brown-ian motion B = {B 0 < t < co} and a continuous nondecreasing processk = {k 0 _5 t < 00} such that, for Pz-a.e. w E

W(w)1 = z - Bt(w) + kt(w); 0 < t < co,

(6.33) ko(o)) = 0, t k,(co) is nondecreasing, and

I EiR,{0}(ws.)dks.)=0.But this has already been accomplished in Problem 6.13 (relations (6.25), (6.26)with a = 0), if we make the identifications

B, Er- - t sgn(Ws)dVV kt = 2L,(0).

6.17 Theorem (P. Levy (1948)). The pairs of processes {(MR' - Mr);0 t < co} and {(1W,1,24(0)); 0 5 t < co} as in (6.27), (6.28) have the samelaws under P°.

PROOF. Because of uniqueness in the Skorohod equation, we have from (6.33)

(6.34) 24(0, co) = MP(to), I No41 = MN(0) - B,(0)); 0 < t < 00

for P ° -a.e. co e S/, upon observing that

(6.35) MP(co) = max lits(w)= 'T,(0; -B(w))o<s<i

(Remark 6.15). The assertion follows, since both W and B are Brownianmotions starting at the origin under P°. We also notice the useful identity,valid for every fixed t e [0, co):

(6.36) AV = lim -1

meas {0 < s < t; Mr - B5 < c}, a.s. P°. Li.i.1) 2s

6.18 Problem. Show that for every pair (a, z)eg12 we have

Pz [co e SI, lim Lf(a,w)= co] = 1.1-.00

D. A Generalized ItO Rule for Convex Functions

The functions fl (x) = (x - a)+, f2(x) = (x - a)- , and f3(x) = Ix - al in theTanaka formulas (6.11)-(6.13) share an important property, namely convexity:

(6.37) f().x + (1 - A)z) Af(x) + (1 - A)f(z); x < z, 0 < A < 1,

which can be put in the equivalent form

(6.38) f (Y)z -y

f(x) +y -x

f(z); x < y < zz -x z -x

upon substituting y = Ax + (1 - 01.)z. Our success in representing f(W) ex-plicitly as a semimartingale, for the particular choices f(x) = (x - a)± andf(x) = Ix - al, makes us wonder whether it might be possible to obtain ageneralized Ito formula for convex functions which are not necessarily twicedifferentiable. This possibility was explored by Meyer (1976) and Wang (1977).We derive the pertinent Ito formula in Theorem 6.22, after a brief digressionon the fundamental properties of convex functions on R.

6.19 Problem. Every convex function f: R -> ER is continuous. For fixed x e ER,the difference quotient

A f(x + h)- f(x)(6.39) Af (x; h) = ; h 0 0

h

is a nondecreasing function of he R \ 101, and therefore the right- and left-derivatives

(6.40) DIf(x) lim1- [f(x + h) - f(x)]

h-o+ h

exist and are finite for every x e R. Furthermore,

(6.41) D' f(x) < D f(y) < D'f(y); x < y,

and WA) (respectively, D-f()) is right- (respectively, left-) continuous andnondecreasing on R.

Finally, there exist sequences fa1,°;'=, and {/3};,°=, of real numbers, such that

(6.42) f(x) = sup (anx + A); x e R.n>1

(Hint: Use (6.38) extensively.)

6.20 Problem. Let the function (p: R O be nondecreasing, and define

x(p+(x) = lim ('(y), 1'(x) = (u) du.y--uc± d 0

(i) The functions cp.,. and (p_ are right- and left-continuous, respectively, with

(6.43) (p_(x) < cp(x) < (p_,_(x); x e R.

(ii) The functions cp., have the same set of continuity points, and equalityholds in (6.43) on this set; in particular, except for x in a countable set N,we have (14 (x) = p(x).

(iii) The function cl) is convex, with

D-(1)(x) = (p_(x) < cp(x) < (p_,.(x) = D+ 10(x); x e R.

(iv) If f: R R is any other convex function for which

(6.44) D- f(x) < cp(x) < D+ f(x); x e R,

then we have f(x) = f(0) + (1)(x); x e R.

6.21 Problem. For any convex function f : O R, there is a countable setN l such that f is differentiable on R \ N, and

(6.45) f '(x) = D+ f(x) = D-f(x); x e R\ N

Moreover

(6.46) f(x) - f(0) = f f '(u) du = J x D-1 f(u) du; x e R.

The preceding problems show that convex functions are "essentially" differ-entiable, but Ito's rule requires the existence of a second derivative. For aconvex function f, we use instead of its second derivative the second derivativemeasure p on (R, AR)) defined by

(6.47) p([a, b)) g D-f(b) - D- f(a); -co < a < b < co.

Of course, if f " exists, then it(dx) = f "(x) dx. Even without the existence of f ",we may compute Riemann-Stieltjes integrals by parts, to obtain the formula

(6.48) g(x)/1(dx) = -f g'(x)D-f(x)dx

for every function g R R which is piecewise and has compact support.

6.22 Theorem (A Generalized Ito Rule for Convex Functions). Let f: Rbe a convex function and it its second derivative measure introduced in (6.47).Then, for every z e R, we have a.s. Pz:

(6.49) f(W) = f(z) + J D f (Ws) digs + J Li(x)it(dx) 0 < t < co.o

PROOF. It suffices to establish (6.49) with t replaced by t n T_ n 1,, and bysuch a localization we may assume without loss of generality that D+f isuniformly bounded on R. We employ the mollifiers of (6.18) to obtainconvex, infinitely differentiable approximations to f by convolution:

CO

(6.50) fn(x) f p(x - y)f(y)dy; n 1.CO

It is not hard to verify that fn(x) = f°200 p(z)f(x - (z /n)) dz and

(6.51) lim fn(x) = f(x), lim f,;(x) = Erf(X)n-,co n-co

hold for every x e R. In particular, the nondecreasing functions D f and{f'} 11_, are uniformly bounded on compact subsets of R. If g: R R is of classC1 and has compact support, then because of (6.48),

lim g(x)f"(x)dx = -lim g'(x)f,,'(x)dxJ-.n- co n-.co -co

co

= - g'(x)D-f(x)dx = J g(x)it(dx).

A continuous function g with compact support can be uniformly approximatedby functions of class 0, so that

(6.52) lim 1 g(x)f"(x)dx = 1 g(x)p(dx).n- co -co -co

We can now apply the change-of-variable formula (Theorem 3.3) to f(Ws),and obtain, for fixed t e (0, co):

`f 1 fin(Wt) - fn(z) = on1WOdWs + -2

0fn" (Ws) ds, a.s. Pz..1

When n -, co, the left-hand side converges almost surely to f(W,) - f(z), andthe stochastic integral converges in L2 to fic, a 1(Ws)d Ws because of (6.51) and

the uniform boundedness of the functions involved. We also have from (6.7)and (6.52):

lira fn"(Ws)ds = 2 lim f,"(x)Li(x)dx = 2 L,(x)p(dx), a.s. P2"OD 0 n-.co

because, for P2-a.e. co GS-2, the continuous function x w) has supporton the compact set [min, Ws(co), Ws(co)]. This proves (6.49) foreach fixed t, and because of continuity it is also seen to hold simultaneouslyfor all t e [0, co), a.s. PZ. El

6.23 Corollary. If f: R R is a linear combination of convex functions, then(6.49) holds again for every z e R; now, p defined by (6.47) is in general a signedmeasure with finite total variation on each bounded subinterval of R.

6.24 Problem. Let a, < a2 < < an be real numbers, and denote D ={a, , . , an} . Suppose that f:111 P is continuous and f' and f" exist and arecontinuous on R \D, and the limits

f ' (a, ±) s lim f '(x), f "(a, ±) = lim f "(x)x-wk + x--*ak ±

exist and are finite. Show that f is the difference of two convex functions and,for every z e R,

(6.53) f(W,) = f(z) + f'(W)dW, +2-1 ft f"(W)ds

0

+ E Li(a,)[f ' (a, +) -f ' (a, - )]]; 0 < t < co, a.s. P .k =1

6.25 Exercise. Obtain the Tanaka formulas (6.11)-(6.13) as corollaries of thegeneralized Ito rule (6.49).

E. The Engelbert-Schmidt Zero-One Law

Our next application of local time concerns the study of the continuous,nondecreasing additive functional

A1(w) = fg f(Wn(co))ds; 0 t < co,0

where f: P [0, co) is a given Borel-measurable function. We shall beinterested in questions of finiteness and asymptotics, but first we need anauxiliary result.

6.26 Lemma. Let f: P [0, cc) be Borel-measurable; fix x e P, and supposethere exists a random time T with

T

Pq0 < T < co] = 1, P° [ I. f(x + Ws) ds < col> O.

o

Then, for some e > 0, we have

(6.54) Lf(x + y)dy < co.

PROOF. From (6.7) and Problem 6.13 (iii), we know there exists an event SPwith infe) = 1, such that for every co en*:

T"I f(x + Ws(w)) ds = 2 a° f(x + y)LT(a,)(y, co) dyo --.

and LT())(0, co) > 0. By assumption, we may choose co e SI* such thatfl-"f(x + Ws(w)) ds < cc as well. With this choice of w, we may appeal tothe continuity of LT(0)( , w) to choose positive numbers c and c such thatLT(.)(y, co) c whenever IA 5 E. Therefore,

2c

I.e T(w)

f(x + y) dy 5 j. f(x + Ws(co))ds < co,-E 0

which yields (6.54). D

6.27 Proposition (Engelbert-Schmidt (1981) Zero-One Law). Let f: l[0, co) be Borel-measurable. The following three assertions are equivalent:

(i) P°[IL f(His)ds < co; V 0 -_ t < co] > 0,(ii) P° [I:, f(Ws)ds < oo; V0 5 t < oo] = 1,(iii) f is locally integrable; i.e., for every compact set K c R, we have

JKf(y)dy < cc.

PROOF. For the implication (i) (iii) we fix be R and consider the firstpassage time Tb. Because P° [Tb < co] = 1, (i) gives P°[ f o Tb f(Ws)ds < co;V 0 < t < co] > 0. But then

,4- ori,coo ,-4- Tb(eol

f(47(0))ds > f(Ws(w))ds = j. f(b + Bs(w))ds,JO fTb(w) 0

where Bs(w) A 14/,+71(.)(co) - b; 0 < s < oo is a new Brownian motion underP°. It follows that for each t > 0, P°[ f 10 f(b + Bs) ds < co] > 0, and Lemma6.26 guarantees the existence of an open neighborhood U(b) of b such thatSu(of (Y1dY < oo. If K R is compact, the family {U (0}4. lc, being an opencovering of K, has a finite subcovering. It follows that Ix f 61 dy < co.

For the implication (iii) (ii) we have again from (6.7), for P°-a.e. w e ft

Jo

mt(co)

f(Ws(w))ds = 2 I f(y)Lt(y, co) dy = 2 I f(y)Lt(y, w) dy-. m,(.)Mt(w)

5[ max 24(y, co)] - .1 f(y)dy; 0 5 t < co,?NO)). A 1 t(0)) m,(0))

3.7. Local Time for Continuous Semimartingales 217

where ?Ow) = min° ss Ws(co), Mt(co) = max05,,, Ws(w). The last integral isfinite by assumption, because the set K = [mi(co), A li(co)] is compact.

6.28 Corollary. For 0 < a < co, we have the following dichotomy:

p0 [f E wdSla< 00; V 0 < t < 00 =

1; if 0 < a < 110; if a _._ 1 f

6.29 Problem. The conditions of Proposition 6.27 are also equivalent to thefollowing assertions:

(iv) P° [po f (Ws) ds < co] = 1, for some 0 < t < co;(v) Px[fo f(Ws) ds < co; V 0 < t < co] = 1, for every x e IR;

(vi) for every x e R, there exists a Brownian motion IB g,; 0 < t < col anda random time S on a suitable probability space (0, , Q), such thatQ [Bo = 0, 0 < S < co] = 1 and

Q[

sf f(x + 13s)ds < ool> O.o

(Hint: It suffices to justify the implications (ii) = (iv) (vi) (iii) (v) (vi),

the first and last of which are obvious.)

630 Problem. Suppose that the Borel-measurable function f: R --9 [0, oo)satisfies: measty e 01; f(y) > 01 > 0. Show that

.3

(6.55) Px[u) e SI; f f (Ws(w))ds = xi= 1

o

holds for every x e Fl. Assume further that f has compact support, and considerthe sequence of continuous processes

1

X:n) A- f(Ws)ds; 0 t < co, n > 1..\/n o

Establish then, under P°, the convergence

(6.56) X (^) --. X

in the sense of Definition 2.4.4, where X, -4 2 V f ll 1 L,(0) and Vfil, Aff. f(y) dy > 0.

3.7. Local Time for Continuous Semimartingalest

The concept of local time and its application to obtain a generalized Ito rulecan be extended from the case of Brownian motion in the previous sectionto that of continuous semimartingales. The significant differences are that

f This section may be omitted on first reading; its results will be used only in Section 5.5.

time-integrals such as in formula (6.7) now become integrals with respectto quadratic variation, and that the local time is not necessarily jointlycontinuous in the time and space variables. We shall use the generalized Itorule developed in this section as a very important tool in the treatment ofexistence and uniqueness questions for one-dimensional stochastic differentialequations, presented in Section 5.5.

Let

(7.1) X, = X0 + M, + Vt; 0 < t < ocbe a continuous semimartingale, where M = IM Ft; 0 < t < col is in dr',V = { V g;; 0 < t < oo} is the difference of continuous, nondecreasing, adaptedprocesses with Vo = 0 a.s., and {,,} satisfies the usual conditions. The resultsof this section are contained in the following theorem and are inspired bya more general treatment in Meyer (1976); they say in particular that con-vex functions of continuous semimartingales are themselves continuous semi-martingales, and they provide the requisite decomposition.

7.1 Theorem. Let X be a continuous semimartingale of the form (7.1) on someprobability space (Q, .0-i-, , P). There exists then a semimartingale local time forX, i.e., a nonnegative random field A = lAt(a,w);(t,a)e [0, co) x R, wen}such that the following hold:

(i) The mapping (t, a, w) F- At(a,w) is measurable and, for each fixed (t, a), therandom variable Ada) is .97,-measurable.

(ii) For every fixed a e R, the mapping ti--11,(a,w) is continuous and non-decreasing with Ao(a, w) = 0, and

(7.2) fom 1 R\ {}(X,(co)) d A,(a, co) = 0, for P-a.e. wen.

(iii) For every Borel-measurable k: IR [0, op), the identityt

(7.3) 1 k(Xs(co))d<M> (co) = 2 1 k(a)A,(a, co) da; 0 < t < coo

holds for P-a.e. we a

(iv) For P-a.e. we Q, the limits

lim At(b, co) = At(a, co) and At(a -, w) '' lim At(b, co)r-,1b.i.a

S -./bt a

exist for all (t, a) e [0, cc) x R. We express this property by saying that Ais a.s. jointly continuous in t and RCLL in a.

(v) For every convex function f:R R, we have the generalized change ofvariable formula

(7.4) f(X,) = f(X0) + 1 Df(Xs)dM, + 1 Df(Xs)dV,t

o o

+ I At(a)tt(da); 0 < t < oo, a.s. P,CO

3.7. Local Time for Contiauous Semimartingales 219

where D- f is the left -hand derivative in (6.40) and II is the second derivativemeasure (6.47).

7.2 Corollary. If f: R -- R is a linear combination of convex functions, (7.4)still holds. Now p defined by (6.47) is a signed measure, finite on each boundedsubinterval of R.

7.3 Problem. Let X be a continuous semimartingale with decomposition (7.1)and let f: R --- FR be a function whose derivative is absolutely continuous. Thenf" exists Lebesgue-almost everywhere, and we have the ItO formula:

t :

fiXt) = MO + f f(Xs)dM, + f r(X0dV,o o

1

2 0+ - f"(Xs)d<M>s; 0 s t< co, a.s. P.

7.4 Remark. In the setting of Theorem 6.17, we observe that the reflectedBrownian motion

X, A- 1W,1 = -B, + 2L,(0); 0 s t < co

is a semimartingale with X0 E-..- 0, M E-..-. -B, V :--.4 2L(0) under P°. The general-ized IVO rule (7.4) applied to this semimartingale with f(x) = Ix I gives inconjunction with (6.26): X, = -B, - 2L,(0) + 2A,(0), and therefore A,(0) =2L,(0); 0 < t < oo, a.s. P°. In other words, the semimartingale local time ofreflected Brownian motion at the origin is twice the Brownian local time at theorigin, as one would expect intuitively.

By way of preparation for the proof of Theorem 7.1, we provide a con-struction similar to that used in the proof of Theorem 6.22. Let f : 01 -. 011 beconvex. Thanks to the usual localization argument, we may assume that D-fis uniformly bounded on R and <M>, and V, are uniformly bounded in0 s t < oo and welt. Applying Ito's rule to the smooth function f, of (6.50),we obtain

(7.5) f(X,) = f,,(X0) + f 'f:(X.,)dMs + ft f,;(Xs)dV, + On); 0 S t < co,o o

where

On) =2-1 ft f,,(Xs)d<M>s; 0 5 t < co

0

is a continuous, nondecreasing (by the convexity of f,,), and {,,}-adaptedprocess. As in Theorem 6.22, we have as n -+ oo:

t t

f(X,)- f(X,) and f f;(Xs)d1/, --4 f D-f(Xs)dVs, a.s.o o

.1of;(Xs)d111, ft D-f(Xs)dM in probabilityo

for every fixed t. It follows that the remaining term CP) in (7.5) must alsoconverge in probability to a limit Ct(f), and

(7.6) f(Xt)= f(X0)+ 1 D-f(Xs)dills+ 1 Df(Xs)dV, + Ct(f); 0 t < oo.t t

o o

Now f(Xt) is continuous in t and both integrals have continuous modifica-tions, so we may and do choose a continuous modification of Ct(f). Each C")is nondecreasing and adapted to {.Ft}; the limit C(f) inherits both theseproperties.

With a e R fixed, we apply (7.6) to the functions fl(x) &` (x - a)+, f2(x) =(x - a)-, and f3(x) = Ix - al to obtain the Tanaka-Meyer formulas

t

(7.7) (Xt - a)± = (X0 - a)± + 1 1.0,,c0)(Xs)dills + 1 1.0,,c0)(Xs)dV, + CC (a)o o

t

(7.8) (Xt - a)- = (X0 - a) -1 lt_com(Xs)dMs- 1 lt_com(Xs)dVs+ Ct-(a)o o

t t

(7.9) I Xt - al = 1 X0 - al + sgn(X, - a)dM, +I

sgn(X, - a)dV,o o

+ 2At(a),

with the conventions (6.14), Ct±(a) Ct(ii), CC (a) .'` Ct(12), and 2At(a) ACt(f3). The processes Ct± (a), At(a) are adapted, continuous, and nondecreasingin t, and the random field At(a, w) will be our candidate for the local time ofX. Now (7.7) and (7.8) yield CC (a) = CC (a) (upon subtraction), as well as

(7.10) At(a) = Ct± (a); 0 t < cx), a e R

(upon addition and comparison with (7.9)).Although the process {At(a); 0 < t < ool is continuous for every fixed a, we

do not yet have any information about the regularity of At(a) in the pair (t, a).We approach this issue by studying the regularity of the other terms appearingin (7.7).

7.5. Lemma. Let X be a continuous semimartingale with decomposition (7.1).Define

It(a) A 1 t 10,,,o(Xs) dMs; 0< t< co, a e R.o

The random field I = {L(a), ,t; 0 < t < co, a e RI has a continuous modification.In other words, there exists a random field r= {rt(a),,t; 0 < t < 00, a e RIsuch that:

(i) For P-a.e. well, the mapping (t, a)I-* 1,(a, co) is continuous on [0, oo) x R.(ii) For every t a [0, co) and a a 11-8, we have 1,(a) = 11(a), a.s. P.

PROOF. By the usual localization argument, we may assume without loss ofgenerality that there is a constant K for which

(7.11) sup I Xr1 K, <MX, 5 K, Pc, < K ,

0 < ( < co

where R., is the total variation of V on [0, co). According to Remark 4.9, wemay choose a Brownian motion B for which we have the equations

(7.12)

(7.13)

I,(a) -4 j.

Hs(a) A

,

1.0,,,o(X)(//11o

s

1(s,))(11)dB.fo

=

=

Of>,

T(s)

o

1(.,o(Y)dB;

14,, 00(X) dM;

0 < t < co,

0 < s < co,

where T() is given by (4.15), lc -A XT(S) for 0 s < <MX, and I', is chosenso that 10,0,1( Ys) = 0 for s <M> b in some neighborhood of a (cf.(4.20)', Remark 4.9). We shall prove the existence of a continuous modificationof H by using the extension of the Kolmogorov-entsov theorem (Problem2.2.9). According to the latter, it suffices to show

(7.14) Ellk(a) - Hs1(b)1' 5 C[(s2 - s1)2 + (b - a)271+s;

0<s, <s2< co, a,beR

for suitable positive constants a, /3, and C. Note that

(7.15) E I 11,2(a) -Hs 1(b)18 < 2' E I 11,2(a) - 11,2(b)11 + 2' E 111, 2(b) -It i(b)18,

and, according to the martingale moment inequality (3.37), we may bound thelatter expectation by

E111,2(b) - 11,,(b)12 C.E[M(b)>s, - <1-1(b)>sj212 5 Cs(s2 - s1)a'2.

When a > 4, this bound is of the type required by (7.14). Thus, it remains onlyto deal with the first expectation on the right-hand side of (7.15), i.e., to show

(7.16) Ellis(a) - 115(b)12 5 C(b - a)2("); 0 < s < co, -co < a 4, /3 > 0, and C is a positive constant. We fix a < b and introducethe convex function

f (x) = f 4,,,i(z)dz dy,o o

for which I f 'I is bounded by b -a and 1)--f' = 10,,bi. In particular, passageto the limit in (7.5) yields

f(X,) = f(X0) + i f'(X0)d1148 + f f'(X0)dV0 +- 10, bi(X8)d<M>o.0 o 2 0 '

J', ,

X

Assumption (7.11) and Problem 1.5.24 show that X, and all the precedingintegrals have limits as t oo, so we may replace t by T(s) (which may beinfinite) to obtain, for every k > 1, the bound

(7.17)

rT(s)

Jo

k

G 6k I f(XT(s))- f (x 0)1k + 6k

+ 6krT(s)

f'(X,i)diiJo

k

j.T(s)

f'(X9)dMok

We bound the terms on the right-hand side of (7.17). The mean-value theoremimplies

I f (x T(o)) -f (x 0)1k < (b - a)k I XT(S) -X Olk2k Kk (b - a)k,

and it is also clear thatT(s) k

f'(X9)d1/9 < Kk(b - a)k.io

Applying the martingale moment inequality (3.37) to the stochastic integralin (7.17), we obtain the bound

EJ.

To)

f ' (X v)ditiv0

k

< lim E fo(s)Atf'(XJdMt,k

oo

G Ck E [1 I f ' (x v)I2 d<M>vo

< CkKki2(b - a)k.

We conclude from these considerations that there exists a constant C, dependingonly on k and on the bound K in (7.11), such that

ET(s) k

1(,,,)(Xt,)d<M>t, C(b - a)k; 0 < s < co, -co < a 2. From (7.12), (7.13) we see that It(a)= Hoo,(a),and since <M> is continuous, the existence of a continuous modification forH implies the same for I.

The Lebesgue-Stieltjes integral

(7.18) J,(a) --4 1[ 1(,00)(Xs)dVso

appearing in (7.7) can fail to be jointly continuous in (t, a); see Exercise 7.9.However, it is jointly continuous in t and RCLL in a; the proof is left as aproblem for the reader.

7.6 Problem. Let X be a continuous semimartingale with decomposition (7.1).For P-almost every co e 1, we have for all (t, a) e [0, co) x

(7.19)

(7.20) limttbfa

lim .1,(b,

b

=

co) =

f lia,c0(X,(0)dVs(co).0

PROOF OF THEOREM 7.1. Using Lemma 7.5 and Problem 7.6, we choose amodification of Cf+ (a) in (7.7) which is jointly continuous in t and RCLL in a.We take Ai(a) and C, ±(a) to satisfy (7.10) for every t, a, and co. In particular,Ai(a, co) is jointly a( [0, co)) 0 MR) F-measurable and satisfies (iv), and theother measurability claims of (i) and (ii) hold. In particular, it follows from(7.19), (7.20) that

A,(a) - A,(a -) = fo 1{}(Xs)dVs.

For the proof of (7.2), consider any two rational numbers 0 < u < v < coand the event

Ha A {co ED; Xs(co) < a, V se [u, v] }.

From relation (7.7) we have on H: Av(a, co) = A (a, co), except for co in anull set N,,,,. Let N M,_, 0 Na and fix co e 12\N. The set S(a, co) A

u,v e Q{0 < t < co; X,(co) < a} is open and, as such, is the countable union of disjointopen intervals. Let (a, /3) be such an interval. If a < u < v < /1, where u and vare rational, then A(a, co) = A (a, co). It follows that 1,.13) d A,(a, co) = 0, andthus

(7.21) IW0

1,_,,o(X,(co))d A,(a, co) = 0.

A similar argument based on (7.8) shows that

(7.22) 1,,)(X,(co))dA,(a, co) = 0, a.s.f.

which establishes (7.2).For the proof of (7.4), we may assume, by the usual localization argument,

that there exists a constant K > 0 such that (7.11) holds. Consequently, wemay also assume without loss of generality that D-f is constant outside(- K, K), so the second derivative measure p has support on [ - K, K). Letx e [ - K, K] be fixed and introduce the function

0; a < -K - 1,(x + K)(a + K + 1); -K - 1 < a < -K,

g(a) = x - a; -K < a < x,0; x < a.

According to (6.48), (6.46), we have for -K < x < K,

(7.23) (x - a) +µ(da) = -f g'(a)D- f (a) daJ

-K= -(x + K) D- f (a) da + J X D- (a) da

-K-1 -K

= -(x K) D-f(- K) + f(x) - f(- K),

and from the definition of it:

(7.24) cc' 1,,,,,,o(x)y(da) = K, x)) = D- f(x) - D-f(- K).

We may now integrate with respect to it in the Tanaka-Meyer formula (7.7)and use (7.23) to obtain

(7.25) f(X,) = (X, - X0) D- f( - K) + f(X0) + J J t 10,,,,)(X0d114,p(da)-co 0

+J

co

_co 0

co

i(,00(Xs)dVsit(da) + f Ai(a)it(da).-co

Fubini's theorem and (7.24) allow us to write

(7.26) fc° f t 10,,00)(Xs)dV, Oda) = f i 1)- f (X ,) dils -V, 1)- f ( - 10.-a) 0 0

A similar interchange of the order of integration in the integral

('t(7.27) 1(..)(X,) dM, y(da) = D- f(Xs) d114, -M, D- f(- K)

00 0

is justified by Problem 7.7 following this proof. Substitution of (7.26), (7.27)into (7.25) results in (7.4).

Finally, let us consider (7.4) in the special case f(x) = x2. Then µ(da) = 2 da,and comparison of (7.4) with the result from the usual Ito rule reveals that

<M>, = 2 J Ai(a)da; 0 < t < co, a.s.00

Thus, for any measurable function h(s, co): [0, co) x [0, co,OD CO OD

h(s, co) d <M>,(co) = 2 h(s, co) d As(a, co) dao -co o

holds for P-a.e. wee. Now if k: -* [0, co) is measurable, we may takeh(s,(o) = 110,11(s)- k(X Jo))) and obtain for P-a.e. to:

3.7. Local Time for Cc ntinuous Semimartingales 225

J.`k(X (w))d<M>s(co)= 2 k(Xs(w))dAs(a,w)da

oJ

s _.

= 2 J k(a)Ai(a, co) da; 0 5 t< co,

thanks to (7.2). This completes the proof.

7.7 Problem. Let X be a continuous semimartingale with decomposition (7.1),it be a 6- finite measure on (DI, .4(01)), and h: R [0, oo) be a continuousfunction with compact support. Then

.1°3 h(a)(1 10,,,)(Xs)dMs)p(da)= (.1 h(a)10,.)(Xs)p(da))01,.

7.8 Remark. The proof of Theorem 7.1 shows that the semimartingale localtime Azi" (a) for a continuous local martingale M is jointly continuous in (t, a),because the possibly discontinuous term .1,(a) of (7.18) is not present. Inparticular, (7.9) becomes then a.s. P:

(7.28) IM, - al =1M0 - + sgn(Ms - a)dMs + 2Ana); 0 < t < co.

Comparison of (7.28) with the Tanaka formula (6.13) shows that the semi-martingale local time A"(a) for Brownian motion W coincides with the localtime L(a) of the previous section. If M e,C, then for any stopping time t,

(7.29) BIM, Arl = 2E[e, r(0)]; 0 < t < co.

7.9 Exercise. Show by example that Jt(a) defined by (7.18) can fail to becontinuous in a.

7.10 Exercise. Let X be a continuous semimartingale with decomposition(7.1). Show that for every a e R,

0

1{.}(Xs)d<M>s= 0, a.s. P.

7.11 Exercise. Show that the semimartingale local time of a continuous processof bounded variation is identically zero.

7.12 Exercise (LeGall (1983)). Let X be a continuous semimartingale withdecomposition (7.1), and suppose that there exists a Borel-measurable functionk: (0, co) (0, co) such that .1(0,0 (du /k(u)) = po,V s > 0, but for every t e (0, co)we have

it d<M>jo k(X5) i{"} < c°'

Then the local time A(0) of X at the origin is identically zero, almost surely.

a.s. P.

7.13 Exercise. Consider a continuous local martingale M and denote St 4maxo <s <, Ms, L, A 2A,(0). Suppose now that f(t,x,y): R3 -0 J is a functionof class C2 (R3) which satisfies

in R3 and

of 102fat

+2 ax2

=0

Of

ax

of-(t, 0, y) +

ay- (t, 0, y) = 0

for every (t, y) e 1112. Show then that the processes f(< MX, 141,4) and .R<M>t,S, - Mt, St) are local martingales.

Deduce also the following:

(i) The process (S, - M,)2 - <M>, is a local martingale.(ii) For every real-valued function g of class C1(R), the processes

g(L) - 141 04)are local martingales.

and g(S,) - (S, - M,) g'(S,)

7.14 Exercise. For a nonnegative, continuous semimartingale X of the form(7.1) with X0 -..= 0, the following conditions are equivalent:

(i) 1/ is flat off {t > 0; X, = 0}.(ii) The process P01{,s00} dXs; 0 < t < co belongs to .ilic I".

(iii) There exists N edr.1" such that X, = maxo <,.,, Ns - Ali.


2.5. (a) It is easily verified that u - (1/2") < 9(u) < u. Consequently, XI") is ,-t-measurable, and since gyp" takes only discrete values, V"51 is simple.

(b) The procedure (2.7) results in measurable (but perhaps not adapted) processes{gopoopm=i 1, such that

T

lim E f.0151im) - XtI2 dt = 0.

m-00

Thus, for given e > 0, we can find m > 1 so that with X' A g(m) we haveE Jo IX: - x(12 dt < O. The Minkowski inequality leads then to

(E fr - Xt_hl2 dE)"20

1/2

(E IX, - XN2 dE) + E - X:_hI2 dtf(E IX _h - dt)

13

< 2E + (E - X:_h12 CIEYI)

We can now let h j 0 and conclude, from the continuity of X' and the boundedconvergence theorem, that

T

liM E j. IX, - dt < 4E2./40 0

(c) Let i be any nonnegative integer. As s ranges over [1/2", (1 + 1)/2"),tp(t - s) + s ranges over [t - (1/2"), t). Therefore,

T T

E - X,I2 ds dt = 2"E IX, - dhdt0 0 JO 0

= 2" s2- [E f T IX, - Xr_hI2 dtidh

T

13 0

max E IX, - X,_12 dt,0112-. 0

which converges to zero as n -0 co because of (b).(d) From (c) there exists a sequence {n, }ck°=, of integers, increasing to infinity

as k co, such that for meas x meas x P-a.e. triple (s,t,co) in [0,1] x[0, T] x SI, where meas means "Lebesgue measure," we have

(8.1) lim IXPk')(co) - X,0)I2 = 0.k-.co

Therefore, we can select se [0,1] such that for meas x P-a.e. pair (t, co) in[0, T] x fl, we have (8.1). Setting X('`) 9 X("ks), we obtain (2.8) from thebounded convergence theorem.

2.18. By assumption, we have

E<Im (X)> = E d<M>, < co.

Uniform integrability and the existence of a last element for IM(X) follow fromProblem 1.5.24, as does uniform integrability of (1m(X))2; similarly for /N( Y).

Applying Proposition 2.14 with X., Y. replaced by X.1{.). 71, Y.1{, T},respectively, we obtain

X. Ysd <M, N>,T+I T+t 1/2(f d<M>. f Y.2 d<N>s) ,

whence

a.s. P. Astherefore,

I XsYsd<M,N>s < (fXS

d<M>s f Ys2 d<N>s)112T

T -0 co, the right-hand side of this inequality converges to zero;

<Im(X), Y)>, = f Xs Ysd<M, N>s

converges as t -0 co and is bounded by the integrable random variable

(f: X d<M>.0

a.s. P. The dominated convergence theorem gives then

lim ER' (X)INY)] = lim E<Im (X), INY)>, = E<Im (X), INY)>s,fro.

= E f X sYsd <M, N>s.

We also have

E[e,f(X)C,(Y)] = E[(1:,4, (X) - r (X))(IVY) -Ii (Y))]

+ E[IP4(X)(IVY) -Ii (Y))]

+ E[I7(Y)(e,o4(X) - ir(x))]

+ E[Ir(X)17(Y)].

We have just shown that the fourth term on the right-hand side converges toE (X), IN (Y)>. as t -0 co. The other three terms converge to zero because ofHolder's inequality and the uniform integrability of Om( X))2 and ON Y))2.

2.27. With X E 9"(M), we construct the sequence of bounded stopping times {Tn}:=1in (2.32). In the notation of (2.33), each )0") is in 2*(M(")) and therefore can beapproximated by a sequence of simple processes PC"'''));,°_, g z in the sense

T

lim E j. - )0")I2 d<M>, = 0, V T < cok-co o

(Proposition 2.8). Let us fix a positive number T < co and consider n > T;we have P 01, I )0") - X,I2 d<M>, > 0] = n 1 > 0]P[T, < T] = P[S < T or 11-,X,2 d<M>, > N], and the last quantity convergesto zero as n -0 oo, by assumption. Now, given any E > 0, we have

TPH. T In") - X,I2 d<M>, > Ei P[f In") - n)I2 d<M>, >2-8]

o

+

P[fT

IX} ")- X,I2d <M >, >0

< -2 E T I Xl"'k) - n')I2 d <M > t + P[T < T]j.E 0

by the ebygev inequality.

For any given ö > 0 we can select 115> T so that P[T,, < T] < 312 for everyn n0, and we can find an integer kr,, 1 so that

J.TE I XP4.k..4) -)0 ,)12 d <Al> < -ei5

o 4

It follows that for e = S = j; j > 1, there exist integers n, and k" such that,with Yul A X('0"), we have

P[f - X,I2 d <Mx >j

Then according o Proposition 2.26, both sequences of random variables

T

YtLI) - X,12 d<M>,, sup 11,(YllI) - /,(X)I0 <t 5T

converge to zero in probability, as j oo, and there exists a subsequence forwhich the convergence takes place almost surely. Having done this constructionfor T fixed, we use a diagonalization argument as in the first paragraph of theproof of Lemma 2.4 to obtain a sequence which works for all T

In the case that M is Brownian motion, we use Proposition 2.6 rather thanProposition 2.8 in this construction.

2.29. We have

1 m-1s,(11) = E - 14c2) + - -Hit )2"ii0 "' i=0

1 1) "= 2147,2 + E -

2- E (wt

"1- W,)2.)2.

i=0

Recalling the discussion preceding Lemma 1.5.9, we may write

m-1 2

E[ E ovi, - wo2 - =E E [(141,+, - 111,, )2 - (t1 +1 ti)]i=0 1=0 12

= E EUK.i -14/02 -of+, - 032i-0

.-1ER,, - W04 - E (ti+, - t1)2

i=0

m-102

i=o

5_ c2t1lnii,

where we have used Problem 2.2.10. This proves (2.37). To see that E = 0 corre-sponds to the Ito integral, consider the (piecewise constant) process

m-1

i=o

in ..r,*(W), for which

..-1 ti+,E

ClInl - Ws12 ds = E Eiwi, -we dt

o 1=0 11

m-1 ti 1 'n-1= E (t - ti) dt = - E 41+1- 02-401=0 ti 2 1=0

as Mill 0. By definition, the HO integral pow; dWs is the L2-limit of Se(II) =Ito Xr dWk.

3.7. The proof is much like that of Theorem 3.3. The Taylor expansion in Step 2 ofthat proof is replaced by

f(tk, Xtd - Xik_,)

= [ffik, Xik) X,k)] + [ffik-1, Xid (tk_l, x,_,)]a- 4_1) + E

OXi=1 i

1d d 32

+ E fitk_i , love - xe,)(xe - xed,j=i Ax,ox;

where tk_i < srk < tk and tik is as before.

3.14. Let Y(i) = af(X,)/axi, so according to Ito's rule (Theorem 3.6),

+ f 32 fixs)dmo+ 32, focs)dBeJ=1 axiax; j=1 OXiOXi

1 ° ° 33s)d 0) M(k)>s.

2 k=1 j=1 aXiaXjaXk

V() =

It follows from (3.9) that

2

vo. = .10

1 t 3vodxv + -E f(Xs) d <Ma), Mo)>s,

2 J.1 aXiax;

and now (3.10) reduces to the HO rule applied to f(X i).

3.15. Let X and Y have the decomposition (3.7). The sum in question is

m-1 1 'n-1E - X11) + E (} - i,)(x+, - X ).1=o 2 1=0

The first term is ro dMs + dBs, where

rn -1

1 =0

and the continuity of Y implies the convergence in probability of Po (Ys' -Y)2 d<M> s and Po [Ysn - Ys] dBs to zero. It follows from Proposition 2.26 that

The other term is

rn-1

i=0E - x) rsdxs.

o

L i=c1(M1. - MOlArt,,E

m-i- + - E - )(C., - C,)2 i=o " ' "

m-i 1 m-i+ -2 Xo (N,, - - Be.) + (B4. - -

which converges in probability to l<M, N>, because of Problem 1.5.14 and thebounded variation of B and C on [0, t].

3.29. For x; > 0, i = 1, d, we have

xr + + d(x + + xd)m dm+1(xr + + x,r).

Therefore

(8.2)

and

(8.3)

104,112m [E (VT] dm E 00)12,.i=i

E 014(i)>7 < d(E <00>T) = dAT.i=1

Taking maxima in (8.2), expectations in the resulting inequality and in (8.3), andapplying the right-hand side of (3.37) to each M('), we obtain

EOM 114;12 m < dm E Eumolti2m dm E KmEum(oyn Kmdm+1E(A7)i=i 1=1

A similar proof can be given for the lower bound on EU M

4.5. (i) The nondecreasing character of T is obvious. Thus, for right-continuity, weneed only show that lim045 T(0) < T(s), for 0 s < S. Set t = T(s). Thedefinition of T(s) implies that for each E > 0, we have A(t + c) > s, and fors < 0 < A(t + E), we have T(0) < t + E. Therefore, lim,4s T(9) < t.

(ii) The identity is trivial for s > S; if s < S, set t = T(s) and choose E > 0. Wehave A(t + e) > s, and letting e l 0, we see from the continuity of A thatA(T(s)) > s. If t = T(s) = 0, we are done. If t > 0, then for 0 < e < t, thedefinition of T(s) implies A(t - e) < s. Letting E l 0, we obtain A(T(s)) < s.

(iii) This follows immediately from the definition of T().(iv) By (iii), T(A(t)) = t if and only if A(T) = A(t), in which case 49(t) = q9(t). Note

that if S < cc and A(t) = A(cc) for some t < co, then T(A(t)) = co and (i9 isconstant and equal to 41(t) on [t, co); hence, (p(co) = lira, (Mu) exists andequals (p(t).

(v) This is a direct consequence of the definition of T and the continuity of A.(vi) For a 5 t1 < t2 5 b, let G(t) = 11,,12,(t). According to (v), t1 5 T(s) < t2 if

and only if A(t,) s < A(t2), soA(b)

G(t)dA(t) = A(t2) - A(t1) = G(T(s))ds.A(a)

The linearity of the integral and the monotone convergence theorem implythat the collection of sets C e .41([a,b]) for which

A(b)

(8.4) 1,(t) dA(t) f 1,(T(s))dsA(o)

forms a Dynkin system. Since it contains all intervals of the form [t1, t2) c[a, b], and these are closed under finite intersection and generate 4([a, b]),we have (8.4) for every C e M([a, b]) (Dynkin System Theorem 2.1.3). Theproof of (vi) is now straightforward.

4.7. Again as before, every <M>, (resp., T(s)) is a stopping time of {s} (resp., {F,}),and the same is true of S <M>, (Lemma 1.2.11). The local martingale /IIhas quadratic variation CAI>, < <M>,(s2)= S n s2 < s2 < co (Problem 4.5 (ii)),so again both a, a2 - <til> are uniformly integrable martingales, and byoptional sampling:

E [AlT(s2) MT(Si)1 T(si )] = 0,

EP-4($2) - ICIT(.0)21,77(.)] = EE<M>T(s2) - <M>T001,71501; a.s. P.

It follows that MoT A {MTN, Ws; 0 < s < co} is a martingale with <M 0 T >s =<M>T(s), and by Problem 4.5 (iv), M 0 T has continuous paths. Now if{ Ws, Ws; 0 < s < co} is an independent Brownian motion, the process

BsAWs-W +M 4- 0_s<cios A S - T(s), - s,

is a continuous martingale with quadratic variation s = s - (s A S) +<M>T(s) = s, i.e., a Brownian motion. For this process, (4.17) is established byusing Problem 4.5 (iv).

4.11. Let (p be a deterministic, strictly increasing function mapping [0, co) onto [0,1),and define M E ir 1°' by

ow owM, --4 f Xs dWs, so <M >1 = f XS ds; 0 < t < co,

o o

and lim, <M>, = co, a.s. on E. According to Problem 4.7, there is a Brownianmotion B such that

Jt X s dWs = Bp(,), where p(t) A <M>,-i(i).

We have limgti p(t) = oo a.s. on E, and so, by the law of the iterated logarithmfor Brownian motion,

lm B,1) = -lim Bp(1) = +co, a.s. on E.it 1 it

4.12. (Adapted from Watanabe (1984).) From Problem 4.7 we have the representation(4.17) for a suitable Brownian motion B. Taking n > 3 max(' xi, pT) and denotingR = inf ft _>.: 0;141 n/31, we have the inclusions

{max I X,' n} g { max 141 .1= {<M>T. R,,} {PT Rn},(:)1,5T 051.sT 3

which lead, via (2.6.1), (2.6.2), and (2.9.20), to

P [axm I Xr1 n] P[R 5 pT] 2P[T13 '- PT] = 4P BpTCIT 3

12 jpdz <

n 2Rexp

18pT2R J

The conclusion follows.

6.12. Let h have support in [0, b], and consider the sequence of partitions

D " = 2"}; n >1

of this interval. Choosing a modification of Po 1()( Ws) dWs which is continuousin a (cf. (6.21)), we see that the Lebesgue (and Riemann) integral on the left-handside of (6.24) is approximated by the sum

2"-I bE -h(b17))( f i(bno")dw) = F(Ws)dWs,k=0 2"

where the uniformly bounded sequence of functions

2"-I bF(x) E -hoe)lor,,,,(x); n 1

k=0 2n

converges uniformly, as n -* co, to the Lebesgue (and Riemann) integral

F(x)

Therefore, the sequence of stochastic integrals {it F(Ws)dWs}`°_i converges inL2 to the stochastic integral Po F(Ws)dWs, which is the right-hand side of (6.24).

6.13. (i) Under any P, B(a) is a continuous, square-integrable martingale withquadratic variation process

<B(a)>, = f [sgn(Ws - a)]2 ds = t; 0 5 t < co, a.s. P .o

According to Theorem 3.16, B(a) is a Brownian motion.(ii) For w in the set Q* of Definition 6.3, we have (6.2) (Remark 6.5), and from

this we see immediately that L o(a, w) = 0 and L,(a, a)) is nondecreasing in t.For each z E R, there is a set n e.57" with p(n) = 1 such that 2°,(a) is closedfor all CO E n. For wenn Ce, the complement of .2°,0(a) is the countableunion of open intervals U. NI I. To prove (6.26), it suffices to show thatSi. dL,(co) = 0 for each a e NI. Fix an index a and let 1 = (u, v). Since W(w) -ahas no zero in (u, v), we know that I W(a)) - al is bounded away from theorigin on [u + (1/n), v - (1/n)], where n > 2/(v - u). Thus, for all sufficientlysmall e > 0,

meas { 0< s < u +1

-; I Ws - al 5 e} = meas {0 < s v --1; I Ws - aI 5 E},

whence 4,(1/)(a, co) = co). It follows that jr,,,(I,,,),-(1/0] dL,(a,w) =0, and letting n oo we obtain the desired result.

(iii) Set z = a = 0 in (6.25) to obtain I = -B,(0) + 24(0); 0 .5 t < oo, a.s. P°.The left-hand side of this relation is nonnegative; B,(0) changes sign infinitelyoften in any interval [0, E], e > 0 (Problem 2.7.18). It follows that L,(0) cannotremain zero in any such interval.

(iv) It suffices to show that for any two rational numbers 0 5 q < r < co, ifW,(co) = a for some t e (q, r) then Lq(a, < w), P-a.e. w. Let T(w)inf.{ t q; W(w) = a}. Applying (iii) to the Brownian motion {Ws ., - a;0 5 s < co} we conclude that

Li-00(a, a)) < LT(0,),(a, w) for all s > 0, Pz-a.e. w,

by the additive functional property of local time (Definition 6.1 and Remark6.5). For every we T < r} we may take s = r - T(w) above, and this yieldsLq(a, w) < L,(a, a)).

6.19. From (6.38) we obtain lim),4f (y) 5 f(x), limyt f(y) 5 f(z) and f(y) 5 lim,ty f(x),f(y) 5_ lin- ir.ty f(z). This establishes the continuity of f on R.

For E l fixed and 0 < h, < h2, we have from (6.38), with x = y = u + h1,z= + h2:(8.5) of(; h1) A.N;h2)

On the other hand, applying (6.38) with x = - h2, y = - hi, and z = yields

(8.6) AN; -h2) 5 4f(; -hi).Finally, with x = - e, y = z = + 6, we have

(8.7) Af(; -E) AN; 6); E, 6 > a

Relations (8.5)-(8.7) establish the requisite monotonicity in h of the differencequotient (6.39), and hence the existence and finiteness of the limits in (6.40).

In particular, (8.7) gives D- f(x) 5 Liff (x) upon letting E 10, 610, which estab-lishes the second inequality in (6.41). On the other hand, we obtain easily from(8.5) and (8.6) the bounds

(8.8) (y - x) to+f(x) 5 f(y) - f(x) 5 (y - x) Df(y); x < y,

which establish (6.41).For the right-continuity of the function Dtf(), we begin by observing the

inequality D+ f (x) 5 limy 4,, D+f(y); x e R, which is a consequence of (6.41). In theopposite direction, we employ the continuity of f, as well as (8.8), to obtain forx < z:

f(z) - f(x) = lim f (z) -f (y)lim Go+ f (y).

z -X y ,i, z -y y 4,.

Upon letting z 1 x, we obtain D+ f(x)_ limy 4 x D+f(y). The left-continuity ofIT f(-) is proved similarly.

From (8.8) we observe that, for any function (p: R - l satisfying

(8.9) D- f(x) < (p(x) S D+f(x); x E ER,

we have for fixed y E R,

(8.10) f(x) Gy(x) f(y) + (x - y)(p(y); x e R.

The function Gy) is called a line of support for the convex function fe). It isimmediate from (8.10) that f(x) = supyERGy(x); the point of (6.42) is that f()can be expressed as the supremum of countably many lines of support. Indeed,let E be a countable, dense subset of R. For any XE R, take a sequence { y }'.1of numbers in E, converging to x. Because this sequence is bounded, so arethe sequences {D±f(yO} 1 (by monotonicity and finiteness of the functionsD±)) and {q)(y)},T=1 (by (8.9)). Therefore, lim, Gyn(x) = f(x), which impliesthat f(x) = supy e EGy(x).

6.20. (iii) For any x < y < z,

(8.11)

This gives

we have

(1)(y) - (1)(x) 1co(u)du

x

f

- 4)(Y)

235

9(y)

9(z)

y-x1

y-x

z yco(u)du -(1)(z)

y

z -y y -x(D(Y) 4)(x) + 4:0(z),z -x z -x

which verifies convexity in the form (6.38). Now let x j y, z 1 y in (8.11), toobtain

9-(y) < D (D(Y) < 9(.1) < D+O(Y) 5- 9+(y); Y e

At every continuity point x of 9, we have 9±(x) = 9(x) = D± (1)(x). Theleft- (respectively, right-) continuity of 9_ and D- co (respectively, andWO) implies tp__(y) = D -1(y) (respectively, 9.,(y) = D+4)(y)) for all ye R.

(iv) Letting x 1 y (respectively, x T y) in (6.44), we obtain

Ir.f(Y) < (PAY) 5 9(Y) 5 9+(Y) 5- D+.1(Y); Y e R.

But now from (6.41) one gets

50+(x) D+ f(x) 5- D.1(Y) -5 9-01 5 OA x <

and letting y 1 x we conclude 9+(x) = 117(x); x e R. Similarly, we conclude9_(x) = D-f(x); xe R. Now consider the function G A f - 410, and simplynotice the consequences D±G(x) = (x) - D±(1)(x) = 0; x E P., of thepreceding discussion; in other words, G is differentiable on P. with derivativewhich is identically zero. It follows that G is identically constant.

6.29. ( iv ) ( vi): Let t e(0, co) be such that P°[1z, f(Ws) ds < co] = 1. For x = 0,just take S = t. For x # 0, consider the first passage time 7; andnotice that P°[0 < Tx < oo] = 1, P°[27; < t] > 0, and that {B, 4='Ws+t, - x, 0 < s < co} is a Brownian motion under P°. Now, forevery we {2T < t}:

2T (m)

f(x + 13,(a)))ds f(W(a)))du f f(Wx(co))du < oo,o LA.)

whence {2 Tx t} g {fp f(x + BOds < co}, a.s. P°. We concludethat this latter event has positive probability under P°, and (vi)follows upon taking S = Tx.Lemma 6.26 gives, for each x e K, the existence of an open neigh-borhood U(x) of x with fu(x)f(y)dy < co. Now (iii) follows fromthe compactness of K.For fixed x e R, define gx(y) = f(x + y) and apply the known impli-cation (iii) (ii) to the function gx.

7.3. We may write f as the difference of the convex functions

x )f,(x) f(0) + xf'(0) + [f"(z)]+ dz dy, f2(x) -4 f 1 [f"(z)]- dz dy,o o o 0

and apply (7.4). In this case, Adx) = f "(x) dx, and (7.3) shows that I% A,(a)m(da) =ifof "(Xs)d <M>s.

7.6. Let 1%,(w) denote the total variation of V(w) on [0, t]. For P-a.e. we a we haveI7,(w) < cc; 0 < t < co. Consequently, for a < b,

131(a) - .1(b)1 - + IJ,(a) - Jr(b)I < I V - VzI + 1(am(XS)dVs,0

and these last expressions converge to zero a.s. as r t and b La. Furthermore,the exceptional set of CO E S2 for which convergence fails does not depend on t ora. Relation (7.20) is proved similarly.

7.7. The solution is a slight modification of Solution 6.12, where now we use Lemma7.5 to establish the continuity in a of the integrand on the left-hand side.

3.9. Notes

Section 3.2: The concept of the stochastic integral with respect to Brownianmotion was introduced by Paley, Wiener & Zygmund (1933) for nonrandomintegrands, and by K. Ito (1942a, 1944) in the generality of the present section.ItO's motivation was to achieve a rigorous treatment of the stochastic differ-ential equation which governs the diffusion processes of A. N. Kolmogorov(1931). Doob (1953) was the first to study the stochastic integral as a martin-gale, and to suggest a unified treatment of stochastic integration as a chapterof martingale theory. This task was accomplished by Courrege (1962/1963),Fisk (1963), Kunita & Watanabe (1967), Meyer (1967), Millar (1968), Doleans-Dade & Meyer (1970). Much of this theory has become standard and hasreceived monograph treatment; we mention in this respect the books byMcKean (1969), Gihman & Skorohod (1972), Arnold (1973), Friedman (1975),Liptser & Shiryaev (1977), Stroock & Varadhan (1979), and Ikeda & Watanabe(1981) and the monographs by Skorohod (1965) and Chung & Williams (1983).Our presentation draws on most of these sources, but is closest in spirit toIkeda & Watanabe (1981) and Liptser & Shiryaev (1977). The approachsuggested by Lemma 2.4 and Problem 2.5 is due to Doob (1953). A majorrecent development has been the extension of this theory by the "Frenchschool" to include integration of left-continuous, or more generally, "predict-able," processes with respect to discontinuous martingales. The fundamentalreference for this material is Meyer (1976), supplemented by Dellacherie &Meyer (1975/1980); other accounts can be found in Metivier & Pellaumail(1980), Metivier (1982), Kopp (1984), Kussmaul (1977), and Elliott (1982).

Section 3.3: Theorem 3.16 was discovered by P. Levy (1948: p. 78); a different

3.9. Notes 237

proof appears on p. 384 of Doob (1953). Theorem 3.28 extends the Burkholder-Davis-Gundy inequalities of discrete-parameter martingale theory; see theexcellent expository article by Burkholder (1973). The approach that we followwas suggested by M. Yor (personal communication). For more informationon the approximations of stochastic integrals as in Problem 3.15, see Yor(1977).

Section 3.4: The idea of extending the probability space in order to accom-modate the Brownian motion W in the representation of Theorem 4.2 is dueto Doob (1953; pp. 449-451) for the case d = 1. Problem 4.11 is essentiallyfrom McKean (1969; p. 31). Chapters II of Ikeda & Watanabe (1981) and XIIof Elliott (1982) are good sources for further reading on the subject matter ofSections 3.3 and 3.4. For a different proof and further extensions of the F. B.Knight theorem, see Cocozza & Yor (1980) and Pitman & Yor (1986)(Theorems B.2, B.4), respectively.

Section 3.5: The celebrated Theorem 5.1 was proved by Cameron & Martin(1944) for nonrandom integrands X, and by Girsanov (1960) in the presentgenerality. Our treatment was inspired by the lecture notes of S. Orey (1974).Girsanov's work was presaged by that of Maruyama (1954), (1955). Kazamaki(1977) (see also Kazamaki & Sekiguchi (1979)) provides a condition differentfrom the Novikov condition (5.18): if exp(14) is a submartingale, then4 = exp(M, - l<M>1) is a martingale. The same is true if E[exp(1111,)] < co(Kazamaki (1978)). Proposition 5.4 is due to Van Schuppen & Wong (1974).

Section 3.6: Brownian local time is the creation of P. Levy (1948), althoughthe first rigorous proof of its existence was given by Trotter (1958). Ourapproach to Theorem 6.11 follows that of Ikeda & Watanabe (1981) andMcKean (1969). One can study the local time of a nonrandom functiondivorced from probability theory, and the general pattern that develops isthat regular local times correspond to irregular functions; for instance, for thehighly irregular Brownian paths we obtained Holder-continuous local times(relation (6.22)). See Geman & Horowitz (1980) for more information on thistopic. On the other hand, Yor (1986) shows directly that the occupation timeB r; (B, co) of (6.6) has a density.

The Skorohod problem of Lemma 6.14, for RCLL trajectories y, wastreated by Chaleyat-Maurel, El Karoui & Marchal (1980).

The generalized Ito rule (Theorem 6.22) is due to Meyer (1976) and Wang(1977). There is a converse to Corollary 6.23: if f( W) is a continuous semi-martingale, then f is the difference of convex functions (Wang (1977), cinlar,Jacod, Protter & Sharpe (1980)). A multidimensional version of Theorem6.22, in which convex functions are replaced by potentials, has been provedby Brosamler (1970).

Tanaka's formula (6.11) provides a representation of the form f(147,) -f(Wo) + f o g(W)dWs for the continuous additive functional L,(a), with a efixed. In fact, any continuous additive functional has such a representation,where f may be chosen to be continuous; see Ventsel (1962), Tanaka (1963).


We follow Ikeda & Watanabe (1981) in our exposition of Theorem 6.17.For more information on the subject matter of Problem 6.30, the reader isreferred to Papanicolaou, Stroock & Varadhan (1977).

Section 3.7: Local time for semimartingales is discussed in the volumeedited by Azema & Yor (1978); see in particular the articles by Azema & Yor(pp. 3-16) and Yor (pp. 23-36). Local time for Markov processes is treatedby Blumenthal & Getoor (1968). Yor (1978) proved that local time AM fora continuous semimartingale is jointly continuous in t and RCLL in a. Hisproof assumes the existence of local time, whereas ours is a step in the proofof existence.

Exercise 7.13 comes from Azema & Yor (1979); see also Jeulin & Yor (1980)for applications of these martingales in the study of distributions of randomvariables associated with local time. Exercise 7.14 is taken from Yor (1979).


CHAPTER 4

Brownian Motion and PartialDifferential Equations

4.1. Introduction

There is a rich interplay between probability theory and analysis, the studyof which goes back at least to Kolmogorov (1931). It is not possible in a fewsections to develop this subject systematically; we instead confine our atten-tion to a few illustrative cases of this interplay. Recent monographs on thissubject are those of Doob (1984) and Durrett (1984).

The solutions to many problems of elliptic and parabolic partial differentialequations can be represented as expectations of stochastic functionals. Suchrepresentations allow one to infer properties of these solutions and, con-versely, to determine the distributions of various functionals of stochasticprocesses by solving related partial differential equation problems.

In the next section, we treat the Dirichlet problem of finding a functionwhich is harmonic in a given region and assumes specified boundary values.One can use Brownian motion to characterize those Dirichlet problems forwhich a solution exists, to construct a solution, and to prove uniqueness. Weshall also derive Poisson integral formulas and see how they are related toexit distributions for Brownian motion.

The Laplacian appearing in the Dirichlet problem is the simplest ellipticoperator; the simplest parabolic operator is that appearing in the heat equa-tion. Section 3 is devoted to a study of the connections between Brownianmotion and the one-dimensional heat equation, and, again, we give prob-abilistic proofs of existence and uniqueness theorems and probabilistic inter-pretations of solutions. Exploiting the connections in the opposite direction,we show how solutions to the heat equation enable us to compute boundarycrossing probabilities for Brownian motion.

Section 4 takes up the study of more complicated elliptic and parabolic


240 4. Brownian Motion and Partial Differential Equations

equations based on the Laplacian. Here we develop formulas necessary forthe treatment of Brownian functionals which are more complex than thoseappearing in Section 2.8.

The connections established in this chapter between Brownian motion andelliptic and parabolic differential equations based on the Laplacian are a fore-shadowing of a more general relationship between diffusion processes andsecond-order elliptic and parabolic differential equations. A good deal of themore general theory appears in Section 5.7, but it is never so elegant and sur-prisingly powerful as in the simple cases of the Laplace and heat equationsdeveloped here. In particular, in the more general setting, one must rely onexistence theorems from the theory of partial differential equations, whereasin this chapter we can give probabilistic proofs of the existence of solutionsto the relevant partial differential equations.

4.2. Harmonic Functions and the Dirichlet Problem

The connection between Brownian motion and harmonic functions is pro-found, yet simply explained. For this reason, we take this connection as ourfirst illustration of the interplay between probability theory and analysis.

Recall that a function u mapping an open subset D of Rd into R is calledharmonic in D if u is of class C2 and Au (52 u/a4) = 0 in D. As weshall prove shortly, a harmonic function is necessarily of class C' and hasthe mean-value property. It is this mean-value property which introducesBrownian motion in a natural way into the study of harmonic functions.

Throughout this section, {147 A; 0 < t < co}, (S2, {F"}ER, is a d-dimensional Brownian family and {A} satisfies the usual conditions. Wedenote by D an open set in Rd and introduce the stopping time (Problem 1.2.7)

(2.1) TD inf It 0; W, e De 1,

the time of first exit from D. The boundary of D will be denoted by ap, andD = D u ap is the closure of D. Recall (Theorem 2.9.23) that each componentof W is almost surely unbounded, so

(2.2) Px[TD < co] = 1; V x eD c Rd, D bounded.

Let Br A {x E Rd; Ilxll < r} be the open ball of radius r centered at the origin.The volume of this ball is

(2.3)

and its surface area is

2r d d/2

d F (d)'2

2rd-1 nal2 d(2.4) Sr -

F(d/2) r

4.2. Harmonic Functions and the Dirichlet Problem 241

We define a probability measure t, on 8B, by

(2.5) po(dx) = P°[W,Bre dx]; r > 0.

A. The Mean-Value Property

Because of the rotational invariance of Brownian motion (Problem 3.3.18),the measure u, is also rotationally invariant and thus proportional to surfacemeasure on BB,. In particular, the Lebesgue integral of a function f over B,can be written in iterated form as

(2.6) f(x)dx = Sp .1 f(x))ip(dx)dp.eo aeo

2.1 Definition. We say that the function u: D -4 IR has the mean-value propertyif, for every a e D and 0 < r < oo such that a + B, c D, we have

u(a) = J u(a + x)Eir(dx).aBr

With the help of (2.6) one can derive the consequence

u(a) = -1 u(a x)dxVr Br

of the mean-value property, which asserts that the mean integral value of uover a ball is equal to the value at the center. Using the divergence theoremone can prove analytically (cf. Gilbarg & Trudinger (1977), p. 14) that a har-monic function possesses the mean-value property. A very simple probabilisticproof can be based on It6's rule.

2.2 Proposition. If u is harmonic in D, then it has the mean-value property there.

PROOF. With a e D and 0 < r < oo such that a + B, c D, we have from Ito'srule

d to ra,Br

U(Wt ) = U(WO) E f (Ws)dKi)i=1 0 uxi

1 A to*,+

2- Au(W.$)ds; 0 < t < oo.

0

Because u is harmonic, the last (Lebesgue) integral vanishes, and since (8u /ax;);1 < i < d, are bounded functions on a + B the expectations under P° of thestochastic integrals are all equal to zero. After taking these expectations onboth sides and letting t oo, we use (2.2) to obtain

u(a) = E° u(W,..r ) = f u(a + x)yr(dx).()Br

2.3 Corollary (Maximum Principle). Suppose that u is harmonic in the open,connected domain D. If u achieves its supremum over D at some point in D, thenu is identically constant.

PROOF. Let M = supxe, u(x), and let DM = {x e D; u(x) = M }. We assume thatDM is nonempty and show that DM = D. Since u is continuous, DM is closedrelative to D. But for a e Dm and 0 < r < co such that a + Br D, we havethe mean value property:

M = u(a) =1- J u(a + x)dx,

V, Br

which shows that u = M on a + Br. Therefore, DM is open. Because D is con-nected, either DM or D \DM must be empty.

2.4 Exercise. Suppose D is bounded and connected, u is defined and con-tinuous on D, and u is harmonic in D. Then u attains its maximum over D onD. If v is another function, harmonic in D and continuous on D, and v = u

on (3D, then v = u on D as well.

For the sake of completeness, we state and prove the converse of Prop-osition 2.2. Our proof, which uses no probability, is taken from Dynkin& Yushkevich (1969).

2.5 Proposition. If u maps D into IR and has the mean value property, then u isof class C° and harmonic.

PROOF. We first prove that u is of class C. For e > 0, let ge: [0, co) bethe C' function

g e(x) =c(e)exp [11x112 - 2 ]; 114 <

0; 114 E,

where c(e) is chosen so that (because of (2.6))

(2.7)18,

g,(x)dx = c(e) f s exp 1

0dp = 1.(p2 E2

For e > 0 and a e D such that a + Be c D, define

ue(a) A J u(a + x)ge(x)dx = f u(y)g e(y - a) dy.B, ad

From the second representation, it is clear that u, is of class C' on the opensubset of D where it is defined. Furthermore, for every a e D there exists e > 0so that a + Be D; from (2.6), (2.7), and the mean-value property of u, we maythen write


u,(a) = J u(a + x)g,(x)dx

Jae= c(e) f 1

E

Sp, u(a + x)exp(P

2 2)1.2,(dx)dp-'

= c(E) j'o

Spu(a)exp(P,

1

- E, dP = u(a),

and conclude that u is also of class C.In order to show that Au = 0 in D, we choose a ED and expand a la Taylor

in the neighborhood a + Bd d d 02

(2.8) u(a + y) = u(a) + E 3,1au 1-

()n

(a) + E E yiy; (a)Xi 2 i=1 J=1 WC;UXJ

+ 0(II312); YEk,

where again e > 0 is chosen so that a + BE c D. Odd symmetry gives us

YtiMdY) = 0, Yiblit(dY) = 0; i j,Jas, Jas,

so upon integrating in (2.8) over OA and using the mean-value property weobtain

(2.9) u(a) = f u(a + y)ue(dy)as,

But

d 02= u(a) +

1

- E -(a) pc(dy) + o(0)2 04 as,

1as, as,

E2

= 71 Y lit(dY) = d

and so (2.9) becomes

p2

+ o(e2) = 0.2d

Dividing by E2 and letting e 0, we see that Au(a) = 0.

B. The Dirichlet Problem

We take up now the Dirichlet problem (D,f): with D an open subset of Rd andf: OD R a given continuous function, find a continuous function u:such that u is harmonic in D and takes on boundary values specified by f; i.e.,u is of class C2 (D) and


(2.10)

(2.11)

Au = 0; in D,

u = f; on aD.

Such a function, when it exists, will be called a solution to the Dirichlet problem(D, f). One may interpret u(x) as the steady-state temperature at x e D whenthe boundary temperatures of D are specified by f.

The power of the probabilistic method is demonstrated by the fact that wecan immediately write down a very likely solution to (D, f), namely

(2.12)

provided of course that

(2.13)

u(x) A Exf(W,.); x

Exlf(W,D)1 < co; V x e D.

By the definition of TD, u satisfies (2.11). Furthermore, for a a D and Br chosenso that a + Br c D, we have from the strong Markov property:

u(a) = Eaf(N)) = Ea {Ea ER wz,,) 1,,,*)}

= Ea lu(W,± )1 = f u(a + x)y,.(dx).aar

Therefore, u has the mean-value property, and so it must satisfy (2.10). Theonly unresolved issue is whether u is continuous up to and including OD. Itturns out that this depends on the regularity of aD, as we shall see later. Wesummarize our discussion so far and establish a uniqueness result for (D,f)which strengthens Exercise 2.4.

2.6 Proposition. If (2.13) holds, then u defined by (2.12) is harmonic in D.

2.7 Proposition. If f is bounded and

(2.14) P`Tri, < cc] = 1; V a eD,

then any bounded solution to (D, f) has the representation (2.12).

PROOF. Let u be any bounded solution to (D,f), and let D A Ix e D;infy cap fix - yll > 1 /n }. From ItO's rule we have

d

u(WtAnAtDn) = u(Wo) + E -(WO dWe;i=i ax,

0 < t < oo, n > 1.

Since (au/axi) is bounded in B n D, we may take expectations and concludethat

u(a) = Ea 14(Wi A TB., TD); 0 < t < co, n 1, a e D.

As t co, n cc, (2.14) implies that u(Wt A 28 A 28 ) converges to f(W,D), a.s. P°.The representation (2.12) follows from the bOunCled convergence theorem.

4.2. Harmonic Functio.is and the Dirichlet Problem 245

2.8 Exercise. With D = {(x,, x2); x2 > 0} and f(x,, 0) = 0; x, e R, show byexample that (D, f) can have unbounded solutions not given by (2.12).

In the light of Propositions 2.6 and 2.7, the existence of a solution to theDirichlet problem boils down to the question of the continuity of u definedby (2.12) at the boundary of D. We therefore undertake to characterize thosepoints a e aD for which

(2.15) lim Exf(W,D) = f(a)x-axeD

holds for every bounded, measurable function f: aD -9 which is continuousat the point a.

2.9 Definition. Consider the stopping time of the right-continuous filtration{,,,} given by aD 9 inf ft > 0; 14/,e Del (contrast with the definition of TD in(2.1)). We say that a point a e aD is regular for D if Pa[crp = 0] = 1; i.e., aBrownian path started at a does not immediately return to D and remain therefor a nonempty time interval.

2.10 Remark. A point a e aD is called irregular if /3/up = 0] < 1; however,the event {o- = 0} belongs to Fo","_, and so the Blumenthal zero-one law(Theorem 2.7.17) gives for an irregular point a: P °[aD = 0] = 0.

2.11 Remark. It is evident that regularity is a local condition; i.e., a e aD isregular for D if and only if a is regular for (a + 13,) n D, for some r > 0.

In the one-dimensional case every point of aD is regular (Problem 2.7.18)and the Dirichlet problem is always solvable, the solution being piecewise-linear. When d > 2, more interesting behavior can occur. In particular, ifD = {x e Rd; 0 < jlx II < 1} is a punctured ball, then for any x e D the Brownianmotion starting at x exits from D on its outer boundary, not at the origin(Proposition 3.3.22). This means that u defined by (2.12) is determined solelyby the values off along the outer boundary of D and, except at the origin,this u will agree with the harmonic function

a(x) o Exf(141,B) =-- Exf(W,D); x e B,.

Now u(0) f(0), so u is continuous at the origin if and only if f(0) =When d > 3, it is even possible for aD to be connected but contain irregularpoints (Example 2.17).

2.12 Theorem. Assume that d > 2 and fix a e aD. The following are equivalent:

(i) equation (2.15) holds for every bounded, measurable function f: aDwhich is continuous at a;

(ii) a is regular for D;

(iii) for all e > 0, we have

(2.16) lim Px[TD > e] = 0.x-qixeD

PROOF. We assume without loss of generality that a = 0, and begin by provingthe implication (i) (ii) by contradiction. If the origin is irregular, thenP ° [6D = 0] = 0 (Remark 2.10). Since a Brownian motion of dimension d > 2never returns to its starting point (Proposition 3.3.22), we have

lim r[WD e Br] = P° [W,D = 0] = 0.40

Fix r > 0 for which P°[W,,, e B,] < (1/4), and choose a sequence On In°., forwhich 0 < 6 < r for all n and ön .1. 0. With ; A inf {t z 0; II Wtli b.}, we haveP° [Tni. 0] = 1, and thus, limn, P° ['s . < CD] = 1. Furthermore, on the eventItn < aDI we have Wz. e D. For n large enough so that r [rn < o-D] > (1/2),we may write

4-1 > r[W,DeB,] > PqW,, e B Tn < api

= E°(1{,<,,,,}P°[Wane 41.9c])

=fDnBonPx [Hitt, e Br] P° Lin < Cfp, Hit. edx]

1

2inf Px[W,D e BA,

z xeDnBon

from which we conclude that Px"[W,D e B,] < (1/2) for some xeD nBon. Nowchoose a bounded, continuous function f: OD IR such that f = 0 outside Bf < 1 inside B and f(0) = 1. For such a function we have

lim Ex.f(W,) lim Px^ [Wt. e B,] <2-1 < f(0),

n-co n-,co

and (i) fails.

We next show that (ii) (iii). Observe first of all that for 0 < ö < c, thefunction

gb(x) A Px[Wse D; .5 < s < e] = Ex(PwITD > E - 6])

PY ['CD > E- 6] Px[WaedY]R.

is continuous in x. But

go(x)1 g(x) A I' [147, e D; 0 < s < = Px [o-D > c]

as 510, so g is upper semicontinuous. From this fact and the inequalitytD G o-D, we conclude that lim,, Px[rD > E] < lim,, g(x) < g(0) = 0, by (ii).

xeDFinally, we establish (iii) (1). We know that for each r > 0,

Px [max° < t < II Wt -Wo ll < r] does not depend on x and approaches one asE 10. But then

Px[liVK - Woll < 11 Px [{ max II W, - Woll < r }n {To El0<t<E

P°[ max II Wtil < rl - Px[rD > E].

0<t<e

Letting x -4 0 (x e D) and e 10, successively, we obtain from (iii)

lim Px[II W. -x II < /1 = 1; 0 < r < co.x-HDxeD

The continuity off at the origin and its boundedness on ap give us (2.15).El

C. Conditions for Regularity

For many open sets D and boundary points a e ap, we can convince ourselvesintuitively that a Brownian motion originating at a will exit from D immedi-ately; i.e., a is regular. We formalize this intuition with a careful discussion ofregularity.

We have already seen that when d = 2, the center of a punctured disc isan irregular boundary point. The following development, culminating withProblem 2.16, shows that, in 112, any irregular boundary point of D must be"isolated" in the sense that it cannot be connected to any other point outsideD by a simple arc lying outside D.

2.13 Definition. Let D c Rd be open and an D. A barrier at a is a continuousfunction v: D R which is harmonic in D, positive on D \ {a}, and equal tozero at a.

2.14 Example. Let D c Br. c R2 be open, where 0 < r < 1, and assume(0, 0) e aD. If a single-valued, analytic branch of log(x1 + ix2) can be definedin D \(0, 0), then

v(x,, x2) A

1

Relog(x + ix2)

log Jxi + xillog(x, + ix2)12'

(x1, x2)eD\(0,0),

0; (x1, x2) = (0,0),

is a barrier at (0,0). Indeed, being the real part of an analytic function, v isharmonic in D, and because 0 < x? + xi < r < 1 in D\(0, 0), v is positiveon this set.

2.15 Proposition. Let D be bounded and a E aD. If there exists a barrier at a,then a is regular.

PROOF. Let v be a barrier at a. We establish condition (i) of Theorem 2.12.With f: ap R bounded and continuous at a, define M = supxeaDChoose a> 0 and let 8 > 0 be such that I f(x) - f(a)1 < a if xeaD andIlx - all < 6. Choose k so that kv(x) > 2M for x e D and 'Ix - all L 6. Wethen have l f(x) - f(a)l < E + kv(x); x e aD, so

1 Ex.i. (WD) - f(a)l < E k Ex v(14/, D) = E k v(x); x E D

by Proposition 2.7. But v is continuous and v(a) = 0, so

lim IP.RWt,)- f(a)I < E.x-oase D

Finally, we let a 1.0 to obtain (2.15). 02.16 Problem. Let D c 1R2 be open, and suppose that a e al) has the propertythat there exists a point b 0 a in 1R2 \ D, and a simple arc in R2 \ D connectinga to b. Show that a is regular.

D D

In three or more dimensions, it is possible to create a cusped region D sothat the boundary point at the end of the cusp is irregular. We illustrate thissituation in R'. In particular, we will construct a region as demarcated inthe following figure by the broken line, so that, when this region is rotatedabout the xraxis, the resulting solid has an irregular boundary point at the

origin. It is a simple matter to replace this solid by an even larger one, havinga smooth boundary except for a cusp at the origin (dotted curve). It is a directconsequence of Definition 2.9 that the origin is also an irregular boundarypoint for this larger solid.

2.17 Example (Lebesgue's Thorn). With d = 3 and {e}°1_, a sequence of posi-tive numbers decreasing to zero, define

E = {(x1, x2, x3); -1 < x1 < 1, xi + xi < 1},

F = {(xi, x2, x3); 2-n < x1 < 2 -n +1 x2 + xi < s ,

D = E\(U F).n=1

Now P° [(W(2), Wi(3)) = (0, 0), for some t > 0] = 0 (Proposition 3.3.22), so theP°-probability that W = (Ww, W(2), W3) ever hits the compact set K 4{(x x2, x3); 2- < x1 < 2-n+1, x2 = x3 = 0} is zero. According to Problem3.3.24, lime = x a.s. P°, so for P°-a.e. co ell, the path t 1-- Wi(u))remains bounded away from K. Thus, if s is chosen sufficiently small, we canensure that P°[W,e F, for some t > 0] < 3-". If W, beginning at the origin,does not return to D immediately, it must avoid D by entering U',., F. Inother words,

Lebesgue's Thorn

cop0 [0.D 0] < p0 ..,[VV, e F, for some t > 0 and n > 1] < E 3-11 < 1.

n=1

If cusplike behavior is avoided, then the boundary points of D are regular,regardless of dimension. To make this statement precise, let us define for

y e \ {0} and 0 5 8 5 1C, the cone C(y, 0) with direction y and aperture 0 by

C(y, 0) = {x e Rd; (x, II cos 0}.

2.18 Definition. We say that the point a e OD satisfies Zaremba's cone con-dition if there exists y 0 and 0 < 0 < it such that the translated conea + C(y, 0) is contained in Rd \ D.

2.19 Theorem. If a point a E ap satisfies Zaremba's cone condition, then it isregular.

PROOF. We assume without loss of generality that a is the origin and C(y, 0) cd \ D, where y 0 and 0 < 0 < n. Because the change of variable z = (x/.1i)

maps C(y, 0) onto itself, we have for any t > 0,

1

P°[ Wt CU, =c(m) (2ittr2

exp [ 11x1121dx

2t

= f 1

exp [ I 21 dz q > 0,2

where q is independent of t. Now P° [6D < t] > P° [W, E C(y, 0)] = q, andletting t j 0 we conclude that Pqo-D = 0] > 0. Regularity follows from theBlumenthal zero-one law (Remark 2.10).

2.20 Remark. If, for a e ar) and some r > 0, the point a satisfies Zaremba'scone condition for the set (a + 13,) r D, then a is regular for D (Remark 2.11).

D. Integral Formulas of Poissont

We now have a complete solution to the Dirichlet problem for a large classof open sets D and bounded, continuous boundary data functions f: OD R.Indeed, if every boundary point of D is regular and D satisfies (2.14), then theunique bounded solution to (D,f) is given by (2.12) (Propositions 2.6, 2.7 andTheorem 2.12). In some cases, we can actually compute the right-hand side of(2.12) and thereby obtain Poisson integral formulas.

2.21 Theorem (Poisson Integral Formula for a Half-Space). With d 2, D ={(x1,..., x); x, > 0} and f: OD O bounded and continuous, the uniquebounded solution to the Dirichlet problem (D,f) is given by

df(/2) xdf (Y)(2.17) u(x) = ird/2 d dy; xeD.

2.22 Problem. Prove Theorem 2.21.

The Poisson integral formula for a d-dimensional sphere can be obtainedfrom Theorem 2.21 via the Kelvin transformation. Let go: Rd VOI Rd \ {0} bedefined by q;o(x) = (x/11.42). Note that cp is its own inverse. We simplifynotation by writing x* instead of co(x).

For r > 0, let B = {x e Rd: llx - Cul < r}, where c = red and e1 is the unitvector with a one in the i-th position. Suppose f: aB R is continuous (andhence bounded), so there exists a unique function u which solves the Dirichletproblem (B, f). The reader may easily verify that

cp(B) = H fx* e IRd; (x*, c) > 11

and (p(OB \ {O}) = aH = {x* e Rd; (x*, c) = -I}. We define u*: H -+ R, theKelvin transform of u, by

1

(2.18) u*(x*) = u(x).oc*Ild-2

A tedious but straightforward calculation shows that Au*(x*) =Au(x), so u* is a bounded solution to the Dirichlet problem (H, f*)

where

(2.19) f *(x*) = x*II d -2 fix);

Because H = (1/2r)ed + D, where D is as in Theorem 2.21, we may apply (2.17)to obtain

The results of this subsection will not be used later in the text.

1

(d/2) f r)f*(Y*)li

(2.20) u*(x*) =xd

d/2 Y* - x*dy*; x* e H.

Formulas (2.18)-(2.20) provide us with the unique solution to the Dirichletproblem (B, f). These formulas are, however, a bit unwieldy, a problem whichcan be remedied by the change of variable y = q)(y*) in the integral of (2.20).This change maps the hyperplane OH into the sphere 5B. The surface elementon aB is Sra,(dy - c) (recall (2.4), (2.5)). A little bit of algebra and calculus onmanifolds (Spivak (1965), p. 126) shows that the proposed change of variablein (2.20) involves

S(2.21) dy* =

rar(dy - c)

1131112(")

(The reader familiar with calculus on manifolds may wish to verify (2.21) firstfor the case y* = e, + (1/2r)ed and then observe that the general case maybe reduced to this one by a rotation. The reader unfamiliar with calculus onmanifolds can content himself with the verification when d = 2, or can referto Gilbarg & Trudinger (1977), p. 20, for a proof of Theorem 2.23 which usesthe divergence theorem but avoids formula (2.21).)

On the other hand,

(2.22)Ilx - y112

ILV* - X*112 =1142

2r).(2.23) r2 - Ilx - cii 2 = 114 2[2(c, x*) - 1] = 2rIke (x: - 1

Using (2.18), (2.19), and (2.21)-(2.23) to change the variable of integration in(2.20), we obtain

(2.24) u(x) = rd-2(r - cii 2) I f011tr(dYx e B.

as I1Y -

Translating this formula to a sphere centered at the origin, we obtain thefollowing classical result.

2.23 Theorem (Poisson Integral Formula for a Sphere). With d > 2, Br ={xe 141d; 114 < r}, and f: aB, continuous, the unique solution to theDirichlet problem (B,, f) is given by

f(Y)fc(dY)(2.25) ; x e B,..1142) foe, Y -

2.24 Exercise. Show that for x E Br, we have the exit distribution

rd-2(r2 X112"1(dY); = r-Px[WtBe dY1 = x

11

Y- Ild

(2.26)

2.25 Problem. Consider as given an open, bounded subset D of Rd andthe bounded, continuous functions g: D -, R and f: 8D -, R. Assume thatu: D -gcR is continuous, of class OD), and solves the Poisson equation

12Au = -g; in D

subject to the boundary condition

u = f ; on aD.

Then establish the representation

,1)

(2.27) u(x) = Ex[ f(W,D) + .1 g(W) dt]; x e D.o

In particular, the expected exit time from a ball is given by

i

(2.28) Ex TB,.

r2

d

ixil 2= ; X E Br.

(Hint: Show that the process {M, gL- u(W, AO + Po^ `D g(W) ds, Ft; 0 < t < colis a uniformly integrable martingale.)

2.26 Exercise. Suppose we remove condition (2.14) in Proposition 2.7. Showthat v(x) A Px [-up = co] is harmonic in D, and if aeaD is regular, thenlimx.a v(x) = 0. In particular, if every point of al) is regular, then with

xe Du(x) = Ex[f(41/01{,D,)}], the function u + Av is a bounded solution to theDirichlet problem (D, f) for any A e R. (It is possible to show that everybounded solution to (D, f) is of this form; see Port & Stone (1978), Theorem4.2.12.)

2.27 Exercise. Let D be bounded with every boundary point regular. Provethat every boundary point has a barrier.

2.28 Exercise. A complex-valued Brownian motion is defined to be a processW = {IV) + ilgt(2), .Ft; 0 < t < co}, where W = {(W('0, W(2)), A; 0 < t < co}is a two-dimensional Brownian motion and i = .\/- 1:

(i) Use Theorem 3.4.13 to show that if W is a complex-valued Brownianmotion and f: C --0 C is analytic and nonconstant, then (under an ap-propriate condition) f(W) is a complex-valued Brownian motion with arandom time-change (P. Levy (1948)).

(ii) With e C \ {0}, show that M, A e''',, 0 < t < co is a time-changed,complex-valued Brownian motion. (Hint: Use Problem 3.6.30.)

(iii) Use the result in (ii) to provide a new proof of Proposition 3.3.22.For additional information see B. Davis (1979).

4.3. The One-Dimensional Heat Equation

In this section we establish stochastic representations for the temperatures ininfinite, semi-infinite, and finite rods. We then show how such representationsallow one to compute boundary-crossing probabilities for Brownian motion.

Consider an infinite rod, insulated and extended along the x-axis of the (t, x)plane, and let f(x) denote the temperature of the rod at time t = 0 and loca-tion x. If u(t, x) is the temperature of the rod at time t > 0 and position x e R,then, with appropriate choice of units, u will satisfy the heat equation

(3.1)314 102uat = 2 0x2'

with initial condition u(0, x) = f(x); x E R. The starting point of our prob-abilistic treatment of (3.1) is furnished by the observation that the transitiondensity

1p(t; x, y)

1

yPx[W,edy] = e-(x-y)212t; t > 0, x, ye R,

of the one-dimensional Brownian family satisfies the partial differentialequation

(3.2)Op 102pat 2 0x2

Suppose then that f: R -+ R is a Borel-measurable function satisfying thecondition

(3.3) f_. e-`"2 I f (x)I dx < co

for some a > 0. It is well known (see Problem 3.1) that

(3.4) u(t, x) A Exf(Wi) = f(y)p(t; x, y) dy

is defined for 0 < t < (1/2a) and x E R, has derivatives of all orders, and satisfiesthe heat equation (3.1).

3.1 Problem. Show that for any nonnegative integers n and m, under theassumption (3.3), we have

ani-m oo ani-m

1

(3.5)Otnaxm

u(t, x) = f(y)OtnOxm

p(t; x, y) dy; 0 < t < x e R.

Iff is bounded and continuous, then rewriting (3.4) as u(t, x) = E °f(x + 147d,we can use the bounded convergence theorem to conclude

4.3. The One-Dimensional Heat Equation 255

(3.6) f(x) = lim u(t, y), V x e IR.viroy -.x

In fact, we have the stronger result contained in the following problem.

3.2 Problem. If f: is a Borel-measurable function satisfying (3.3) andf is continuous at x, then (3.6) holds.

A. The Tychonoff Uniqueness Theorem

We shall call p(t; x, y) a fundamental solution to the problem of finding afunction u which satisfies (3.1) and agrees with the specified function f at timet = 0.

We shall say that a function u: has continuous derivatives up to acertain order on a set G, if these derivatives exist and are continuous in theinterior of G, and have continuous extensions to that part of the boundaryaG which is included in G. With this convention, we can state the follow-ing uniqueness theorem. For nonnegative functions, a substantially strongerresult is given in Exercise 3.8.

3.3 Theorem (Tychonoff (1935)). Suppose that the function u isstrip (0, T] x FE and satisfies (3.1) there, as well as the conditions

(3.7) lim u(t, y) = 0; x e

y -.x

(3.8) sup I u(t, x)I < Keax2; x e0<t<T

for some positive constants K and a. Then u = 0 on (0, T] x

3.4 Remark. If u, and u2 satisfy (3.1), (3.8) and

lim u (t, y) = lim u2(t,t4.0

0,2

then Theorem 3.3 applied to u, - u2 asserts that u, = u2 on (0, T) x R.

on the

3.5 Remark. Any probabilistic treatment of the heat equation involves a time-reversal. This is already suggested by the representation (3.4), in which theinitial temperature function f is evaluated at W, rather than K. We shall seethis time-reversal many times in this section, beginning with the followingprobabilistic proof of Theorem 3.3.

PROOF OF THEOREM 3.3. Let Ty be the passage time of W to y as in (2.6.1). Fixx e R, choose n > Ix!, and let R = 7' n T_. With t e [0, T) fixed and

v(0, x) u(T - t - 0, x); 0 < 0 < T - t,

we have from Ito's rule, for 0 5 s < T - t,

(3.9) u(T - t, x) = v(0, x) = Ex v(s R, Ws R,,)

= [v(s, Ws)1{,<R,,}] + Ex [v(R, Witn)l{s, J].

Now I v(s, Ws)11{,<R} is dominated by

max I u(T - t - Ke"2,< s<T-tIA

and v(s, Ws) converges Px-a.s. to zero as s j T - t, thanks to (3.7). Likewise,I v(R, WR.)11{, R,,} is dominated by Ke°"2. Letting sIT-t in (3.9), we obtainfrom the bounded convergence theorem:

u(T- t, x) = Ex [v(R, WR,,)1{R<r-,}

Therefore, with 0 < t < T, Ix' < n,

I u(T - t, x)I < Kew° Px [R < T - t]

Ke°"2 (P° [T, < + P° [Tn+x < T])

f< Kea 2 x2I2 dz + e- '212 dz ,2

(n-x)/Ii (n+x)hiT

where we have used (2.6.2). But from (2.9.20) it is evident thaten2 e-z212 dz = 0, provided a < 1/2T

Having proved the theorem for a < (1/2T), we can easily extend it to thecase where this inequality does not hold by choosing To = 0 < T, < <T,, = T such that a < (1/2(Ti - i = 1, n, and then showing succes-sively that u = 0 in each of the strips (T,_,, Ti]; i = 1, n.

It is instructive to note that the function

(3.10) h(t, x) -x p(t; x, 0) = --a

xp(t; x, 0); t > 0, x E R,

a

solves the heat equation (3.1) on every strip of the form (0, x R; further-more, it satisfies condition (3.8) for every 0 < a < (1/2T), as well as (3.7) forevery x 0. However, the limit in (3.7) fails to exist for x = 0, although wedo have lima° h(t, 0) = 0.

B. Nonnegative Solutions of the Heat Equation

If the initial temperature f is nonnegative, as it always is if measured on theabsolute scale, then the temperature should remain nonnegative for all t > 0;this is evident from the representation (3.4). Is it possible to characterize thenonnegative solutions of the heat equation? This was done by Widder (1944),

4.3. The One-Dimensicnal Heat Equation 257

who showed that such functions u have a representationCO

u(t, x) = J p(t; x, y) dF(y); x e

where F: is nondecreasing. Corollary 3.7 (i)', (ii)' is a precise state-ment of Widder's result. We extend Widder's work by providing probabilisticcharacterizations of nonnegative solutions to the heat equation; these appearas Corollary 3.7 (iii)', (iv)'.

3.6 Theorem. Let v(t, x) be a nonnegative function defined on a strip (0, T) xwhere 0 < T < cc. The following four conditions are equivalent:

(i) for some nondecreasing function F:CO

(3.11) v(t, x) = J p(T - t; x, y)dF(y); 0 < t < T, X E fR;-co

(ii) v is of class C1'2 on (0, T) x IR and satisfies the "backward" heat equation

av 1 02 v(3.12) - + -

2 axe= 0

at

on this strip;(iii) for a Brownian family {W. ..,;; 0 < s < op}, (S2, ikPxxe aindeaacrh_

fixed t e (0, T), x e 0:2, the process {v(t + s, Ws), .mss; 0 s T - }tingale on (SI, Px);

(iv) for a Brownian family {Ws, .; 0 s < co}, {Px}xefra we have

(3.13) v(t, x) Ex v(t + s,VV,); 0 < t < t + s < T, xeEJ.

PROOF. Since (a/at)p(T - t; x, y) + (1/2)(02/ex2)p(T - t; x, y) = 0, the impli-cation (i) (ii) can be proved by showing that the partial derivatives ofv can be computed by differentiating under the integral in (3.11). For a > 1/2Twe have

j_ooe-aY2dF(y) = -v (T - < cc.

This condition is analogous to (3.3) and allows us to proceed as in Solution3.1.

For the implications (ii)(iii) and (ii) (iv), we begin by applying Ito'srule to v(t + s, Ws); 0 < s < T - t. With a < x < b, we consider the passagetimes To and Tb as in (2.6.1) and obtain:

sATanTh av(t +(s n Ta n Tb), Ws A T. A = V(t, Wo) -04 ± 0-, Wcy)dHfc,

axsAToTb

2

a 1 a 2-

x2± W(,) do-.

Under assumption (ii) the Lebesgue integral vanishes, as does the expectationof the stochastic integral because of the boundedness of (0/ax)v(t + o, y) whena < y < b and 0 < a <s<T-t. Therefore,

(3.14) v(t, x) = Exv(t + (s A T:, A 70, Ws A T. A TO'

Now let a I -co, b i Go and rely on the nonnegativity of v and Fatou's lemmato obtain

(3.15) v(t, x) Exv(t+s, Ws); 0 <t<t+s< T, x e R.

Inequality (3.15) implies that for fixed t e (0, T) and x e R, the process{v(t + s, Ws), ,Fs; 0 < s < T - t} is a supermartingale on (0,.r,, , Px). Indeed,for 0 < s1 < s2 < T - t, the Markov property yields

(3.16) Ex [v(t + s2, Ws2)1A,]((a) = f(Ws i(w)),for Px-a.e. w e 0,

where

(3.17) f(y) A EY v(t + s2, W,2_,i)

(see Proposition 2.5.13). From (3.15), we have

EY v(t + s2, W,,) v(t + si, y),

and so for 0 < t < t + si < t + s2 < T, xe R:

(3.18) v(t + s1, WO > Ex[v(t + s2, 14/s2)1,,,], a.s. Px.

It is clear from this argument that if equality holds in (3.15), then {v(t + s, iv,.; 0 < s < T - t} is a martingale. To complete our proof of ( ii) ( iii) and(ii) (iv), we must establish the reverse of inequality (3.15).

Returning to (3.14), we may write

v(t, x) = Ex[v(t + s, Ws)i{s< T A To] + Ex[v(t + T, a)1{1.<., A TO]

+ Ex[v(t + Tb,b)1{Tb<s A T.}]

< Exv(t + s, Ws) + Ex[v(t + T, a)l{r<s}i

+ Ex[v(t + Tb, b)1{Tb<s}i.

We will have established (3.13) as soon as we prove

(3.19) lim Ex[v(t + 'Tb,b)1{Tb<s}] = 0b-..co

(a dual argument then shows that lima__. Ex[v(t + 'T,a)1,-.<4] = 0). For(3.19), it suffices to show that with B > 0 large enough, we have

J: Ex[v(t + Tb,b)1{Tb<s}] db < o o .

We choose x e R, 0 < t < T, and 0 < s < t so that s + t < T From (2.6.3) and(3.10) we have

Px[Tb Edo] = h(o-; b - x)da; b > x, a > 0.

i

For B _.- x sufficiently large, h(a; b - x) is an increasing function of a e (0,$),provided b > B. Furthermore, for r e (s, t) and B perhaps larger, we have

h(s,b - x) ._-_ p(r; x, b); b _>._ B.

It follows that

Ex[v(t + T6, b)1{Tb<s}] db = .133 fs v(t + a,b)h(a,b - x)da dbB 0

fsly

s

J v(t + o-,b)p(r; x, b) db do-o B

-113- I Ex v(t + a,W,)do-o

r is0

where the next to last inequality is a consequence of (3.15). This proves (3.13)for x e fR, 0 <t<t+s<T, as long as s < t.

We now remove the unwanted restriction s < t. We show by induction onthe positive integers k that if

(3.20)

then

(3.21)

0<t<t+s<T, s<kt,

v(t, x) = Ex v(t + s, Ws); x e R.

This will yield (3.13) for the range of values indicated there. We have justestablished that (3.20) implies (3.21) when k = 1. Assume this implicationfor some k > 1, so {v(t + s, Ws), Fs; 0 < s < kt} is a martingale. Choose s2 e[kt,(k + 1)t) and si e [0, kt) so that 0 < s2 - s, < t. Then

Ex v(t + s2, Ws) = Ex {Ex[v(t + S2, W011])

= Ex v(t + si, WO = v(t, x),

where we have used (3.16), (3.17), and the induction hypothesis in the form

E" v(t + s2, 147,2_0 = v(t + s 1, y).

Finally, we take up the implication (iv) =(i). For 0 <s < (T/4), (T/2) < t < T,(3.13) gives

v(t - s, x) .-- Exv(T - e, Wr-t) =i' p(T - t; x, y)

dF,(y),i..P (- -27= ; 0 , Y)

where F, is the nondecreasing function

FE(x) -4- fx p ET ; 0, y)v(T - s, y)dy; x e IR._,, 2

L


Again from (3.13), FE(co) = ev( T - e, WT /2) = v(( T/2) - e, 0), and thus

sup FE(co) < max v(s,0) < co.0 <e<(T14) (T14),s(T12)

By Helly's theorem (Ash (1972), p. 329), there exists a sequences, > E2 > >Ek 10 and a nondecreasing function F*: R -, [0, ao), such that lim, F,k(x) =F*(x) for every x at which F* is continuous. Because for fixed x e R andt e((T/2), T) the ratio (p(T - t; x, y)/p((T/2); 0, y)) is a bounded, continuousfunction of y, converging to zero as IyI -, co, we have

1)

iv(t, x) = lim v(t - Ek, x) =k-oco -o0 T

dF*(y)

2;(")by the extended Helly-Bray lemma (Loeve (1977), p. 183). Defining F(x) =If:, (dF*(y)/p((T/2); 0, y)), we have (3.11) for (T/2) < t < T, x e R.

If 0 < t _<_ (T/2), we choose t, e((T /2), T) and use (3.13) to write

v(t, x) = J p(t, - t; x, y)v(t y)dy-0

= p(t, - t; x, y)p(T - t,; y, z)dy dF(z)

=

1.p(T - t; x, z)dF (z). CI

3.7 Corollary. Let u(t, x) be a nonnegative function defined on a strip (0, T) x R,where 0 < T < co. The following four conditions are equivalent:

(i)' for some nondecreasing function F: R - R,

CO

(3.22) u(t, x) = J p(t; x, y) dF(y); 0 < t < T, x e R;

(ii)' u is of class C'.2 on (0, T) x R and satisfies the heat equation (3.1) there;(iii)' for a Brownian family {Ws, Fs; 0 -_ s < co}, (SI, F), {PlxeR and each

fixed t e (0, T), xeR, the process {u(t - s, Ws), F; 0 s < t} is a mar-tingale on (0,F , Px);

(iv)' for a Brownian family {Ws, Fs; 0 < s < op}, (0, F), {Px}x. R we have

(3.23) u(t, x) = Exu(t - s, Ws); 0 <s<t<'T,xe R.

PROOF. If T is finite, we obtain this corollary by defining v(t, x) = u(T - t, x)and appealing to Theorem 3.6. If T = co, then for each integer n 1 we setv(t, x) = u(n - t, x); 0 < t < n, x e R. Applying Theorem 3.6 to each v we seethat conditions (ii)', (iii)', and (iv)' are equivalent, they are implied by (i)', andthey imply the existence, for any fixed n > 1, of a nondecreasing functionF: R -, R such that (3.22) holds on (0, n) x R. For t > n, we have from (3.23):

u(t, x) = Exu(--n2, 01-/2)) = c c ) 1 4

zZ)p(t -12

2'x, z) dz

CO c0

PG; z,y)p (t - 2; x, dz (y)

= f p(t; x, y) dF(y).

3.8 Exercise (Widder's Uniqueness Theorem).

(i) Let u(t, x) be a nonnegative function of class defined on the strip(0, T) x R, where 0 < T < co, and assume that u satisfies (3.1) on this stripand

lim u(t, y) = 0; x e R.110

Show that u = 0 on (0, T) x R. (Hint: Establish the uniform integrabilityof the martingale u(t - s, Ws); 0 5 s < t.)

(ii) Let u be as in (i), except now assume that lima° u(t, y) = f(x); x e R.Show that

u(t, x) = f p(t; x, y)f(y)dy; 0 < t < T, x e R.-co

CO

Can we represent nonnegative solutions v(t, x) of the backward heat equa-tion (3.12) on the entire half-plane (0, co) x R, just as we did in Corollary 3.7for nonnegative solutions u(t, x) of the heat equation (3.1)? Certainly thiscannot be achieved by a simple time-reversal on the results of Corollary 3.7.Instead, we can relate the functions u and v by the formula

j2ir (x2) (1 x(3.24) v(t, x) = exp u -

"

0 < t < co , x e R.

The reader can readily verify that v satisfies (3.12) on (0, co) x R if and onlyif u satisfies (3.1) there. The change of variables implicit in (3.24) allows us todeduce the following proposition from Corollary 3.7.

3.9 Proposition (Robbins & Siegmund (1973)). Let v(t,x) be a nonnegativefunction defined on the half-plane (0, co) x R. With T= co, conditions (ii), (iii),and (iv) of Theorem 3.6 are equivalent to one another, and to (i) ":

(i)" for some nondecreasing function F: R R,

(3.25) v(t, x) = J exp1

yx - y2 t) dF(y); 0 < t < oo, x e R.

PROOF. The equivalence of (ii), (iii), and (iv) for T = co follows from theirequivalence for all finite T If v is given by (3.25), then differentiation underthe integral can be justified as in Theorem 3.6, and it results in (3.12). If vsatisfies (ii), then u given by (3.24) satisfies (ii)', and hence (i)', of Corollary 3.7.But (3.24) and (3.22) reduce to (3.25). 0

C. Boundary-Crossing Probabilities for Brownian Motion

The representation (3.25) has rather unexpected consequences in the computa-tion of boundary-crossing probabilities for Brownian motion. Let us consider apositive function v(t, x) which is defined and of class C`2 on (0, co) x IR, andsatisfies the backward heat equation. Then v admits the representation (3.25)for some F, and differentiating under the integral we see that

a(3.26) -

atv(t,x)< 0; 0 < t < oo, x E R

and that v(t, ) is convex for each t > 0. In particular, lim,4,0 v(t, 0) exists. Weassume that this limit is finite, and, without loss of generality (by scaling, ifnecessary), that

(3.27) lim v(t, 0) = 1.ti.o

We also assume that

(3.28) lim v(t, 0) = 0,r00

(3.29) lim v(t, x) = co; 0 < t < co,x-co

(3.30) lim v(t, x) = 0, 0 < t < co.

It is easily seen that (3.27)-(3.30) are satisfied if and only if F is a probabilitydistribution function with F(0+) = 0. We impose this condition, so that (3.25)becomes

c° 1

(3.31) v(t, x) = 1 exp yx - 2y2 t)dF(y); 0 < t < oo, X E R,0+

where F(co) = 1, F 0 + ) = 0. This representation shows that v(t, .) is strictlyincreasing, so for each t > 0 and b > 0 there is a unique number A(t, b) suchthat

(3.32) v(t, A(t, b)) = b.

It is not hard to verify that the function A(- , b) is continuous and strictlyincreasing (cf. (3.26)). We may define A(0, b) = lima° A(t, b).

We shall show how one can compute the probability that a Brownian pathW, starting at the origin, will eventually cross the curve A(.,b). The problem of

computing the probability that a Brownian motion crosses a given, time-dependent continuous boundary {/i(t); 0 < t < co} is thereby reduced to find-ing a solution v to the backward heat equation which also satisfies (3.27)-(3.30)and v(t, 0)) = b; 0 < t < co, for some b > 0. In this generality both problemsare quite difficult; our point is that the probabilistic problem can be tradedfor a partial differential equation problem. We shall provide an explicit solu-tion to both of them when the boundary is linear.

Let { W gt; 0 < t < co}, (CI, .x), {Px}. pl be a Brownian family, and define

Z, = v(t, Wt); 0 < t < oo.

For 0 < s < t, we have from the Markov property and condition (iv) ofProposition 3.9:

EqZ,1";] = f(14/5) = v(s, Ws) = Z a.s. P °,

where f(y) A EY v(t, Wi_s). In other words, {Z Ft; 0 < t < co} is a continuous,nonnegative martingale on (0, P°). Let {tn} be a sequence of positivenumbers with tn.j. 0, and set Z0 = Z. This limit exists, P ° -a.s., andis independent of the particular sequence {t.} chosen; see the proof of Prop-osition 1.3.14(i). Being "O",1-measurable, Zo must be a.s. constant (Theorem2.7.17).

3.10 Lemma. The extended process Z A {Z gi; 0 < t < co} is a continuous,nonnegative martingale under P° and satisfies Z0 = 1, 4 = 0, P°-a.s.

PROOF. Let {t.} be a sequence of positive numbers with t l O. The sequence{4},D=1 is uniformly integrable (Problem 1.3.11, Remark 1.3.12), so by theMarkov property for W, we have for all t > 0:

E°[Z,IF0] = E°Z, = lim E°Zi. = E°Z0 = Z0.n-,co

This establishes that {Z .a.07,; 0 < t < co} is a martingale.Since Zi exists P°-a.s. (Problem 1.3.16), as does Z0 lima° 4,

it suffices to show that lima° 4 = 1 and lim, Z, = 0 in P °- probability. Forevery finite c > 0, we shall show that

(3.33) lim sup 1v(t,x) - 11 = 0.

Indeed, for t > 0, 1x1 <cf:

1

(3.34) exp - yc.,/ t - -2

y2 t)dF(y) v(t, x)Jo+

CO1

< exp (yc - -2y2 t)dF(y).

Because ± yc.,/t - y2 t/2 5 c2 /2; V y > 0, the bounded convergence theo-rem implies that both integrals in (3.34) converge to 1, as ti 0, and (3.33)follows. Thus, for any a > 0, we can find tc, depending on c and a, such that

l - E < v(t, x) < 1 ± E; I XI < C-s/t, 0 < t < tc,,

Consequently, for 0 < t < tc,

P°[14 - 11> e] = P°[ 1v((, Wt) - 11> E] P°[1Wt1> c-s/i] = 2[1 - (I)(e)],

where

(1)(x) 1=--` ,1 xi' e-2212 dz.N/27r -.

Letting first t 10 and then c cc, we conclude that Z, 1 in probability ast 10. A similar argument shows that

(3.35) lim sup v(t, x) = 0,too ixi .c.,./i

and, using (3.35) instead of (3.33), one can also show that Z, -0 in probabilityas t - co. 111

It is now a fairly straightforward matter to apply Problem 1.3.28 to themartingale Z and obtain the probability that the Brownian path {W,(w);0 < t < co} ever crosses the boundary {A(t, b); 0 < t < co}.

3.11 Problem. Suppose that v: (0, cc) x FR - (0, cc) is of class C' '2 and satisfies(3.12) and (3.27)-(3.30). For fixed b > 0, let A( , b): [0, co) -, R be the con-tinuous function satisfying (3.32). Then, for any s > 0 and Lebesgue-almostevery a e R with v(s, a) < b, we have

(3.36) P° [Wt > A(t, b), for some t > sl Ws = a] =v(s, a)

b'(3.37) P° [W A(t, b), for some t > s]

A(s,b)) 1 f (A(s,b)= 1 -(I)( + (I) yNfi)dF (y),. , / s u o+ Is

where F is the probability distribution function in (3.31).

3.12 Example. With it > 0, let v(t, x) = exp(px - p2 t /2), so A(t,b) = fit + y,where 13 = (it/2), y = (1/12)log b. Then F(y) = 11,,,00(y), and so for any s > 0,13 > 0, y e R, and Lebesgue-almost every a < y + 13s:

(3.38) P° [W, > fit + y, for some t > sI W., = a] = e-"('''' I's),

and for any s > 0, iq > 0, and y e R:

(3.39) P°[W, fit + y, for some t > s] = 1 -(I)( + /3is-s

+ e-211Y(13( , I3A.. , / s

The observation that the time-inverted process Y of Lemma 2.9.4 is aBrownian motion allows one to cast (3.38) with y = 0 into the followingformula for the maximum of the so-called "tied-down" Brownian motion or"Brownian bridge":

(3.40) P° [ max W > /3(:1 (57.

1,177. ] e-2p(p-a)/T

for T > 0, > 0, a.e. a < 13, and (3.39) into a boundary-crossing probabilityon the bounded interval [0, T]:

(3.41) P °[W > /3 + yt, for some t e [0, Ti]

= 1 - (1)(7.,j+ 13,) + e-2137,1)(VT > 0,y e R..\/ T ,/ T

3.13 Exercise. Show that P° [W, > ft + y, for some t > 0] = e-211Y, for 13 > 0and y > 0 (recall Exercise 3.5.9).

D. Mixed Initial/Boundary Value Problems

We now discuss briefly the concept of temperatures in a semi-infinite rod andthe relation of this concept to Brownian motion absorbed at the origin. Supposethat f: (0, oo) R is a Borel-measurable function satisfying

(3.42) co° e-"21f(x)Idx < oo

for some a > 0. We define

(3.43) ui(t, x) -4 Ex[f(Wf)1(ro>t)]; 0 < t < x > 0.

The reflection principle gives us the formula (2.8.9)

Px [ W, e dy, To > t] = p At; x, y)dy A [p(t; x, y) - p(t; x, -y)] dy

for t > 0, x > 0, y > 0, and so

(3.44) ui(t, x) = I f(y)p(t; x, y)dy -0

o

j AP(t; x, AdY,

which gives us a definition for t. valid on the whole strip (0, 1/2a) x R.This representation is of the form 9.4), where the initial datum f satisfiesf(y) = -f( -y); y > 0. It is clear then that u, has derivatives of all orders,satisfies the heat equation (3.1), satisfies (3.6) at all continuity points of f,and

1lim ui (t, 0) = 0; 0 < t < -2a .

x10

We may regard u (t, x); 0 < t < (1/2a), x > 0, as the temperature in a semi-infinite rod along the nonnegative x-axis, when the end x = 0 is held at aconstant temperature (equal to zero) and the initial temperature at y > 0 isf(v)

Suppose now that the initial temperature in a semi-infinite rod is identicallyzero, but the temperature at the endpoint x = 0 at time t is g(t), whereg: (0, 1/2a) R is bounded and continuous. The Abel transform of g, namely

(3.45) u2(t, x) -4 Ex [g(t -To)1{,0,}]

= J r g(t - r)h(r, x) dr

.1= g(s)h(t - s, x)ds; 0 < t < -1 , x > 0o 2a

with h given by (3.10), is a solution to (3.1) because h is, and h(0, x) = 0 forx > 0. We may rewrite this formula as

u2(t, x) = E° [g(t -7'x)1{,,,,}]; 0 < t < 1 , x > 0,

and then the bounded convergence theorem shows that

lim u2(s, x) = g(t); 0 < t < -1 ,

s-t 2a

lim u2(t, y) = 0; 0 < x < co.tloy -.x

We may add ul and u2 to obtain a solution to the problem with initial datumf and time-dependent boundary condition g(t) at x = 0.

3.14 Exercise (Neumann Boundary Condition). Suppose that f: (0, co) Ris a Borel-measurable function satisfying (3.42), and define

1

u(t, x) Exf(11411); 0 < t < - , x > 0.2a

Show that u is of class C1'2, satisfies (3.1) on (0, 1/2a) x (0, co) and (3.6) at allcontinuity points of f, as well as

lim0

u(s, x) = 0; 0 < t <1-.

s_.t uX 2ax4,0

3.15 Exercise (Finite Rod). Suppose that g, k are bounded, continuous func-tions from (0, co) into R, and f is a bounded, continuous function from (0, b)into R. We seek a function u which is of class 0.2 on (0, co) x (0, b) and whichhas a continuous extension to the boundaries {0} x (0, b), (0, co) x {0}, and

4.4. The Formulas of Feynman and Kac 267

(0, co) x {b}, such that

814 102uat 2 0 x2

u(0, x) = f(x);

u(t, 0) = g(t);

u(t, b) = k(t);

on (0, co) x (0, b),

0 < x < b,

0 < t < oo,

0 < t < co .

Show that the unique bounded solution to this problem is given by theexpression

(3.46) u(t, x) = F r tiw)i-x -{t<To TO g(t TO)1{7.0: A TO

+ k(t - Tb)1{Tb<t To}l; 0 < t < oo , 0 < x < b.

(Hint: Use Proposition 2.8.10 and Formulas (2.8.25), (2.8.26).)

4.4. The Formulas of Feynman and Kac

We continue our program of obtaining stochastic representations for solu-tions of partial differential equations. In the first subsection, we introduce theFeynman-Kac formula, which provides such a representation for the solutionof the parabolic equation

au 1(4.1) + ku =

2- Au + g; (t, x) e (0, oo) x Rd

subject to the initial condition

(4.2) u(0, x) = f(x); x e Rd

for suitable functions k: Rd [0, co), g: (0, co) x R4 IR and f: Rd R.In the special case of g = 0, we may define the Laplace transform

z(x) A f e'u(t, x)dt; x e Rd,

and using (4.1), (4.2), integration by parts, and the assumption thatlim, e'u(t, x) = 0; a > 0, x e Rd, we may compute formally

(4.3) 1-1

2 2 0f° Cat Au dt = (a + k)z,, -f.

The stochastic representation for the solution z of the elliptic equation (4.3)is known as the Kac formula; in the second subsection we illustrate its usewhen d = 1 by computing the distributions of occupation times for Brownianmotion. The second subsection may be read independently of the first one.

Throughout this section, {W, .; 0 s t < {-Px}xeRd is a d-dimensional Brownian family.

A. The Multidimensional Formula

4.1 Definition. Consider continuous functions f : R, k: Rd [0, co), andg: [0, T] x Rd R. Suppose that v is a continuous, real-valued function on[0, T] x Rd, of class C" on [0, T) x Rd (see the explanation precedingTheorem 3.3), and satisfies

av 1(4.4) --at + kv = Av + g; on [0, T) x Rd,

(4.5) v(T, x) = f(x); x e Rd.

Then the function v is said to be a solution of the Cauchy problem for thebackward heat equation (4.4) with potential k and Lagrangian g, subject to theterminal condition (4.5).

4.2 Theorem (Feynman (1948), Kac (1949)). Let v be as in Definition 4.1 andassume that

(4.6) max I v(t, x)I + max Ig(t, x)I < Kell x112; V x e Rd,05t5T 0<t<T

for some constants K > 0 and 0 < a < 1/2Td. Then v admits the stochasticrepresentation

T-t(4.7) v(t, = EX J.( Wr-i) exP - J k(147s)ds}

T-:g(t + 0, W0) exp - k(llis) ds} di 05t5T,xERd.

In particular, such a solution is unique.

43 Remark. If g > 0 on [0, T] x Rd, then condition (4.6) may be replaced by

(4.8) max I v(t, x)I < Ke°14112; V x e Rd.

This leads to the following maximum principle for the Cauchy problem: if thecontinuous function v: [0, T] x Rd R is of class C1'2 on [0, T) x Rd andsatisfies the growth condition (4.8), as well as the differential inequality

av 1

at+

2kv > - Av on [0, T) x Rd-

with a continuous potential k: Rd [0, oo), then v > 0 on {T} x Rd impliesv > 0 on [0, T] x Rd.

4.4 Remark. If we do not assume the existence of a 0.2 solution to the Cauchyproblem (4.4), (4.5), then the function defined by the right-hand side of (4.7)need not be C1.2. The reader is referred to Friedman (1964), Chapter 1,Friedman (1975), p. 147, or Dynkin (1965), Theorem 13.16, for conditionsunder which the Cauchy problem of Definition 4.1 admits a solution. This isthe case, for example, if k is bounded and uniformly Holder-continuous oncompact subsets of Rd, g is continuous on [0, T] x l and Holder-continuousin x uniformly with respect to (t, x) e [0, T] x Rd, f is continuous, and for someconstants L and v > 0 we have

max I g(t, x)I + I f(x)I _< L(1 + 11x11"); X E Rd.OS.t

PROOF OF THEOREM 4.2. We obtain from Ito's rule, in conjunction with (4.4):

d[v(t + 0, Wo)exp k(W) ds}]

d

= exp -f0 ax

k(W)ds}[- g(t + 0, W0)d0 + E v(t + 0, Wo)dWoalt

Let S = inf {t 0; II 14/,11 > n,/d}; n > 1. We choose 0 < r < T - t and in-tegrate on [0, r n Se]; the resulting stochastic integrals have expectation zero,so

rAs,,v(t, x) = g(t + 0, Wo)exp {- f k(W)ds} d0

o

+ Ex [ s"[v(t + Sn, Ws.) exp -f k(Ws) ds} 1{sr}]o

+ Ex[v(t + r, W.) exp k(W)ds} 1 {sn,r}].

The first term on the right-hand side converges toT-t

Ex J g(t + 0, Wo) exp k(VV,)ds} dB

as n oo and r j T - t, either by monotone convergence (if g > 0) or by dom-inated convergence (it is bounded in absolute value by f o -`fg(t + 0, W0)Id0,which has finite expectation by virtue of (4.6)). The second term is dominatedby

Ex [Iv(t + S, Wsn)11(snr--1)] < Kea"' Px[S < T]

< Keadn2 E px [ max IV! >j=1 0.t5T

d

2Kead"2 E {px[wp) > n]

+ Px[ -wp) n]

where we have used (2.6.2). But by (2.9.20),

eadn, px[+ W- ) > n]T 1

eadn2 e(n -T- x()))212T

2n n -T

which converges to zero as n oo, because 0 < a < (1/2Td). Again by thedominated convergence theorem, the third term is shown to converge toEx [v(T, W -i) exP -g-k(Ws)dsl] as n co and rT T - t. The Feynman-Kac formula (4.7) follows.

4.5 Corollary. Assume that f: Rd -> R, k: Rd -> [0, oo), and g: [0, oo) x Rd -> Rare continuous, and that the continuous function u: [0, co) x Rd -R is of classC''2 on (0, co) x Rd and satisfies (4.1) and (4.2). If for each finite T > 0 thereexist constants K > 0 and 0 < a < (1/2Td) such that

max lu(t,x)1 + max Ig(t,x)I < Kea11x112; V x e R,OStST 0 <t <T

then u admits the stochastic representation

(4.9)

u(t, x) = Ex[f(140exp j -f t k(Ws)ds}

+ j'i g(t - 0, Wo) exp - k(Ws)ds} Al 0 t < co, x e Rd.

In the case g = 0 we can think of u(t, x) in (4.1) as the temperature at timet > 0 at the point x e Rd of a medium which is not a perfect heat conductor,but instead dissipates heat locally at the rate k (heat flow with cooling).The Feynman-Kac formula (4.9) suggests that this situation is equivalent toBrownian motion with annihilation (killing) of particles at the same rate k: theprobability that the particle survives up to time t, conditional on the path{ Ws; 0 < s < t }, is then exp{ -f o k(Ws) ds}

4.6 Exercise. Consider the Cauchy problem for the "quasilinear" parabolicequation

av(4.10) = AV 1

ilVVII2 k; in (0, co) x Rd,

(4.11) V(0, x) = 0; x e Rd

(linear in (0 V/0t) and the Laplacian AV, nonlinear in the gradient VV),where k: [0, co) is a continuous function. Show that the only solutionV: [0, co) x fed -> R which is continuous on its domain, of class CI.2 on(0, 0o) x Rd, and satisfies the quadratic growth condition for every T > 0:

- V(t, x) C ± a Ilx112 ; (t, x)e [0, x Rd

4.4. The Formulas of Fe; nman and Kac 271

where T > 0 is arbitrary and 0 < a < 1/2Td, is given by

(4.12) V(t, x) = - log Elexp j -f k(147,)dsfl; 0 < t < co, x e Rd.0

We turn now our attention to equation (4.3). The discussion at the beginningof this section and equation (4.9) suggest that any solution z to the equation

(4.13) (a + k)z = 2-Az + f; on Rd

should be represented as

(4.14) z(x) = Ex f f (WE) exp { - at -f k(Ws)ds} dt.

4.7 Exercise. Let f: Rd -R and k: Rd - [0, oo) be continuous, with

.0

(4.15) Ex f if(NI exp {- at -f k(Ws)d.s} dt < co; VxeRd,0 0

for some constant a > 0. Suppose that 1//: Rd -+ R is a solution of class C2 to(4.13), and let z be defined by (4.14). If ti/ is bounded, then tai = z; if tk is non-negative, then ti/ > (Hint: Use Problem 2.25).

B. The One-Dimensional Formula

In the one-dimensional case, the stochastic representation (4.14) has the re-markable feature that it defines a function of class C2 when f and k are con-tinuous. Contrast this, for example, to Remark 4.4. We prove here a slightlymore general result.

4.8 Definition. A Borel-measurable function f: R -R is called piecewise-continuous if it admits left- and right-hand limits everywhere on R and it hasonly finitely many points of discontinuity in every bounded interval. Wedenote by Df the set of discontinuity points of f. A continuous functionf R R is called piecewise Ci, j > 1, if its derivatives f"), 1 < i < j - 1 arecontinuous, and the derivative f4-1) is piecewise-continuous.

4.9 Theorem (Kac (1951)). Let f : R -0 R and k: R - [0, co) be piecewise-continuous functions with

(4.16) J W I f(x + y)le-lYkrix dy < co; V x E R,

for some fixed constant a > 0. Then the function z defined by (4.14) is piecewiseC2 and satisfies

1(4.17) (a + k)z = -2z" + f; on 01\(Df u Dk).

4.10 Remark. The Laplace transform computationT00

1 1e' , e-42/2` dt = e-I41; a> 0, e Ro /2nt / 2a

enables us to replace (4.16) by the equivalent condition

(4.16)' Ex f e-alf1WtHdt < co,V x e R.OD

PROOF OF THEOREM 4.9. For piecewise-continuous functions g which satisfycondition (4.16), we introduce the resolvent operator G. given by

1

OD

I(G.g)(x) A Ex e.' a g(W)dt =1

e-Iv-xl ' g(y) dyJo .,./2a -03

1 x= e(Y-x) + I

03

e(x-Y g(y) dy1; x e R..,./Tot -cc x

Differentiating, we obtain

(Gag)'(x) = J e(x-v1 Nr2xg(y)dy - J x e(Y-x1 (y)dy; x e R,

(4.18) (Gag)" (x) = - 2g(x) + 2a(Gag)(x); x e Dg.

It will be shown later that

(4.19) G7(kz) = Gaf -z

and

(4.20) G2(1kz1)(x) < co; V x e R.

If we then write (4.18) successively with g = f and g = kz and subtract, weobtain the desired equation (4.17) for x e R\(Df u DkZ), thanks to (4.19). Onecan easily check via the dominated convergence theorem that z is continuous,so DkZ g Dk. Integration of (4.17) yields the continuity of z'.

In order to verify (4.19), we start with the observation

0 < k(Ws) exp - J k(WJdul ds = 1 - exp - J k(W,,) du} < 1; t > 0,

and so by Fubini's theorem and the Markov property:

(Gaf - z)(x) = Ex e't(1 - e-ftok(ws)ds)f(W,)dtJo

= f e-"f(147,) fo k(Ws)exp -f k(W) du} dsdto

= Ex [k(Ws) exp - - k(VV.) du} f(W,) dtids

= Cc'sEx[t k(W.,) exp ca -f k(Ws,..) du} f(Ws,)dtids

= Ex e'k(Ws) Ex[f exp - at - J k(Ws,) du} f(W54.,)dt ds

= Ex f k(Ws)z(WS)ds = (G(kz))(x); x fR,

which gives us (4.19). We may replace fin (4.14) by If I to obtain a nonnegativefunction > Izi, and just as earlier we have

G,r(ikz1)(x) (G,z(k1))(x) = (G(I f I) - i)(x) < oo; X E

Relation (4.20) follows.

Here are some applications of Theorem 4.9.

4.11 Proposition (P. Levy's Arc-Sine Law for the Occupation Time of (0, co)).Let 1"+(t) rol(0,03)(W.$)ds. Then

el'

fo r,/s(1 - s)(4.21) P°[F,(t) lc

ds0] = =

2arc sin < < t.

PROOF. For a > 0, fl > 0 the function

z(x) = Ex J exp - at - f 1(0,03)(Ws) ds) dtCO

(with potential k = # 1(0,,) and Lagrangian f = 1) satisfies, according toTheorem 4.9, the equation

acz(x) = +z "(x) - #2(x) + 1; x > 0,

az(x) = Zz "(x) + 1; x < 0,

and the conditions

z(0 +) = z(0 -); z'(0 +) = z'(0 -).

The unique bounded solution to the preceding equation has the form

1+

a + "

x > 0

Bex/1

i-`" + -; x < 0.a

The continuity of z( ) and z'() at x = 0 allows us to solve for

A = (.\ / + # - .1,-2)/(cc SO, so

fco

e1

' E° e-"i)dt = z(0) = ; a > 0, f > 0.o ,/a(a + /3)

We have the related computation

CI" ' CI' " e-atCa' d0 dt = dt dB

1o 1o ik/0(t - 0) fo 7r/B 8 ./t -01 c°e-(24-4" ..1°D e-.

= ds d0It o /13 o

since

1

(4.22)e-7`

dt =Jo

y; y > 0.

The uniqueness of Laplace transforms implies (4.21).

4.12 Proposition (Occupation Time of (0, co) until First Hitting b > 0). For> 0, > 0, we have

Tb

(4.23) exp [ - 131-_,(Tb)] A E °exp [- 1(0,.)(Ws) (Li

1

cosh

PROOF. With 1-6(t) A Po 1(b,c0)(Ws)ds, positive numbers cc, 13, y and

we have

z(x) EX J 1(0,00)(W,)exp(- at - $F (t) - yl-b(t))dt,

Z(0) = E° exp(- at - #1-+(t))dr+(t)

+ exp( -ca - 13r+(t) - yrb(t))dr+(t)Ty

Since Tb(t) > 0 a.s. on { Tb < t} (Problem 2.7.19), we have

Tb

lim z(0) = E° J exp(- at - 131-+(t)) dr+(t)rt co

('o(4.24) lim lim z(0) = E° J b exp(-I3F+(t))dr+(t)

rt cc)

1

= -[1 - E° exp(- #1-+(Tb))]

According to Theorem 4.9, the function z() is piecewise C2 on R and satisfiesthe equation (with a = a + p3):

az(x) = z" (x); x < 0,

crz(x) = lz"(x) + 1; 0 < x < b,

(a- + y)z(x) = lz"(x) + 1; x > b.

The unique bounded solution is of the form

Aex12;; x < 0,

z(x) =Bex zQ + Ce-xN/ia +

a0 < x < b,

1x > b.+ ;

a y

Matching the values of z() and z'() across the points x = 0 and x = b, weobtain the values of the four constants A, B, C, and D. In particular, z(0) = Ais given by

sinh b.,/2a a + y [-cosh b - 1 1

a L ,/2a ,/2(a + y)2

(,./2ot + \/2(a + y)) cosh 62a + (.12ot ± Y) + sinh bN/2a

whence

and

lim z(0) =yt N/2(oz + 13) cosh b.,/2(a + /3) + ,./2a sinh b.\/2(« + fl)

2(cosh b,./2(ot + - 1)

lim lim z(0) =1-[1 1

a4,0 ytco /3 cosh b

The result (4.23) now follows from (4.24).


2.16. Assume without loss of generality that a = O. Choose 0 < r < fl b II A 1. It sufficesto show that a is regular for Br c D (Remark 2.11). But there is a simple arc C inOld\D connecting a = 0 to 6, and in 13,\C, a single-valued, analytic branch oflog(x, + ix2) can be defined because winding about the origin is not possible.Regularity of a = 0 is an immediate consequence of Example 2.14 and Proposition2.15.

2.22. Every boundary point of D satisfies Zaremba's cone condition, and so is regular.It remains only to evaluate (2.12).

Whenever Y and Z are independent random variables taking values in mea-surable spaces (G, W) and (H, le), respectively, and f: G x H R is bounded andmeasurable, then

Ef(Y,Z) = f f f(y,z)P[Y dy]P[Z dz].H G

We apply this identity to the independent random variables TD = inf {t 0;

Wt(d) = 01 and {(W,"), W(d-1)); 0 < t < oo 1, the latter taking values inC[0, cold'. This results in the evaluation (see (2.8.5))

cc

E'fiwt,,) =J

/(Y1, , Yd-1, 0)P[(14/t(1), , Wt(d-1))E (dY I, ,dYd-i)]la ad-t

P"[TDE dt]

xdfol- x112]dt

dy,t(2rztr2exP 2tao

and it remains only to verify that

1Az 2,42 r (do)

(5.1) ed+2)/2 exP [ A > 0.2t dt = Ad

For d = 1, equation (5.1) follows from Remark 2.8.3. For d = 2, (5.1) can beverified by direct integration. The cases of d > 3 can be reduced to one of thesetwo cases by successive application of the integration by parts identity

Jo

r"r1 1

e-''212` dt =2(ot 1)

ta" 1

e dt, a > 1.Az+

2.25. Consider an increasing sequence {D};,°_, of open sets with /5 c D; V n 1

and U,T=, D = D, so that the stopping times zn = inf {t > 0; IV, D} satisfylimn-. tn = TD, a.s. Px. It is seen from ItO's rule that

tnrMr) g(Ws)ds, 0 < t <

is a P"-martingale for every n > 1, X ED; also, both I Mg(w)1, I M,")(0)1 are boundedabove by

max lul + (t A TD(co)) max WI, for Px-a.e. w E

By letting n cc and using the bounded convergence theorem, we obtain themartingale property of the process M; its uniform integrability will follow assoon as we establish

(5.2) E'rD < cc, V x ED.

Then the limiting random variable Mm = Mt of the uniformly integrablemartingale M is identified as

M = u(W,D) + g(141,)dt, a.s. P' (Problem 1.3.20)

and the identity E"Mo = PA/co yields the representation (2.27).As for (5.2), it only has to be verified for D = B,. But then the function

v(x) = (r2 - 1142)/d of (2.28) satisfies (1/2)Av = -1 in B, and v = 0 on OB,,and by what we have already shown, {v(Wt,,,) + (t A TO, Ft; 0 t < op) isa P"-martingale. So v(x) = Eqv(W,.,) + (t -ra)] ET A TB), and uponletting t co we obtain Ex TB v(x) < GC .

3.1. (Copson (1975)): Fix fi > 0, E > 0, 0 < to < t, s (1/2(a + E)), and set B = {(t, x);to < t < tI,Ixj < fi}. For (t, JOE B, ye R, we have

(x Y)2 ay2 - -2t

11 1

ay2t

(Ix1 - ly1)2 aye --2ty2 + -t,

IxIlyI, ,

R2E E

2Y2 - -

2Y2 +

#-t,IYI +

2 2Et?

For any nonnegative integers n and m, there is a constant C(n, m) such that

on+m

Ot"Ox 'np(t; x, y)

and so

(5.3)

C(n, m)(1 + 1Y12"-"") exP

an+m

et"Ox'"p(t; x, y)

(x - y)2}2t

; (t, x)e B, ye

ly12"-"")expl-(a + -2)3/2 + -2SE2t1}5- IBA m)(1 +

D(n,m)If(y)le-aY2; (t, x)e B, y E R,

where D(n, m) is a constant independent of t, x, and y. It follows from (3.3) thatthA integral in (3.5) converges uniformly for (t, x) e B, and is thus a continuousfunction of (t, x) on (0,1/2a) x

We prove (3.5) for the case n = 0, m = 1; the general case is easily establishedby induction. For (t, x) E B, (t, x + h) e B, we have

h

1

-[u(t, x + h) - u(t, x)] = Jh-[p(t; x + h, y) - p(t, x, yflf(y)dy,a

= -pct; oho, 31, YV(Y)dy,Ox

where, according to the mean-value theorem, O(t, y) lies between x and x + h.We now let h 0, using the bound (5.3) and the dominated convergence theorem,to obtain (3.5).

3.2. (Widder (1944)): We suppose f is continuous at xo and assume without loss ofgenerality that f(xo) = 0. For each E > 0, there exists (5 > 0 such that If(y)I < Efor Iy - xo 15 S. We have for X E [x0 - (5/2), -X0 (6/2)],


so -6 so+6(5.4) I u(t, x)I f I f( APO; -; A dy + I fly)1P(t; x, y)dy

so-6

+

fco

I f(Y)IP(t; x, Ady...+6

The middle integral is bounded above by e; we show that the other two convergeto zero, as t 10. For the third integral, we have the upper bound

by1r (y - xo -ze'Y If(y)1 exp ay 2(5.5)

27cJt J so+6 2tdy.

For t sufficiently small, exp[ay2 - (y - xo - 6/2)2 /2t] is a decreasing functionof y for y xo + 6 (it has its maximum at y = (x0 + 6/2)/(1 - 2at)). Therefore,the expression in (5.5) is bounded above by

p t; 0, -2) exp[a(x, + 6)2] j. e "2 If(Al dY.,

xo+ 6

which approaches zero as t10. The first integral in (5.4) is treated similarly.

6

4.6. Notes

Section 4.2: The Dirichlet problem has a long and venerable history (see,e.g., Poincare (1899) and Kellogg (1929)). Zaremba (1911) was the first toobserve that the problem was not always solvable, citing the example of apunctured region. Lebesgue (1924) subsequently pointed out that in three ormore dimensions, if D has a sufficiently sharp, inward-pointing cusp, then theproblem can fail to have a solution (our Example 2.17). Poincare (1899) usedbarriers to show that if every point on ap lies on the surface of a sphere whichdoes not otherwise intersect D, then the Dirichlet problem can be solved inD. Zaremba (1909) replaced the sphere in Poincare's sufficient condition by acone. Wiener (1924b) has given a necessary and sufficient condition involvingthe capacity of a set.

The beautiful connection between the Dirichlet problem and Brownianmotion was made by Kakutani (1944a, b), and his pioneering work laidthe foundation for the probabilistic exposition we have given here. Hunt(1957/1958) studied the links between potential theory and a large class oftransient Markov processes. These matters are explored in greater depth inIto & McKean (1974), Sections 7.10-7.12; Port & Stone (1978); and Doob(1984).

Section 4.3: The representation (3.4) for the solution of the heat equationis usually attributed to Poisson (1835, p. 140), although it was known to bothFourier (1822, p. 454) and Laplace (1809, p. 241). The heat equation for thesemi-infinite rod was studied by Widder (1953), who established uniqueness


4.6. Notes 279

and representation results similar to Theorems 3.3 and 3.6. Hartman &Wintner (1950) considered the rod of finite length. For more examples andfurther information on the subject matter of Subsection C, including applica-tions to the theory of statistical tests of power one, the reader is referred toRobbins & Siegmund (1973), Novikov (1981), and the references therein.

Section 4.4: Theorem 4.2 was first established by M. Kac (1949) for d = 1;his work was influenced by the derivation of the Schrodinger equation achievedby R. P. Feynman in his doctoral dissertation. Kac's results were strengthenedand extended to the multidimensional case by M. Rosenblatt (1951), who alsoprovided Holder continuity conditions on the potential k in order to guaranteea C1.2 solution. Proposition 4.12 is taken from ItO & McKean (1974).

Let k: l -4 [0, co) be continuous and satisfy k(x) = co; then theeigenvalue problem

(k(x) - A)tli(x) =2- tr(x); x e

with e L2(R), has a discrete spectrum Al < A2 < ... and corresponding eigen-functions ti/J;17_1 g L2(R). Kac (1951) derived the stochastic representation

1

(6.1) Al - lirn - log Ex[exp k(14(,) ds}],_) t

for the principal eigenvalue, by combining the Feynman-Kac expression

u(t,x) = Ex[exp { -Jo k(141s)ds}]

for the solution of the Cauchy problem

Ou 1

ot+ k u =

2-Au; (0, c3o) x

(6.2)u(0, x) 1; xel

(Corollary 4.5), with the formal eigenfunction expansion

00u(t,x) E cie-Aittp,(x).i=1

for the solution of (6.2). Recall also Exercise 4.6, and see Karatzas (1980) fora control-theoretic interpretation (and derivation) of this result.

Sweeping generalizations of (6.1), as well as an explanation of its connectionwith the classical variational expression

(6.3) A, = infJ.

k(x)tly 2 (X) dx + -_co

( /(x))2dx2

if°, 14,2(x) dx=1.

} ,

are provided in the context of the theory of large deviations of Donsker &


Varadhan (1975), (1976). This theory constitutes an important recent develop-ment in probability theory, and is overviewed succinctly in the monographsby Stroock (1984) and Varadhan (1984).

The reader interested in the relations of the results in this section withquantum physics is referred to Simon (1979).

Alternative approaches to the arc-sine law for fi.(t) can be found inExercise 6.3.8 and Remark 6.3.12.


CHAPTER 5

Stochastic DifferentialEquations

5.1. Introduction

We explore in this chapter questions of existence and uniqueness for solutionsto stochastic differential equations and offer a study of their properties. Thisendeavor is really a study of diffusion processes. Loosely speaking, the termdiffusion is attributed to a Markov process which has continuous sample pathsand can be characterized in terms of its infinitesimal generator.

In order to fix ideas, let us consider a d-dimensional Markov family X ={X A; 0 < t < co}, (52, ,F), Px 1xe Rd, and assume that X has continuouspaths. We suppose, further, that the relation

(1.1) lim1

- [ef(X,) -f (x)] = (cif )(x); dxePdrlo t

holds for every f in a suitable subclass of the space C2(Rd) of real-valued,twice continuously differentiable functions on Rd; the operator s/f in (1.1) isgiven by

d d a2f( d

(1.2) (sif )(x)I

.E1 kE1 a ik(x)x)

+ofi x)

2

for suitable Borel-measurable functions aik: Rd R, 1 5 i, k S d. The left-hand side of (1.1) is the infinitesimal generator of the Markov family, applied tothe test function f. On the other hand, the operator in (1.2) is called the second-order differential operator associated with the drift vector b = (1)1, . , bd) andthe diffusion matrix a = {aik} <i,ksd, which is assumed to be symmetric andnonnegative-definite for every x e Rd.

The drift and diffusion coefficients can be interpreted heuristically in thefollowing manner: fix X E Rd and let fi(y) -4 yj, fik(y) A (yi - xi)(yk - xk);

282 5. Stochastic Differential Equations

y e Rd. Assuming that (1.1) holds for these test functions, we obtain

(1.3) Ex[Xls) - xi] = tb,(x) + o(t)

(1.4) Ex [(XiI) - xj)()Ok) - xk)] = ta,k(x) + o(t)

as t J, 0, for 1 < i, k < d. In other words, the drift vector b(x) measures locallythe mean velocity of the random motion modeled by X, and a(x) approximatesthe rate of change in the covariance matrix of the vector X, - x, for smallvalues of t > 0. The monograph by Nelson (1967) can be consulted for adetailed study of the kinematics and dynamics of such random motions.

1.1 Definition. Let X = IX ,57;,; 0 < t < co }, (12, g;),dimensional Markov family, such that

{1'1. Esd be a d-

(i) X has continuous sample paths;(ii) relation (1.1) holds for every f e C2(Rd) which is bounded and has bounded

first- and second-order derivatives;(iii) relations (1.3), (1.4) hold for every x e Rd; and(iv) the tenets (a)-(d) of Definition 2.6.3 are satisfied, but only for stopping

times S.

Then X is called a (Kolmogorov-Feller) diffusion process.

There are several approaches to the study of diffusions, ranging from thepurely analytical to the purely probabilistic. In order to illustrate the traditionalanalytical approach, let us suppose that the Markov family of Definition 1.1has a transition probability density function

(1.5) Px [X, e dy] = F (t; x, y)dy; V x e t > 0.

Various heuristic arguments, with (1.1) as their starting point, can then beemployed to suggest that r(t; x, y) should satisfy the forward Kolmogorovequation, for every fixed x e Rd:

0(1.6) -

Otf(t; x, y) = ..il*F(t; x, y); (t, y)E(0, 00) x Rd,

and the backward Kolmogorov equation, for every fixed y e Rd:

0(1.7) -

atI-(t; x, y) = sir(t; x, y); (t, x) e (0, co) x Rd.

The operator d* in (1.6) is given by

(1.8) (.91*.f)(Y) uy (4,i=i k=1[aik(Y)f(Y)]

1

Ebt(Y)f(Y)1,=1

the formal adjoint of .4 in (1.2), provided of course that the coefficients k, askpossess the smoothness requisite in (1.8). The early work of Kolmogorov(1931) and Feller (1936) used tools from the theory of partial differential

5.1. Introduction 283

equations to establish, under suitable and rather restrictive conditions, theexistence of a solution C(t; x,y) to (1.6), (1.7). Existence of a continuousMarkov process X satisfying (1.5) can then be shown via the consistencyTheorem 2.2.2 and the entsov-Kolmogorov Theorem 2.2.8, very much in thespirit of our approach in Section 2.2. A modern account of this methodologyis contained in Chapters 2 and 3 of Stroock & Varadhan (1979).

The methodology of stochastic differential equations was suggested byP. Levy as an "alternative," probabilistic approach to diffusions and wascarried out in a masterly way by K. ItO (1942a, 1946, 1951). Suppose that wehave a continuous, adapted d-dimensional process X = {X 3,7,; 0 < t < cc)which satisfies, for every x e Rd, the stochastic integral equation

(1.9)r

AT ft

) = xi + bi(Xs)ds + Eo J =1

' t0-1;(xjawch;

00 < t < co, I R; 1 < i < d, 1 < j < rare Borel-measurable. Then it is reasonable to expect that, under certainconditions, (1.1)-(1.4) will indeed be valid, with

(1.10) au,(x) Er aii(90"ki(X).i=1

We leave the verification of this fact as an exercise for the reader.

1.2 Problem. Assume that the coefficients bi, au are bounded and continuous,and the Or-valued process X satisfies (1.9). Show that (1.3), (1.4) hold for everyx e Or, and that (1.1) holds for every f e C2(Rd) which is bounded and hasbounded first- and second-order derivatives.

Ito's theory is developed in Section 2 under the rubric of strong solutions.A strong solution of (1.9) is constructed on a given probability space, withrespect to a given filtration and a given Brownian motion W. In Section 3we take up the idea of weak solutions, a notion in which the probability space,the filtration, and the driving Brownian motion are part of the solution ratherthan the statement of the problem. The reformulation of a stochastic differentialequation as a martingale problem is presented in Section 4. The solutionof this problem is equivalent to constructing a weak solution. Employingmartingale methods, we establish a version of the strong Markov property-corresponding to (iv) of Definition 1.1-for these solutions; they thereby earnthe right to be called diffusions.

The stochastic differential equation approach to diffusions provides apowerful methodology and the useful representation (1.9) for a very largeclass of such processes. Indeed, the only important strong Markov processeswith continuous sample paths which are not directly included in such adevelopment are those which exhibit "anomalous" boundary behavior (e.g.,reflection, absorption, or killing on a boundary).

Certain aspects of the one-dimensional case are discussed at some lengthin Section 5; a state-space transformation leads from the general equationto one without drift, and the latter is studied by the method of randomtime-change. The notion and properties of local time from Sections 3.6, 3.7play an important role here, as do the new concepts of scale function, speedmeasure, and explosions.

Section 6 studies linear equations; Section 7 takes up the connections withpartial differential equations, in the spirit of Chapter 4 but not in the samedetail.

We devote Section 8 to applications of stochastic calculus and differen-tial equations in mathematical economics. The related option pricing andconsumption/investment problems are discussed in some detail, providingconcrete illustrations of the power and usefulness of our methodology. Inparticular, the second of these problems echoes the more general themes ofstochastic control theory.

The field of stochastic differential equations is now vast, both in theory andin applications; we attempt in the notes (Section 10) a brief survey, but weabandon any claim to completeness.

5.2. Strong Solutions

In this section we introduce the concept of a stochastic differential equationwith respect to Brownian motion and its solution in the so-called strong sense.We discuss the questions of existence and uniqueness of such solutions, as wellas some of their elementary properties.

Let us start with Borel-measurable functions Mt, x), x); 1 < i < d,1 < j < r, from [0, co) x Rd into R, and define the (d x 1) drift vector b(t, x) ={13,(t, x)} and the (d x r) dispersion matrix o-(t, x) = {6;;(t, x) }, The

sjSrintent is to assign a meaning to the stochastic differential equation

(2.1) dX, = b(t, X ,) dt + o(t, X,)dW

written componentwise as

(2.1)' = bat, X,)dt + E xt) dwto; 1 < i < d,.i=i

where W = {W; 0 < t < oo } is an r-dimensional Brownian motion and X ={X,; 0 < t < co} is a suitable stochastic process with continuous sample pathsand values in Rd, the "solution" of the equation. The drift vector b(t, x) andthe dispersion matrix o-(t, x) are the coefficients of this equation; the (d x d)matrix a(t, x) A x)o-T(t, x) with elements

(2.2) aik(t, x) A E 0-,;(t,x)o-kp,x); 1 < i, k < d1=1

will be called the diffusion matrix.

A. Definitions

285

In order to develop the concept of strong solution, we choose a probabilityspace (0, P) as well as an r-dimensional Brownian motion W = .317,w;

0 < t < co} on it. We assume also that this space is rich enough to accom-modate a random vector taking values in Rd, independent of 'cow, and withgiven distribution

u(r) = ell; re ARd)-We consider the left-continuous filtration

a (0 v = cr(, Ws; 0 s t); 0 < t < op,

as well as the collection of null sets

P-4 {Ns O.; 3 G e W., with N G and P(G)= 0},

and create the augmented filtration

(2.3) a (g, u .41), 0 < t < oo; (ut 0

by analogy with the construction of Definition 2.7.2. Obviously, { W A;o < t < co} is an r-dimensional Brownian motion, and then so is { W0 < t < co} (cf. Theorem 2.7.9). It follows also, just as in the proof ofProposition 2.7.7, that the filtration {",} satisfies the usual conditions.

2.1 Definition. A strong solution of the stochastic differential equation (2.1),on the given probability space (S2, F, P) and with respect to the fixed Brownianmotion W and initial condition is a process X = {X,; 0 .5 t < oo} withcontinuous sample paths and with the following properties:

(i) X is adapted to the filtration {..F,} of (2.3),(ii) P[X0 = ] = 1,

(iii) P[Po Xs)1 + Xs)} ds < co] = 1 holds for every 1 < i < d,1 j r and 0 t < co, and

(iv) the integral version of (2.1)

(2.4) X, = X0 + J b(s, X s) ds + ft o-(s, Xs) dWs; 0 < t < o ,

or equivalently,

a

(2.4)' )01) = Xg) + J bi(s, X 5) ds + E I 0-,;(s, X s)dWP);J=1 Jo

0 < t < co, 1 < i d,

holds almost surely.

2.2 Remark. The crucial requirement of this definition is captured in con-dition (i); it corresponds to our intuitive understanding of X as the "output"

of a dynamical system described by the pair of coefficients (b, Q), whose "input"is W and which is also fed by the initial datum

input W

1

b, a X output

The principle of causality for dynamical systems requires that the output X,at time t depend only on c and the values of the input Ws; 0 < s < t} up tothat time. This principle finds its mathematical expression in (i).

Furthermore, when both c and { W; 0 < t < ool are given, their specificationshould determine the output {Xt; 0 < t < oo} in an unambiguous way. Weare thus led to expect the following form of uniqueness.

2.3 Definition. Let the drift vector b(t, x) and dispersion matrix °It, x) be given.Suppose that, whenever W is an r-dimensional Brownian motion on some(0, P), c is an independent, d-dimensional random vector, {A} is givenby (2.3), and X, 31 are two strong solutions of (2.1) relative to W with initialcondition then P[X, = ; 0 < t < oo] = 1. Under these conditions, we saythat strong uniqueness holds for the pair (b, Q).

We sometimes abuse the terminology by saying that strong uniqueness holdsfor equation (2.1), even though the condition of strong uniqueness requires usto consider every r-dimensional Brownian motion, not just a particular one.

2.4 Example. Consider the one-dimensional equation

dX, = b(t, X,) dt + dW,

where b: [0, co) x IR E is bounded, Borel-measurable, and nonincreasing inthe space variable; i.e., b(t, x) < b(t, y) for all 0 < t < oo, -oo < y <x < oo.For this equation, strong uniqueness holds. Indeed, for any two processes X(1),X(2) satisfying P-a.s.

1Xi) = X0 + t b(s, XV) ds + W; 0 < t < oo and i = 1, 2,o

we may define the continuous process A, = XP) - XP) and observe that

1,6, = 2 t (X11) - V))[b(s, XP)) - b(s, X!2))] ds < 0; 0 < t < oo, a.s. P.o

B. The ItO Theory

If the dispersion matrix °(t, x) is identically equal to zero, (2.4) reduces tothe ordinary (nonstochastic, except possibly in the initial condition) integral

equation

(2.5) X, = X, + ft b(s,

287

In the theory for such equations (e.g., Hale (1969), Theorem 1.5.3), it iscustomary to impose the assumption that the vector field b(t, x) satisfies a localLipschitz condition in the space variable x and is bounded on compact subsetsof [0, op) x Rd. These conditions ensure that for sufficiently small t > 0, thePicard-Lindelof iterations

(2.6) Xr) Xo; X:"+" = X0 + b(s, Xl"))ds, n > 0,0

converge to a solution of (2.5), and that this solution is unique. In the absenceof such conditions the equation might fail to be solvable or might have acontinuum of solutions. For instance, the one-dimensional equation

(2.7) Xt = f t !Xs!' ds

has only one solution for a > 1, namely Xt -.E. 0; however, for 0 < a < 1, allfunctions of the form

0 < t < s,

; s < t < oo,

with # = 1/(1 - a) and arbitrary 0 < s < co, solve (2.7).It seems then reasonable to attempt developing a theory for stochastic dif-

ferential equations by imposing Lipschitz-type conditions, and investigatingwhat kind of existence and/or uniqueness results one can obtain this way.Such a program was first carried out by K. ItO (1942a, 1946).

25 Theorem. Suppose that the coefficients b(t, x), o-(t, x) are locally Lipschitz-continuous in the space variable; i.e., for every integer n 1 there exists aconstant K > 0 such that for every t 0, 114 n and Ilyll n:

(2.8) b(t, - Mt, + 110'(t, x) - Y)11 < KnIlx - .3111.

Then strong uniqueness holds for equation (2.1).

2.6 Remark on Notation. For every (d x r) matrix a, we writed r

(2.9) 116112 -4 E E 6.i =1 j =1

Before proceeding with the proof, let us recall the useful Gronwall inequality.

2.7 Problem. Suppose that the continuous function g(t) satisfies

(2.10) 0 < g(t) < a(t) + # J r g(s) ds; 0 < t < T,0

with fl > 0 and a: [0, T] integrable. Then

(2.11) g(t) < a(t) + 13 f a(s)e"-s) ds; 0 < t < T0

PROOF OF THEOREM 2.5. Let us suppose that X and 2 are both strong solutions,defined for all t > 0, of (2.1) relative to the same Brownian motion W andthe same initial condition on some (SI, P). We define the stopping timesrn = inf ft > 0; IX, II > n} for n > 1, as well as their tilded counterparts, andwe set S T A in. Clearly lim, S = co, a.s. P, and

t A Su

Xt A S Xt A S {NU, Xu) b(u, gu)} du

tAsu+ f

Using the vector inequality Ilv, + + vkIIZk2(Ilvi 112

+ 1114112), theHolder inequality for Lebesgue integrals, the basic property (3.2.27) of sto-chastic integrals, and (2.8), we may write for 0 < t T:

tAS,,

E 11X A - jet A sull2 [f Ilb(u, X) - b(u, gu)II du12o

r t A S

Xu) - u)) d W utni=1

12+ 4E [E fo

4tE

fI A

Ilb(u, Xu) - b(u, g)112 du0

t A Su

+ 4E .1 Xu) - a(u, 5tu)Il2 du

4(T + 1)K,; ft E Xu A - luAsull2 du.

We now apply Problem 2.7 with g(t) A E II X t A s, 21A 5112 to conclude that{X, A S; 0 < t < 00 } and {X n 5; 0 < t < co} are modifications of one another,and thus are indistinguishable. Letting n oo, we see that the same is true for{X,; 0 _< t < co } and {)-1,; 0 t < co }.

2.8 Remark. It is worth noting that even for ordinary differential equations,a local Lipschitz condition is not sufficient to guarantee global existence of asolution. For example, the unique (because of Theorem 2.5) solution to theequation

5.2. Strong Solutions 289

X, = 1 + f t X1 dso

is X, = 1/(1 - t), which "explodes" as t T 1. We thus impose stronger conditionsin order to obtain an existence result.

2.9 Theorem. Suppose that the coefficients b(t, x), a(t, x) satisfy the globalLipschitz and linear growth conditions

(2.12) 116(t, x) - b(t, y)II + lio(t, x) - alt, AII Lc. K 11x - yil,

(2.13) ilb(t, x)O2 + ki(t, X)Il 2 < K2(1 + 0x112),

for every 0 < t < oo, x e Rd, y e Rd, where K is a positive constant. On someprobability space (0, ..F, P), let be an Rd-valued random vector, independent ofthe r-dimensional Brownian motion W = {Hit, ,F,w ; 0 < t < co}, and with finitesecond moment:

(2.14) Ell92 < 00.

Let {A} be as in (2.3). Then there exists a continuous, adapted process X ={X .57;,; 0 < t < col which is a strong solution of equation (2.1) relative to W,with initial condition . Moreover, this process is square-integrable: for everyT > 0, there exists a constant C, depending only on K and T, such that

(2.15) E Vie C(1 + E g 112 )ect; 0 t T

The idea of the proof is to mimic the deterministic situation and to constructrecursively, by analogy with (2.6), a sequence of successive approximations bysetting Xr --- and

t t

(2.16) Xr1) A- + f b(s, XP)ds + 5 o(s, XP)dWs; 0 < t < co,o o

for k > 0. These processes are obviously continuous and adapted to thefiltration {,,}. The hope is that the sequence {X(k)}1°_, will converge to asolution of equation (2.1).

Let us start with the observation which will ultimately lead to (2.15).

2.10 Problem. For every T > 0, there exists a positive constant C dependingonly on K and T, such that for the iterations in (2.16) we have

(2.17) E Il V)II2 C (1 + E g 112)ect; OT,k 0.

PROOF' OF THEOREM 2.9. We have Xr1) - V) = B, + M, from (2.16), where

B, -4 t {b(s V)) - b(s, Xlk-1))1 ds, M, A 5 z fa(s, X!k)) - Os, Xlk-1))1 dWs.o o

Thanks to the inequalities (2.13) and (2.17), the process {M, = (MP ), ... , Mr),

..97t; 0 < t < co is seen to be a vector of square-integrable martingales, forwhich Problem 3.3.29 and Remark 3.3.30 give

E[max ilMs112 A, E X?)) - 0-(s, Xr1))11 2 ds0 <s<t 0

< AIK2E 1t

IV) - Xr1)112ds.o

On the other hand, we have E 1113t II 2 < K2 t f o Ell X?) - there-fore, with L = 4K2(A1 + T),

X.!k-1) 2 ds, and there-11

(2.18) E[ max linc+1) nc). 2 < L J t E X?) - Xr1)112 ds; 0 < t < T.o<s<t

Inequality (2.18) can be iterated to yield the successive upper bounds

(2.19) E[ max(Lt)kxr-i) noil2 < c* . 0 t T,

- k!o<s<t

where C* = maxo<t<T E 11 X}1)1 2,) - a finite quantity because of (2.17).

Relation (2.19) and the Cebygev inequality now give

+

1

(2.20) P[ max xr-1) X(k) ,.II > 1k2 4C*

(4L T)k; k = 1, 2, ... ,

O<t<T k!

and this upper bound is the general term in a convergent series. From theBorel-Cantelli lemma, we conclude that there exists an event Kr e .F withP(S1*) = 1 and an integer-valued random variable N(w) such that for everyxr-i)(0 xpo(co en*: maxo<t<T co) < 2 -(k +1) V k > N(w). Consequently,

(2.21) max II Xrn)(w) - )0k)(0))11 < V m > 1, k > N(w).0<t<T

We see then that the sequence of sample paths 00k)(w); 0 < t < T1 ic°_, is

convergent in the supremum norm on continuous functions, from whichfollows the existence of a continuous limit IXt(w); 0 < t < T} for all w eSince T is arbitrary, we have the existence of a continuous process X = {X t;0 < t < col with the property that for P-a.e. w, the sample paths Ir)(w)},`,°_,converge to X Jo)), uniformly on compact subsets of [0, co). Inequality (2.15)is a consequence of (2.17) and Fatou's lemma. From (2.15) and (2.13) we havecondition (iii) of Definition 2.1. Conditions (i) and (ii) are also clearly satisfiedby X. The following problem concludes the proof.

2.11 Problem. Show that the just constructed process

(2.22) Xt P-- lim V); 0 < t < ook-.co

satisfies requirement (iv) of Definition 2.1.

2.12 Problem. With the exception of (2.15) and the square-integrability of X,the assertions of Theorem 2.9 remain valid if the assumption (2.14) is removed.

C. Comparison Results and Other Refinements

In the one-dimensional case, the Lipschitz condition on the dispersioncoefficient can be relaxed considerably.

2.13 Proposition (Yamada & Watanabe (1971)). Let us suppose that the coeffi-cients of the one-dimensional equation (d = r = 1)

(2.1)

satisfy the conditions

(2.23)

(2.24)

dX, = b(t, Xddt + o(t, XddW,

ib(t, x) - b(t, y)i K ix - yi,

I - h(lx - .Y1),

for every 0 < t < oo and x eR, yeR, where K is a positive constant andh: [0, oo) [0, co) is a strictly increasing function with h(0) = 0 and

(2.25) h-2(u) du = oo; V s > 0.

Then strong uniqueness holds for the equation (2.1).

2.14 Example. One can take the function h in this proposition to be h(u) =(1/2).

PROOF OF PROPOSITION 2.13. Because of the conditions imposed on the functionh, there exists a strictly decreasing sequence {a},T=0 g (0, 1] with ao = 1,lim, a = 0 and fa:-' h-2(u)du = n, for every n > 1. For each n > 1, thereexists a continuous function p on fJ with support in (a, a_,) so that0 < p(x) < (2/nh2(x)) holds for every x > 0, and 117-' p(x) dx = 1. Then thefunction

(2.26) 4 /(x) p,,(u)dudy; x e Ro o

is even and twice continuous y differentiable, with tir,;(x)1 < 1 and0(x) = Ixi for x e rt. Furthermore, the sequence {tp}-_, is

nondecreasing.Now let us suppose that there are two strong solutions x(') and X(2) of (2.1)

with n) = Xo a.s. It suffices to prove the indistinguishability of xo) andX(2) under the assumption

(2.27) E XV)12 ds < oo; 0 t < co, i = 1, 2;

otherwise, we may use condition (iii) of Definition 2.1 and a localizationargument to reduce the situation to one in which (2.27) holds. We have

A, A- XP V) - ) = ft {b(s, X.P)) - b(s, V))} dso

+ft {Os, X11)) - Os, V))1 dWs,o

and by the ItO rule,

(2.28) *At) = fo tk(As)[b(s, XS')) - b(s, X12))] ds

t+ -1 I ti,;(As)[a(s, Vs1)) - o-(s, X12))]2 ds02

+ it IMA.,)[a(s, X,P)) - o(s, X,12))] dWs.0

The expectation of the stochastic integral in (2.28) is zero because of assumption(2.27), whereas the expectation of the second integral in (2.28) is boundedabove by ES ip"(As)h2(14,1) ds < 2t/n. We conclude that

(2.29) Eik(A,) E ft ii/;,(A.,)[b(s, XV)) - b(s, X52))] ds + -to n

_.K.1EI.A.,lds +-t ; t 0, n 1.

o n

A passage to the limit as n -> co yields EIA,1 < K Po E IN ds; t z 0, andthe conclusion now follows from the Gronwall inequality and sample pathcontinuity. 0

2.15 Example (Girsanov (1962)). From what we have just proved, it followsthat strong uniqueness holds for the one-dimensional stochastic equation

(2.30) X,= 1 t I Xsix dWs; 0 t < co,o

as long as a > (1/2), and it is obvious that the unique solution is the trivialone X, -a 0. This is also a solution when 0 < a < (1/2), but it is no longer theonly solution. We shall in fact see in Remark 5.6 that not only does stronguniqueness fail when 0 < a < (1/2), but we do not even have uniqueness in theweaker sense developed in the next section.

2.16 Remark. Yamada & Watanabe (1971) actually establish Proposition 2.13under a condition on b(t, x) weaker than (2.23), namely,

(2.23y I b(t, x) - b(t, Al K(1 x - YI); 0 t < co, x e R, y e R,

where K: [0, co) -+ [0, oo) is strictly increasing and concave with K(0) = 0 andf(0.0(du/K(u)) = co for every s > 0.

2.17 Exercise (Ito & Watanabe (1978)). The stochastic equation

X, = 3 f tX:I3 ds + 3 dW,

has uncountably many strong solutions of the form

0 t <:") =

(47,3; fle < t < co ,

where 0 < O 5 oo and Q® inf {s > W, = 01. Note that the functiono(x) = 3x213 satisfies condition (2.24), but the function b(x) = 3x" fails tosatisfy the condition of Remark 2.16.

The methodology employed in the proof of Proposition 2.13 can be usedto great advantage in establishing comparison results for solutions of one-dimensional stochastic differential equations. Such results amount to a certainkind of "monotonicity" of the solution process X with respect to the driftcoefficient b(t, x), and they are useful in a variety of situations, including thestudy of certain simple stochastic control problems. We develop some com-parison results in the following proposition and problem.

2.18 Proposition. Suppose that on a certain probability space (1-2, .F, P) equippedwith a filtration {,,} which satisfies the usual conditions, we have a standard,one-dimensional Brownian motion {W 37;,; 0 < t < oo} and two continuous,adapted processes Xth; j = 1, 2, such that

fti

(2.31) Xli) = XVI + bf(s, Xn ds + o- (s, XV)) dW.,; 0 < t < coo o

holds a.s. for j = 1, 2. We assume that

(i) the coefficients o-(t, x), k(t,x) are continuous, real-valued functions on[0, co) x R,

(ii) the dispersion matrix o-(t, x) satisfies condition (2.24), where h is as describedin Proposition 2.13,

(iii) XV) < XV) a.s.,(iv) bi(t, x) < b2(t, x), V 0 < t < oo, x e R, and(v) either b1(t, x) or b2(t, x) satisfies condition (2.23).

Then

(2.32) P[XP) < XP), V 0 < t < co] = 1.

PROOF. For concreteness, let us suppose that (2.23) is satisfied by b1(t,x).Proceeding as in the proof of Proposition 2.13, we assume without loss of

generality that (2.27) holds. We recall the functions kfr(x) of (2.26) and createa new sequence of auxiliary functions by setting ton(x) = tfr(x) 1,0,)(x); x e R,n > 1. With A, = XP) - V), the analogue of relation (2.29) is

Eq),,(60 - 5 E .1 (p'(As)[bi(s, x!") - b2(s, X12)11 dsn o

= E ii co;,(As)[bi(s, X11)) - bi(s, X?))] ds0

+ Eit cp',,(As)[bi(s, X?)) - b2(s, V))] ds < K ft WO ds,o o

by virtue of (iv) and (2.23) for 61(t, x). Now we can let n -' co to obtain E(A,+) <K Po E(45-1-)ds; 0 < t < co, and by the Gronwall inequality (Problem 2.7), wehave E(A,+) = 0; i.e., XP) < XP) a.s. P.

2.19 Exercise. Suppose that in Proposition 2.18 we drop condition (v) butstrengthen condition (iv) to

(iv)' bi(t, x) < b2(t, x); 0 < t < co, x e 41.

Then the conclusion (2.32) still holds. (Hint: For each integer m > 3, constructa Lipschitz-continuous function b,,,(t, x) such that

(2.33) bi (t, x) 5 b,(t, x) < b2(t, x); 0 < ( < m, ixi < m).

It should be noted that for the equation

(2.34) X,= + f i b(s, X s) ds + Wr; 0 5 t < co,/3

with unit dispersion coefficient and drift b(t, x) satisfying the conditions ofTheorem 2.9, the proof of that theorem can be simplified considerably. Indeed,since there is no stochastic integral in (2.34), we may fix an arbitrary w e SIand regard (2.34) as a deterministic integral equation with forcing function{ W,(co); 0 5 t < co}. For the iterations defined by (2.16), and with

Er)(w) A max IIV k-i)(w) - )(OE k = 1, 2, ...,osstwe have the bound ak)(co) < K Pc, Dr1)(co)ds; 0 < t < co, valid for everyw e e. The latter can be iterated to prove convergence of the scheme (2.16),path by path, to a continuous, adapted process X which obeys (2.34) surely.This is the standard Picard-Lindeliif proof from ordinary differential equa-tions and makes no use of probabilistic tools such as the martingale inequalityor the Borel-Cantelli lemma.

Lamperti (1964) has observed that, under appropriate conditions on thecoefficients b and o-, the general, one-dimensional integral equation

(2.4)" X, = + f b(Xs)ds + f 6(X0dWs; 0 :5_ t < cc,

can be reduced by a change of scale to one of the form (2.34); see the followingexercise.

2.20 Exercise. Suppose that the coefficients a: IR - (0, oo) and b: R R are ofclass C2 and C1, respectively; that b' - (1/2)66" - (bof/o-) is bounded; and that(1/a) is not integrable at either +oo. Then (2.4)" has a unique, strong solutionX. (Hint: Consider the function f(x) = 1,I(du/o-(u)) and apply Ito's rule tof(x,))

A second important class of equations that can be solved by first fixing theBrownian path and then solving a deterministic differential equation wasdiscovered by Doss (1977); see Proposition 2.21.

D. Approximations of Stochastic Differential Equations

Stochastic differential equations have been widely applied to the study of theeffect of adding random perturbations (noise) to deterministic differentialsystems. Brownian motion offers an idealized model for this noise, but in manyapplications the actual noise process is of bounded variation and non-Markov.Then the following modeling issue arises.

Suppose that { V}°°_, is such a sequence of stochastic processes whichconverges, in an appropriate strong sense, to the Brownian motion W ={W .57;,; 0 < t < co}. Suppose, furthermore, that {X},7_, is a correspondingsequence of solutions to the stochastic integral equations

(2.35) )0") = + J b(Xr))ds + ft o-(X!"))dVs("); n > 1,

where the second integral is to be understood in the Lebesgue-Stieltjes sense.As n -- co, will {X(")},7_, converge to a process X, and if so, what kind ofintegral equation will X satisfy? It turns out that under fairly general condi-tions, the proper equation for X is

(2.36) X, = + f b(Xs)ds + f o-(X )odWs,

where the second integral is in the Fisk-Stratonovich sense.Our proof of this depends on the following result by Doss (1977).

2.21 Proposition. Suppose that a is of class C2(R) with bounded first andsecond derivatives, and that b is Lipschitz-continuous. Then the one-dimensionalstochastic differential equation

1

(2.36)' X, = + f fb(X ) + -o-(X )cr'(X )} ds + f cr(Xs)dly,s 2 "

has a unique, strong solution; this can be written in the form

X t(w) = u(W(w), Yt(co)); 0 < t < oo, co E

for a suitable, continuous function u: R2 R and a process Y which solves anordinary differential equation, for every w el/

2.22 Remark. Under the conditions of Proposition 2.21, the process {cr(X,),.97E; 0 < t < co} is a continuous semimartingale with decomposition

1 1

o-(X,) = cr(0 + J [b(Xs)er'(Xs) + 2cr(X5)(cr'(X5))2 + -2

o-"(Xs)cr2(X5)1ds

+ f r a(X3)0=(X,)dll7s,

and so, according to Definition 3.3.13,

fo cr(XJ0 dW, = -1 ft o-(Xs)o-'(Xs)ds + f o-(Xs)dW.s.2 0

In other words, equations (2.36) and (2.36)' are equivalent.

PROOF OF PROPOSITION 2.21. Let u(x, y): R2 - R be the solution of the ordinarydifferential equation

(2.37)Ou-= o-(u), u(0, y) = y;Ox

such a solution exists globally, thanks to our assumptions. We have then2 02 aU a

(2.38) -Ox2

= a(u)o-'(u), = ce(u)-0y,yu(0,

y) = 1,x0y

which give

(2.39) -oyu(x, y) = exp cr'(u(z, y)) dz} Ap(x, Y).

0 1

Let A > 0 be a bound on o' and a". Then e-Alx1 < p(x, y) < eAlx1, and (2.39)implies the Lipschitz condition

Iu(x,y1) - ti(x,Y2)1 < eAl'IlYi - y21.

If L is a Lipschitz constant for b, then

lb(u(x, Yi)) - b(u(x,Y2))1 < LeAlx1 1y1 Y21

and consequently, for fixed x, b(u(x, y)) is Lipschitz-continuous and exhibitslinear growth in y. Using the inequality le l - ez21 < (ez' v ez2)lz - z21, we

may writelx1

I p(x, yi) - p(x, y2)I [P(x, Yi) v P(x,Y 2)] is Id(u(z, Y in - a' (u(z, Y2))1 dz0

'xieAlx1 f A lu(ztY 1) - u(zt Y Adz03

< Alx1e2A1'11Yi- Y21.

For fixed x, p(x, y) is thus Lipschitz-continuous and bounded in y. It followsthat the product f(x, y) A p(x, y) b(u(x, y)) satisfies Lipschitz and growthconditions of the form

(2.40) I f (x, y 1) -f (x, Y 2)1 <Lkly1 -3, 21; -k x, Yi, y2 <k

(2.41) Ifix, Al K1 + KklYI; Ix' k, y E R,

where the constants Lk and Kk depend on k.We may fix co E I/ and let Yi(co) be the solution to the ordinary differential

equationd

(2.42) -dt Y(w) = f (NO, Y(w)) with Y0(w) = ((.0)

Such a solution exists globally and is unique because of (2.40) and (2.41).The resulting process Y is adapted to {,t}, and the same is true of Xt(w) Au(W,((o), Y(o))). An application of Ito's rule shows that X satisfies (2.36)'. 0

We turn our attention to the integral equation (2.35).

2.23 Lemma. Let P-a.e. path V.(co) of the process V = {11,; 0 < t < co} becontinuous and have finite total variation V;((o) on compact intervals of the form[0, t]. If b, a: IR --, R are Lipschitz-continuous, then the equation (2.35) withV" V possesses a unique solution.

PROOF. Set )0°) = c; 0 < co and define recursively for k > 1:

(2.43) Xr1) -A + f t b(XP) ds + ft o-(X.1k))dVs; 0 < t < co.0 o

Then Mk+1).-- maxo<sst IXr"- xyol satisfies

mk-1-1) < L f t Levis dpo,0

where Lisa Lipschitz constant for b and o. Iteration of this inequality leads to

Lk(t ± Vt)k,0 ...- t < 00, k 0.mk-1-1) < m.)

, v

For 0 < t < co fixed, the right-hand side of this last inequality is summable

over k, so Inc); 0 < s 5 tIp_o is Cauchy in the supremum norm on C[0, aSince t is arbitrary, X(k) must converge to a continuous process X, and theconvergence is uniform on bounded intervals. Letting k oo in (2.43), we seethat X solves (2.35).

If Y is another solution to (2.35), then Dt A maxo <s<tlxs - Ys must satisfy

Lk(t + lt)kDt < Dt

k!, 0 < t < oo, k > 0,

which implies that D = 0.

2.24 Proposition. Suppose that o- is of class C2 (R)with bounded first and secondderivatives, and that b is Lipschitz-continuous. Let {Vt("); 0 < t < Inc°_1 be a

sequence of processes satisfying the same conditions as V in Lemma 2.23, andlet {W, ,Ft; 0 < t < I be a Brownian motion with

lim sup IV?) -W = 0, a.s.,rm O<s<t

for every 0 < t < oo. Then the sequence of (unique) solutions to the integralequation (2.35) converges almost surely, uniformly on bounded intervals, to theunique solution X of equation (2.36).

PROOF. Let u and f be as in the proof of Proposition 2.21; let y(n)(w) be thesolution to the ordinary differential equation (cf. (2.42)):

dt Yt(n)1(0) = f117i(n)1(0), Yi(n)1(0»; Ye%) = ( (o),

and define )qn)(w) A u(Vt(n)(w), y(")(w)). Ordinary calculus shows that X(") isthe (unique) solution to (2.35). It remains only to show that with Y defined by(2.42), we have for every 0 < t < oo:

(2.44) lim sup Ys(") - ysl = 0, a.s.n--.a O<s<t

Let us fix w e SZ, 0 < t < oo, and a positive integer k. With Lk as in (2.40), wemay choose c > 0 satisfying c < e Lkt. Let rk(w) = t n inf {0 < s < t; I Ys(w)1

k - 1 or 1Ws(w)1 > k - 1 }, Tin%) = t n inf {0 < s < t; I Ys(n)(0)1 > k }. For fixedw e [R, we may choose n sufficiently large, so that If (Vs' q(0), Ys(w)) -f (ws(w),Ys(w))I < E2 and I vs(n)(0))1 < k hold for every se [0, rk(w) rk")(con, and thus

1Y(n)(w) - Ys(w)) If (vsn(0), Ys' (a))) -f (vsn(0), Ys((0))1ds

+ If (Vs' q(0), Ys(w)) -f (w.s(w), Y.s(w))I

Lk 1177 qa)) Ys(w)1 E2.

An application of Gronwall's inequality (Problem 2.7) yields

Ys(n)(0) - Y.s(01 < E2eLkt < c; 0 < s < rk(w) A 1)(W).

This last inequality shows that tk(w) < tin)(w), so we may let first n -> co andthen e 10 to conclude that

lim sup 11;( ")(co) - l's(co)1 = 0.n-co 0 5s5rk(c))

For k sufficiently large we have Tk(w) = t, and (2.44) follows. El

2.25 Remark. The proof of Proposition 2.24 works only for one-dimensionalstochastic differential equations with time-independent coefficients. In higherdimensions there may not be a counterpart of the function u satisfying (2.37)(see, however, Exercise 2.28 (i)), and consequently Proposition 2.24 does nothold in the same generality as in one dimension. However, if the continuousprocesses { V(")}'_, of bounded variation are obtained by mollification orpiecewise-linear interpolation of the paths of a Brownian motion W, thenthe multidimensional version of the Fisk-Stratonovich stochastic differentialequation (2.36) is still the correct limit of the Lebesgue-Stieltjes differentialequations (2.35). The reader is referred to Ikeda & Watanabe (1981), ChapterVI, Section 7, for a full discussion.


2.26 Exercise (The Kramers-Smoluchowski Approximation; Nelson (1967)).Let b(t, x): [0, co) x R -÷ R be a continuous, bounded function which satisfiesthe Lipschitz condition (2.23), and for every finite a > 0 consider the stochasticdifferential system

d.)02) = V') dt; XV =

dY,(1) = 016(t, X``))dt - ca,(') dt + a dW; ye = n,

where , ri are a.s. finite random variables, jointly independent of the Brownianmotion W.

(i) This system admits a unique, strong solution for every value of a E (0, CO).(ii) For every fixed, finite T > 0, we have

lim sup 1,02) - XtI = 0, a.s.,ago 0.T

where X is the unique, strong solution to (2.34).

2.27 Exercise. Solve explicitly the one-dimensional equation

dX, = (,/1 + X,2 + PC,)dt + .,/1 + X, MT,.

2.28 Exercise.

(i) Suppose that there exists an Rd-valued function u(t, y) = (1410,11, ,ud(t,17))of class C "2([0, co) x Rd), such that

au; aui(t, y) = bi(t,u(t, y)), (t, y) = o- ;At, u(t, y)); 1 j d

at y,

hold on [0, co) x Rd, where each bi(t, x) is continuous and each o-,;(t, x) isof class C1'2 on [0, co) x Rd. Show then that the process

X, u(t, W); 0 < t < oo,

where W is a d-dimensional Brownian motion, solves the Fisk-Stratonovichequation

(2.36)" dX, = b(t, X,) dt + cr(t, X,) 0 dW,.

(ii) Use the above result to find the unique, strong solution of the one-dimensional Ito equation

d X -- [1 +

2t

X, a(1 + t)21dt + a(1 + t)2 dW; 0 < t < oo.

5.3. Weak Solutions

Our intent in this section is to discuss a notion of solvability for the stochasticdifferential equation (2.1) which, although weaker than the one introducedin the preceding section, is yet extremely useful and fruitful in both theoryand applications. In particular, one can prove existence of solutions underassumptions on the drift term b(t, x) much weaker than those of the previoussection, and the notion of uniqueness attached to this new mode of solvabilitywill lead naturally to the strong Markov property of the solution process(Theorem 4.20).

3.1 Definition. A weak solution of equation (2.1) is a triple (X, W), P),{A}, where

(i) (f2, P) is a probability space, and {F,} is a filtration of sub-a-fieldsof .97; satisfying the usual conditions,

(ii) X = IX A; 0 < t < col is a continuous, adapted Rd-valued process,W = { W 0 < t < oo} is an r-dimensional Brownian motion, and

(iii), (iv) of Definition 2.1 are satisfied.

The probability measure p(T) A P[X0 e 11, F e 4(1?) is called the initialdistribution of the solution.

The filtration {.57,} in Definition 3.1 is not necessarily the augmentation ofthe filtration = o-() v 0 < t < oo, generated by the "driving Brownianmotion" and by the "initial condition" = X0. Thus, the value of the solutionX,(w) at time t is not necessarily given by a measurable functional of theBrownian path { Ws(w); 0 < s < t} and the initial condition (w). On the other

5.3. Weak Solutions 301

hand, because W is a Brownian motion relative to {A}, the solution X,(w)at time t cannot anticipate the future of the Brownian motion; besides { Ws(a));0 < s < t} and (co), whatever extra information is required to compute Xt(w)must be independent of { Wo(w) - W(w); t < 0 < col.

One consequence of this arrangement is that the existence of a weaksolution (X, W), (fl, P), } does not guarantee, for a given Brownianmotion { l , A; 0 < t < cc } on a (possibly different) probability space(n,,,P), the existence of a process )7 such that the triple (51, 1,P), (n, F),{A} is again a weak solution. It is clear, however, that strong solvabilityimplies weak solvability.

A. Two Notions of Uniqueness

There are two reasonable concepts of uniqueness which can be associatedwith weak solutions. The first is a straightforward generalization of stronguniqueness as set forth in Definition 2.3; the second, uniqueness in distribution,is better suited to the concept of weak solutions.

3.2 Definition. Suppose that whenever (X, W), .F, P), {A}, and (X, W),(SI, y, P), {A}, are weak solutions to (2.1) with common Brownian motionW (relative to possibly different filtrations) on a common probability space

P) and with common initial value, i.e., P[X0 = 5C0] = 1, the two pro-cesses X and X are indistinguishable: P[X, = V0 < t < co] = 1. We saythen that pathwise uniqueness holds for equation (2.1).

3.3 Remark. All the strong uniqueness results of Section 2 are also valid forpathwise uniqueness; indeed, none of the proofs given there takes advantageof the special form of the filtration for a strong solution.

3.4 Definition. We say that uniqueness in the sense of probability law holds forequation (2.1) if, for any two weak solutions (X, W), P), {A}, and (g, 11),

c, F), with the same initial distribution, i.e.,

P[X 0 e = i3 [g0 e 1]; V I- e R(Rd),

the two processes X, g have the same law.

Existence of a weak solution does not imply that of a strong solution,and uniqueness in the sense of probability law does not imply pathwiseuniqueness. The following example illustrates these points amply. However,pathwise uniqueness does imply uniqueness in the sense of probability law(see Proposition 3.20).

35 Example. (H. Tanaka (e.g., Zvonkin (1974))). Consider the one-dimen-sional equation

(3.1)

where

X, = f sgn(Xs) d Ws; 0 < t < co,

1; x > 0,sgn(x) =

1 -1; x <0.

If (X, W), P), {A} is a weak solution, then the process X = {X .F,;0 < t < co} is a continuous, square-integrable martingale with quadraticvariation process <X>, = jo sgn2(Xs) ds = t. Therefore, X is a Brownianmotion (Theorem 3.3.16), and uniqueness in the sense of probability law holds.On the other hand, (- X, W), (SI, P), {A} is also a weak solution, so oncewe establish existence of a weak solution, we shall also have shown thatpathwise uniqueness cannot hold for equation (3.1).

Now start with a probability space (0, . P) and a one-dimensionalBrownian motion X = {X .519;ix; 0 < t < co} on it; we assume P[X0 = 0] = 1and denote by IR the augmentation of the filtration } under P. Thesame argument as before shows that

sgn(Xs)dXs; 0 < t < co0

is a Brownian motion adapted to {.3,' }. Corollary 3.2.20 shows that (X, W),(1, P), { x} is a weak solution to (3.1). With {#,w} denoting the augmen-tation of {Fr}, this construction gives Ax, which is the oppositeinclusion from that required for a strong solution!

Let us now show that equation (3.1) does not admit a strong solution. Assumethe contrary; i.e., let X satisfy (3.1) on a given (0, P) with respect to a givenBrownian motion W, and assume ..9%x Ary for every t > 0. Then X isnecessarily a Brownian motion, and from Tanaka's formula (3.6.13) witha = 0, we have

W, = sgn(Xs)dXs = I X ,I - 2Lx, (0)0

= - lim -1 meas{0 s t; I Xsj e}, 0 < t < co , a.s. P,40 2e

where a(0) is the local time at the origin for X. Consequently, Aw cand thus also Ax Axl holds for every t > 0. But this last inclusion isabsurd.

B. Weak Solutions by Means of the Girsanov Theorem

The principal method for creating weak solutions to stochastic differentialequations is transformation of drift via the Girsanov theorem. The proof ofthe next proposition illustrates this approach.

3.6 Proposition. Consider the stochastic differential equation

(3.2) d X, = b(t, X,) dt + dW,; 0 < t < T,

where T is a fixed positive number, W is a d-dimensional Brownian motion, andb(t, x) is a Borel-measurable, Rd-valued function on [0, T] x O which satisfies

(3.3) ilb(t,x)11 K(1 + DM; 0 .t,T,xe9;Rd

for some positive constant K. For any probability measure u on (Rd,.4(Rd)),equation (3.2) has a weak solution with initial distribution II.

PROOF. We begin with a d-dimensional Brownian family X = {X <F,; 0 <t < T}, (11, {P}xE Rel. According to Corollary 3.5.16,

Et I ft4 A- exp E f ys, xs)dxv) -

2-II b(s, X0112 ds}

J=1 o o

is a martingale under each measure 13', so the Girsanov Theorem 3.5.1 impliesthat, under Qx given by (dQxldr) = ZT, the process

(3.4) W, A X, - X0 - b(s, X ,)ds; 0 < t T

is a Brownian with Qx[W0 = 0] = 1, Vx e Rd. Simply rewriting (3.4) as

X, = Xo + i't b(s, Xs)ds + Wt; 0 t < 'T,

we see that, with Q"(A) fad Qx(A)ti(dx), the triple (X ,W), Q"), {Ft}is a weak solution to (3.2).

3.7 Remark. If we seek a solution to (3.2) defined for all 0 < t < co, we canrepeat the preceding argument using the filtration {,Fix } instead of {,57;,} andciting Corollary 3.5.2 rather than Theorem 3.5.1. Whereas {,57;,} in Proposition3.6 can be chosen to satisfy the usual conditions, {....F,x} does not have thisproperty. Thus, as a last step in this construction, we take to be thecollection of null sets of (0, ,F,, QP), set A = o(.Ftx u ./11) and W, = A+. Thefiltration {/,} satisfies the usual conditions.

3.8 Remark. It is apparent from Corollary 3.5.16 that Proposition 3.6 canbe extended to include the case

(3.5) X, = X0 + ft b(s, X)ds + Wt; 0 < t < T,

where b(t, x) is a vector of progressively measurable functionals on C[0, co)d;see Definition 3.14.

3.9 Remark. Even when the initial distribution /..t degenerates to unit pointmass at some x e Rd, the filtration {,Fr} of the driving Brownian motion in

(3.5) may be strictly smaller than the filtration {,F Z` } of the solution process(see the discussion following Definition 3.1). This is shown by the celebratedexample of Cirel'son (1975). Nice accounts of Cirel'son's result appear inLiptser & Shiryaev (1977), pp. 150-151, and Kallianpur (1980), pp. 189-191.

The Girsanov theorem is also helpful in the study of uniqueness in law ofweak solutions. We use it to establish a companion to Proposition 3.6.

3.10 Proposition. Assume that (X(1), W(l)), (Sr), .5"), P(i)), {.F(i)}; i = 1, 2, areweak solutions to (3.2) with the same initial distribution. If

(3.6) P") [1 11 b(t, X:1))112 dt < oo] = 1; i = 1, 2,

then (X"), Wu)) and (X(2), W2)) have the same law under their respectiveprobability measures.

PROOF. For each k > 1, let

(3.7) 41) T inf {0 < t < T; ft Ms, XV)112 ds = k} .

03

According to Novikov's condition (Corollary 3.5.13),tA tip 1

(3.8) 1`)(X(I)) -A exp {- (b(s, r), d1V)) -0

116(s, )0112 ds}io

is a martingale, so we may define probability measures Pr on ,F19, i = 1, 2,according to the prescription (dPe/dP(i)) = VIc)(X")). The Girsanov Theorem3.5.1 states that, under Pe), the process

J(3.9) Xft tip = Xg) + b(s, XV) ds + W,(i,,) tip; 0 < t < To

is a d-dimensional Brownian motion with initial distribution it, stopped attime TV). But TV), {IV); 0 < t < 4)}, and Vc)(X(`)) can all be defined in termsof the process in (3.9) (see Problem 3.5.6). Therefore, for 0 = to < t1 < <t = T and F e,l(R2d("+1)) we have

(3.10) pol[poi), W(0l ) XLI. )5 Wt(1 )) tv.)

Si(x(i))1{0410),),..., xv.), wocv4)= di3,(c1)ry

L2),,)(x(2)) ,,(x,20),w,02),...,x,,2))c,;,,)=T}d,2)

p(2)[(x102), W(02), X: 2), W(2)) r; 42) T].

By assumption (3.6), limk 13(l)[4,4 = T] = 1; i = 1, 2, so passage to the limitas k co in (3.10) gives the desired conclusion.

3.11 Corollary. If the drift term b(t, x) in (3.2) is uniformly bounded, thenuniqueness in the sense of probability law holds for equation (3.2). Furthermore,with 0 5 t1 < t, < < t < T and with the notation developed in the proof ofProposition 3.6, we have then

(3.11) X,n)er] = J Ex[1{(x,I

.x, )E ,--}ZT]p(dx); F e R(Rd").Qgd

3.12 Exercise. According to Proposition 3.6 and Corollary 3.11, the one-dimensional stochastic differential equation

(3.12) dX, = -sgn(X,)dt + dW X0 = 0

possesses a weak solution which is unique in the sense of probability law.Show that if (X, W), P), {",} is such a solution and L,(0) is the localtime at the origin for the Brownian family 1X,, 1, (n,"), {px}ER, then

(3.13) Q[X,ef] = e-1/2 E°[1{,G exP( - I Xt1 2L,(0))]; e *IR).

(An explicit formula for the right-hand side of (3.13) can be derived fromTheorem 3.6.17 and relation (2.8.2). See Problem 6.3.4 and Exercise 6.3.5.)

3.13 Problem. Consider the stochastic differential equation (2.1) with a(t, x)a (d x d) nonsingular matrix for every t z 0 and x e Rd. Assume that b(t, x) isuniformly bounded, the smallest eigenvalue of a(t, x) is uniformly boundedaway from zero, and the equation

(3.14) dX, = o-(t, X,)dW,; 0 < t < T

has a weak solution with initial distribution y. Show that (2.1) also has a weaksolution for 0 < t < T with initial distribution p. (We shall have more to sayabout existence and uniqueness of solutions to (3.14) in Sections 4 and 5.)

3.14 Definition. Let b,(t, y) and o-ti(t, y); 1 < i < d, 1 < j < r, be progressivelymeasurable functionals from [0, co) x C[0, co )d into ER (Definition 3.5.15). Aweak solution to the functional stochastic differential equation

(3.15) dX, = b(t, X) dt + a(t, X) dW,; 0 < t < co,

is a triple (X, W), (12, .9,, P), {.9,} satisfying (i), (ii) of Definition 3.1, as well as

(iii)Jr

X)I + cr,21(s, X)} ds < co;0

1 i d, 1 5 j r, t > 0,

(iv) X, = X0 + b(s, X)ds + a(s, X)dWs; 0 t < co,

almost surely.

3.15 Problem. Suppose bi(t, y) and o-(t, y); 1 < i < d, 1 5 j r, are progres-sively measurable functionals from [0, co) x C[0, oo)d into ER satisfying

(3.16) II b(t,Y)II2 + 110(t,Y)112 K (1 + max II Yls/112);o<stV 0 5 t< co, y e C[0, °o r,

where K is a positive constant. If (X, W), (Si, F, P), {A} is a weak solutionto (3.15) with E II X oil' < co for some m > 1, then for any finite T > 0, we have

(3.17) E ( max 11Xs1I2m) < co + EllX0112m)ect; 0 < t < T,0<s<t

(3.18) E 11X, - Xs1121" C(1 + E 11X01121")(t - s)m; 0 _s<tT,where C is a positive constant depending only on m, T, K, and d.

C. A Digression on Regular Conditional Probabilities

We know that indistinguishable processes have the same finite-dimensionaldistributions, and this causes us to suspect that pathwise uniqueness impliesuniqueness in the sense of probability law. The remainder of this section isdevoted to the confirmation of this conjecture. As preparation, we need tostate certain results about regular conditional probabilities, in the spirit ofDefinition 2.6.12, but in a form better suited to our present needs. We referthe reader to Parthasarathy (1967), pp. 131-150, and Ikeda & Watanabe(1981), pp. 12-16, for proofs and further information.

3.16 Definition. Let (S1, F, P) be a probability space and a sub-a-field of F.A function Q(co; A): 0 x F -> [0,1] is called a regular conditional probabilityfor F given 4 if

(i) for each w en, Q(w; -) is a probability measure on (0.,,F),(ii) for each A e , F , the mapping wi- Q(w; A) is 4-measurable, and(iii) for each A e 9; , Q(w; A) = P[A14](co); P-a.e. w e e.

Suppose that, whenever Q' (co; A) is another function with these properties,there exists a null set N e ,y such that Q(w; A) = Q' (co; A) for all A e .97 andw e n\N. We then say that the regular conditional probability for F given 4is unique.

Note that if X in Definition 2.6.12 is the identity mapping, then conditions(i)-(iii) of that definition coincide with those of Definition 3.16.

3.17 Definition. Let (Si, F) be a measurable space. We say that F is countablydetermined if there exists a countable collection of sets di F such that,whenever two probability measures agree on dt, they also agree on F. We

say that .9" is countably generated if there exists a countable collection of sets(6' g such that 6((6).

In the space C[0, oor, for an arbitrary integer m > 1, let us introduce theo--fields

(3.19) ajcp, con A o(z(s); 0 < s < t) = (a(cp, co)'"))

for 0 < t < cc, where tpt: (x))'" C[0, oor is the truncation mapping(ptz)(s) A z(t A s); z e C[0, oor, 0 5 s < co. As in Problem 2.4.2, it is shownthat A(C[0, corn) = oM), where (6, is the countable collection of all finite-dimensional cylinder sets of the form

C = {ze C[0, co)'"; (z(ti),..., z(t,,))e AI

with n > 1, tie [0, t] n Q, for 1 5 i < n, and A e AR") equal to the productof intervals with rational endpoints. The continuity of ze C[0, corn allows usto conclude that a set of the form C is in a(%), even if the points t, are notnecessarily rational.

It follows that a t(c [0, cor) is countably generated. On the other hand, thegenerating class (6; is closed under finite intersections, so any two probabilitymeasures on A(C[0, con which agree on (6, must also agree on ai(c 0, con,by the Dynkin System Theorem 2.1.3. It follows that A(c[o, con is alsocountably determined.

More generally, Theorem 2.1.3 shows that if a 6 -field .97 is generated bya countable collection of sets (6' which happens to be closed under pairwiseintersection, then ,9-", is also countably determined. In a topological space witha countable base (e.g., a separable metric space), we may take this to be thecollection of all finite intersections of complements of these basic open sets.

3.18 Theorem. Suppose that SI is a complete, separable metric space, and denotethe Borel 6 field y = WI). Let P be a probability measure on (0,,), and let

be a sub-o--field of F. Then a regular conditional probability Q for F givenexists and is unique. Furthermore, if dr is a countably determined sub-o--field

of then there exists a null set N e such that

(iv) Q(o); A) = Wu)); A e , w e SI\N.

In particular, if X is a -measurable random variable taking values in anothercomplete, separable metric space, then with ° denoting the a-field generatedby X, (iv) implies

(iv)' Q(w; {w' e X(&) = X(01) = 1; P-a.e.

When the a-field is generated by a random variable, we may recast theassertions of Theorem 3.18 as follows.

3.19 Theorem. Let (SI, P) be as in Theorem 3.18, and let X be a measurablemapping from this space into a measurable space (S, 91, on which it induces the

distribution P X-1 (B) -A- P [w e O.; X (co) e B], B e 9'. There exists then a functionQ(x; A): S x 07", [0, 1], called a regular conditional probability for y givenX, such that

(i) for each xeS, Q(x; ) is a probability measure on (52, g"),(ii) for each A e ", the mapping xi- Q(x; A) is Y-measurable, and

(iii) f o r each A e 9-;, Q(x; A) = P[Al X = x], P X -1-a.e. xeS.

If Q'(x; A) is another function with these properties, then there exists aset Neff with PX-1(N) = 0 such that Q(x; A) = Q'(x; A) for all A e .f; andxe S\N. Furthermore, if S is also a complete, separable metric space and

= M(S), then N can be chosen so that we have the additional property:

(iv) Q(x; Ico e O.; X (co)e BI) = 1,(x); B e ff, x e S\N.

In particular,

(iv)' Q(x; 1w e O.; X(w) = x }) = 1; PX-1-a.e. xeS.

D. Results of Yamada and Watanabe onWeak and Strong Solutions

Returning to our initial question about the relation between pathwise unique-ness and uniqueness in the sense of probability law, let us consider two weaksolutions (X-', W-'), (51i, vi), }; j = 1, 2, of equation (2.1) with

(3.20) y(B) A vi[n) e 13] = v2[X,1321e 13]; B e M(f?).

We set Ic(f) = - X,(3j); 0 < t < co, and we regard the j-th solution asconsisting of three parts: Xe, W(i), and 17"). This triple induces a measure Pi on

(0, .4(0)) A Old x C[0, cc)' x C[0, co)d,

M(Rd) © .4(C[0, cor) 0 .4(C [0, co)d))

according to the prescription

(3.21) pi(A) A vi[(XW), W"), Yu) e A]; A e .4(13), j = 1, 2.

We denote by B = (x, w, y) the generic element of O. The marginal of each Pion the x-coordinate of B is the marginal on the w-coordinate is Wienermeasure P,, and the distribution of the (x, w) pair is the product measure

x P1 because X,(3-(*) is .F1)-measurable and W") is independent of .9-1)(Problem 2.5.5). Furthermore, under Pi, the initial value of the y-coordinateis zero, almost surely.

The two weak solutions (X"), W")) and (X(2), W(2)) are defined on (possibly)different sample spaces. Our first task is to bring them together on the same,canonical space, while preserving their joint distributions. Toward this end,we note that on (0, .4(0), Pi) there exists a regular conditional probability for.4(0) given (x, w). We shall be interested only in conditional probabilities of

sets in MO) of the form Rd x C[0, coy x F, where F e 4(C[0, °or). Thus, witha slight abuse of terminology, we speak of

Qi(x, w; F): x C[0, coy x .4(C[0, cor) [0,1]

as the regular conditional probability for M(C[0, oor) given (x, w). Accordingto Theorem 3.19, this regular conditional probability enjoys the followingproperties:

(3.22) (i) for each x e Rd, w e Cr[0, co), Q3(x, w; ) is a probability measure on(C[0, cc)", MC [0, cor)),

(3.22) (ii) for each F e [0, cor), the mapping (x,w)i--. Q;(x,w; F) is MO)AC [0, con-measurable, and

(3.22) (iii) PJ(G x F) = $G Qi(X w; F)p.(dx)P,,,(dw); F e M(C [0, cor),

G e .4(01d) ®M(C[0, coy).

Finally, we consider the measurable space (0,F), where SI = O x C[0, co)dand F is the completion of the o--field Af(0) 0 AC[0, cor) by the collection..)1( of null sets under the probability measure

(3.23) P(dco) A Qi(x, w; dyi )Q2(x, w;dy2)p(dx)P,(dw).

We have denoted by w = (x, w4142) a generic element of a In order toendow (SI, P) with a filtration that satisfies the usual conditions, we take

A -4 Q { (x, w(s), y (s), y2(s)); 0 < s < t} , u

,A ,+; 0 < t < co.

It is evident from (3.21), (3.22) (iii), and (3.23) that

(3.21)' P [co e SI; (x, w, yj) e = vi[(4-1), Wu), ri))e A]; A G.4(:3), j = 1, 2,

and so the distribution of (x w) under P is the same as the distribution of(Xu), WLD) under vi. In particular, the w-coordinate process {w(t), A; 0 < t < co}is an r-dimensional Brownian motion on (SI, P), and it is then not difficultto see that the same is true for {w(t), Ft; 0 < t < co}.

3.20 Proposition (Yamada & Watanabe (1971)). Pathwise uniqueness impliesuniqueness in the sense of probability law.

PROOF. We started with two weak solutions (Xth, Wu)), (C/J, {..FP)};

j = 1, 2, of equation (2.1), with (3.20) satisfied. We have created two weaksolutions (x + yj, w), j = 1, 2, on a single probability space (II, F, P), {",},such that (X), WU)) under v.; has the same law as (x + yj, w) under P. Pathwiseuniqueness implies P[x + y1 (t) = x + y2(t), V 0 < t < co] = 1, or equivalently,

(3.24) no) = (x5w41,Y2)E CI; yi = y2] = 1.

It develops from (3.21)', (3.24) that

viun),14/0), Y(1))e A] = P[co ei); (x, w, yi )e A]

= P[to e 0; (x, w, y2) e in

= v2[(X,(32), W(2), Y(2)) e A]; A e R(D),

and this is uniqueness in the sense of probability law. CI

Proposition 3.20 has the remarkable corollary that weak existence andpathwise uniqueness imply strong existence. We develop this result.

3.21 Problem. For every fixed t > 0 and F e .4,(C[0, co)d), the mapping(x, w)i- Qi(x, w; F) is 4,-measurable, where IA 1 is the augmentation of thefiltration 1.4(Rd) ® at(c [0, con} by the null sets of p(dx)P,(dw).

(Hint: Consider the regular conditional probabilities Z(x, w; F): Rd xC[0, co)' x .4,(C[0, co)d) -* [0, 1] for RAC [0, co)d), given (x, cp,w). Theseenjoy properties analogous to those of Qi(x,w; F); in particular, for everyF e .4,(C[0, co)d), the mapping (x, w) r- Ql(x, w; F) is .4(Rd) 0 .4,(C [0, con-measurable, and

(3.25) Pj(G x F) = 1 Vi(x,w; F) p(dx)P,(dw)G

for every G e .4(Rd) 0 .4,(C[0, con. If (3.25) is shown to be valid for everyG e .4(Rd) 0 .4(C[0, con, then comparison of (3.25) with (3.22) (iii) showsthat Qi(x, w; F) = Q;(x,w; F) for it x P,-a.e. (x, w), and the conclusion follows.Establish (3.25), first for sets of the form

(3.26) G = G1 x (pt.' G2 n o-,-1 G3); G1 e .4(Rd), G2, G, e .4(C [0, co)'),

where (o-,w)(s) A w(t + s) - w(t); s > 0, and then in the generality required.)

3.22 Problem. In the context of Proposition 3.20, there exists a functionk: Rd x C[0, co)' C[0, co)d such that, for it x P,-a.e. (x, w) e Rd x C[0, co)",we have

(3.27) Q 1 (x, w; {k(x, w)} ) = Q2(x, w; {k(x, w)} ) = 1.

This function k is .4(Rd) 0 .4(C [0, con/ .4(C [0, co)d)-measurable and, for each0 < t < co, it is also A/.4,(C[0, co)d)-measurable (see Problem 3.21 for thedefinition of A). We have, in addition,

(3.28) P[o) = (x, w, yi, Y2)6 n; Y1 = Y2 = k(X, W)] = 1.

3.23 Corollary. Suppose that the stochastic differential equation (2.1) has a weaksolution (X, W), (0, .F, P), 1.0 with initial distribution 12, and suppose thatpathwise uniqueness holds for (2.1). Then there exists a .4(Rd) 0 .4(C [0, con/.4(C[0, co)d)-measurable function h: Rd x C[0, co)' C[0, co)d, which is alsoA/at(c [o, oo)d)-measurable for every fixed 0 < t < co, such that

(3.29) X. = h(X0, W), a.s. P.

5.4. The Martingale Problem of Stroock and Varadhan 311

Moreover, given any probability space P) rich enough to support anRd-valued random variable with distribution p and an independent Brownianmotion {ll, ; 0 < t < ool, the process

(3.30) X. h(, W.)

is a strong solution of equation (2.1) with initial condition

PROOF. Let h(x, w) = x + k(x, w), where k is as in Problem 3.22. From (3.28)and (3.21)' we see that (3.29) holds. For c and 14, as described, both (X0, W)and ITO induce the same measure p x P,, on Rd x [0, con, and since(X. = h(X 0, W) satisfies (2.1), so does (X. = Pr.). The process )7 isadapted to {A} given by (2.3), because h is / MAC [0, co)d)- measurable.

The functional relations (3.29), (3.30) provide a very satisfactory formulationof the principle of causality articulated in Remark 2.2.

5.4. The Martingale Problem of Stroockand Varadhan

We have seen that when the drift and dispersion coefficients of a stochasticdifferential equation satisfy the Lipschitz and linear growth conditions ofTheorem 2.9, then the equation possesses a unique strong solution. For moregeneral coefficients, though, a strong solution to the stochastic differentialequation might not exist (Example 3.5); then the questions of existence anduniqueness, as well as the properties of a solution, have to be discussed ina different setting. One possibility is indicated by Definitions 3.1 and 3.4:one attempts to solve the stochastic differential equation in the "weak" senseof finding a process with the right law (finite-dimensional distributions), andto do so uniquely. A variation on this approach, developed by Stroock &Varadhan (1969), formulates the search for the law of a diffusion process withgiven drift and dispersion coefficients in terms of a martingale problem. Thelatter is equivalent to solving the related stochastic differential equation in theweak sense, but does not involve the equation explicitly. This formulation hasthe advantage of being particularly well suited for the continuity and weakconvergence arguments which yield existence results (Theorem 4.22) and"invariance principles", i.e., the convergence of Markov chains to diffusionprocesses (Stroock & Varadhan (1969), Section 10). Furthermore, it casts thequestion of uniqueness in terms of the solvability of a certain parabolicequation (Theorem 4.28), for which sufficient conditions are well known.

This section is organized as follows. First, the martingale problem is for-mulated and its equivalence with the problem of finding a weak solution tothe corresponding stochastic differential equation is established. Using thismartingale formulation and the optional sampling theorem, we next establishthe strong Markov property for these solution processes. Finally, conditions

for existence and uniqueness of solutions to the martingale problem areprovided. These conditions are different from, and not comparable to, thosegiven in the previous section for existence and uniqueness of weak solutionsto stochastic differential equations.

4.1 Remark on Notation. We shall follow the accepted practice of denotingby Ck(E) the collection of all continuous functions f: E gl which havecontinuous derivatives of every order up to k; here, E is an open subset of someEuclidean space Rd. If f(t, x): [0, T) x E gl is a continuous function, wewrite f e C([0, T) x E), and if the partial derivatives (afia(),(afiaxi),(a2fiaxiaxi);1 < i,j < d, exist and are continuous on (0, T) x E, we write f e C"2((0, T) xE). The notation f e C" 2 ( [0, T) x E) means that f e C" 2 ((0, T) x E) and theindicated partial derivatives have continuous extensions to [0, T) x E. Weshall denote by Ct(E), Ct(E), the subsets of Ck(E) of bounded functions andfunctions having compact support, respectively. In particular, a function inCt(E) has bounded partial derivatives up to order k; this might not be true fora function in ct(E).

A. Some Fundamental Martingales

In order to provide motivation for the martingale problem, let us supposethat (X, W), P), {.??7,} is a weak solution to the stochastic differentialequation (2.1). For every t > 0, we introduce the second-order differentialoperator

(4.1) (.91, f )(x)d d a2foo

+ b.(t, x)af(x)

; f e C2 (Rd),A1

- atk(t, x)2 i=i k=1 NOXi 10Xk 1=1 a Xi

where aik(t, x) are the components of the diffusion matrix (2.2). If, as in thenext proposition, f is a function of t e [0, oo) and x e gld, then (.91,f)(t, x), isobtained by applying sr', to f(t, -).

4.2 Proposition. For every continuous function f(t, x): [0, oo) x gld -, gl whichbelongs to C"2((0, oo) x Rd), the process Mf = {Mir , .F;; 0 < t < op} given by

(4.2) Ng A f(t, Xt) - f(0, X0) -Jo

t (a-fas

+ sisf)(s, Xs)dsis a continuous, local martingale; i.e., Mf je, loc. If g is another member ofC([0, co) x Rd) C. , 2 ( (05 oo) x glg), then Mg e .41'''" and

d d 't a a(4.3) <Mf, Mg>, = 1 1 afk(s, Xs) f(s, Xs) g(s, XS) ds.

i=1 k=1 0 ax; ax,,

Furthermore, if f e C0([0, oo) x Rd) and the coefficients r; 1 5 i s d, 1 s j 5 r,are bounded on the support of f, then Mf e JP',

PROOF. The Ito rule expresses M1 as a sum of stochastic integrals:

d r(4.4) Mir = E E gi,J), with

i=1 j=1MI" n r aii(s, Xs)f(s, X s)dWo.

ax,

Introducing the stopping times

S inf > 0; X,11 > n or a5(s, Xs) n for some (i,j)}

and recalling that a weak solution must satisfy condition (iii) of Definition 2.1,we see that limn, S = co a.s. The processes

d r t A S 0Al(4.5) i f(n) AlfAs = E E J

i=1 j=1a ii(S, X s)- f(s XS) d Wsti); n > 1,

Oxi

are continuous martingales, and so M1 e dr'. The cross-variation in (4.3)follows readily from (4.5). If f has compact support on which each au isbounded, then the integrand in the expression for M" in (4.4) is bounded, soMf

With the exception of the last assertion, a completely analogous result isvalid for functional stochastic differential equations (Definition 3.14). Weelaborate in the following problem.

4.3 Problem. Let bi(t, y) and c(t, y); 1 < i < d, 1 < j < r, be progressivelymeasurable functionals from [0, co) x C[0, oo)d into R. By analogy with (2.2),we define the diffusion matrix a(t, y) with components

(4.6) ak(t, y) A E aiy,y)o-,;(t,y); 0 < t < co, y e C[0, co )d..J=1

Suppose that (X, W), {;}, is a weak solution to the functionalstochastic differential equation (3.15), and set

(s t'u)(Y) =1

E E aik1t,02u(y(t))

+ bi(t, y)Ou(y(t))d d

if2 i=i k=1 Oxiax, i.1 oxi

0 < t < oo, u e C2(Rd), y e C[0, oo)d.

Then, for any functions f, g e C([°, co) x Rd) cr. ,2(0, co. x Rd), the process

(4.2)' mif 4 f(t, Xt) - f(0, X0) - fto os

+ .4.11(s, X) ds, Ft; 0 <

is in .11`.1", and

t < 00

d d t a(4.3)' <M1, M9 >, = E E aik(s, X) f(s, X,)-g(s, XS) ds.

i=1 k=1 o oXi Oyck

Furthermore, if f e Co([0, co) x Rd) and for each 0 < T < co we have

(4.7) Haft, KT, 0 t T, }le CLO, 00)d,

where K T is a constant depending on T, then f e ./11`2.

The simplest case in Proposition 4.2 is that of a d-dimensional Brownianmotion, which corresponds to k(t, x) 0 and o-u(t, x) Su; 1 j < d. Thenthe operator in (4.1) becomes

1 1 d 82fsif =2-Af =

2 i=1E

ax?;fe C2(g1°).

4.4 Problem. A continuous, adapted process W = {W, ,Ft; 0 < t < co} is ad-dimensional Brownian motion if and only if

f(w) - fiwo) - -20

Ai(Ws) ds, <Ft;

is in .1r1" for every f e C2(01").

0 < t < co,

B. Weak Solutions and Martingale Problems

Problem 4.4 provides a novel martingale characterization of Brownian motion.The basic idea in the theory of Stroock & Varadhan is to employ AV of (4.2)in a similar fashion to characterize diffusions with general drift and dispersioncoefficients.

To explain how this characterization works, we shall find it convenientto deal temporarily with progressively measurable functionals bi(t, y),o-ii(t, y): [0, co) x C[0, oo)d R, 1 < i < d, 1 < j < r. We recall the family ofoperators I.go71 of (4.1)'.

4.5 Definition. A probability measure P on (C[0, co)°, .R(C [0, cor)), underwhich

(4.8) MI = f (y(t)) -f (y(0)) - J t (s4f)(y)ds, ..Ft; 0 < t < co,

is a continuous, local martingale for every f e C2(l ."), is called a solution tothe local martingale problem associated with {,207}. Here <F, = A±, and {A}is the augmentation under P of the canonical filtration ', .4,(C[0, cor) asin (3.19).

According to Problem 4.3, a weak solution to the functional stochasticdifferential equation (3.15) induces on (C[0, oo)°,.4(C[0, cor)) a probabilitymeasure P which solves the local martingale problem associated with {Al.The converse of this assertion is also true, as we now show.

4.6 Proposition. Let P be a probability measure on (C[0, oor,.4(C[0, oor))under which the process M1 of (4.8) is a continuous, local martingale for thechoices f(x) = x, and f(x) = xixk; 1 < i, k < d. Then there is an r-dimensionalBrownian motion W = {W, .; 0 < t < oo} defined on an extension (n, P)of (C[0, oo)d, -.4(C[0, oo)d), P), such that (X, A y(t), W,), P), {A}, is a weaksolution to equation (3.15).

PROOF. By assumption,

MP) =° XP) - n) _ f ' b,(s, X) ds, ..Ft; 0 t < ooo

is a continuous, local martingale under P. In particular,t

(4.9) P[f lbas, X)I ds < cx); 0 ._ t < op] = 1; 1 < i < d.o

With f(x) = xixk, we see that

MP.° '=1- XP)r) - Xg f)n) - [XP bk(s, X) + XY`)bi(s, X) + aik(s, X)] dso

is also a continuous, local martingale. But one can express

(4.10) MP)/e) -J

a,k(s, X) ds

as the sum of the continuous local martingale mi,k) - xg)A4:0 - nogi) andthe process

(4.11) (X() X(`))bk

(s'X)ds + ()CY') - XP'))bi(s, X) ds

fo s

t

+ f t k(s, X)ds f bk(s, X)dso o

=ft (Mr - MP)bk(s, X) ds + ft (MY') - M1k))bi(s, X) dso o

= -ft Lis bk(u, X) du &lir - ot os Mu, X) du d le).o o

The last identity may be verified by applying ItO's rule to both processesclaimed to be equal. We see then that the process of (4.11) is a continuous,local martingale of bounded variation; it follows from Exercise 1.5.21 that itis identically equal to zero. Therefore, the process of (4.10) is in Jr)", and

(4.12) M(k)>, = J aik(s, X)ds; 0 < t < co, a.s.

We may now invoke Theorem 3.4.2 to conclude the existence of a d-dimensionalBrownian motion {1,V A; 0 < t < co} on an extension (n, g, P) of (CEO, oo)d,.-1(C[0, ao)d), P) endowed with a filtration which satisfies the usual condi-tions, as well as the existence of a matrix p = pp), A; 0 5 t < <1.i<dof measurable, adapted processes with

(4.13) P[ ft pl(s)ds < ad= 1; 1 s j d, 0 t < oo

such that

d

(4.14) = E pu(s)daisti); 1 < i < d, 0 < t < ooj =1 0

holds a.s. P. This last equation can be rewritten as

(4.15) X, = X0 + b(s, X) ds + J p(s)dais; 0 5_ t < oo.

In order to complete the proof, we need to establish the existence of anr-dimensional Brownian motion W = {W A; 0 < t < co} on (n, P), suchthat

(4.16) p(s) dais = ft a(s, X) d4; 0 < t < cofo

holds P-almost surely. From (4.12), (4.14) and with the notation (4.6), it willthen be clear that

Pd

E pii(t)pki(t) = aik(t, X), for a.e. t > 0 = 1; 1 < i, k < di=

and (4.13) will imply

(4.17) P[i. ots, X) ds < 1; 1 < i < d, 1 < j < r, OSt<oo.o

The relations (4.9), (4.15)-(4.17) will then yield (X, W), {A} as aweak solution to (3.15).

It suffices to construct W satisfying (4.16) under the assumption r = d.Indeed, if r > d, we may augment X, b, and a by setting ,qi) = 61(t5 y) =o-ii(t, y) = 0; d + 1 < i < r, 1 < j < r. This r-dimensional process X satisfiesan appropriately modified version of (4.8), and we may proceed as beforeexcept now we shall obtain a matrix p which, like a, will be of dimension(r x r). On the other hand, if r < d, we need only augment a by setting

y) = 0; 1 < i < d, r + 1 < j < d, and nothing else is affected. Both p anda are then (d x d) matrices.

According to Problem 4.7 following this proof, there exists a Borel-measurable, (d x d)-matrix-valued function R( p, a) defined on the set

(4.18) D A {(p,o-); p and a are (d x d) matrices with pT aT

such that a = pR(p, a) and R(p, a) RT (p, a) = I, the (d x d) identity matrix.We set

W A f a(s, X))dit 0 < t <

Then W") e dr)"; 1 , = Mu; 1 < i,j < d, 0 < t < co.

It follows from Levy's Theorem 3.3.16 that {W,, .P; 0 < t < co} is ad-dimensional Brownian motion. Relation (4.16) is apparent.

4.7 Problem. Show that there exists a Borel-measurable, (d x d)-matrix-valuedfunction R(p, a) defined on the set D of (4.18) and such that

a = pR(p, o-), R(p, a)RT (p, a) = I; (p, a) e D.

(Hint: Diagonalize ppT = ao-T and study the effect of the diagonalizationtransformation on p and a.)

4.8 Corollary. The existence of a solution P to the local martingale problemassociated with {sit'} is equivalent to the existence of a weak solution (X, W),(5,., P), {A} to the equation (3.15). The two solutions are related by P =Px-i; i.e., X induces the measure P on (C[0, cor,,It(C[0, oo)°)).

4.9 Corollary. The uniqueness of the solution P to the local martingale problemwith fixed but arbitrary initial distribution

P[y e C[0, °or; y(0) ell = p(T); F e M(Rd)

is equivalent to uniqueness in the sense of probability law for the equation (3.15).

Because of the difficulty in computing expectations for local martingales, itis helpful to introduce the following modification of Definition 4.5.

4.10 Definition (Martingale Problem of Stroock & Varadhan (1969)). Aprobability measure P on (C[0, oo)°,4(C[0, °o)°)) under which M1 in (4.8)is a continuous martingale for every f e CROV) is called a solution to themartingale problem associated with {34}.

Given progressively measurable functionals bi(t,y), o-u(t, y): [0, oo) xC[0, co)" R; 1 < i < d, 1 < j < r, the associated family of operators {s47},and a probability measure p on .4(Rd), we can consider the following threeconditions:

(A) There exists a weak solution to the functional stochastic differentialequation (3.15) with initial distribution p.

(B) There exists a solution P to the local martingale problem associated withwith P[y(0) e n = p(r); r e M(01°).

(C) There exists a solution P to the martingale problem associated withwith P [y(0) e = it(F); F e .4(Fe).

4.11 Proposition. Conditions (A) and (B) are equivalent and are implied by (C).Furthermore, (A) implies (C) under either of the additional assumptions:

(A.1) For each 0 < T < co, condition (4.7) holds.(A.2) Each o-i(t, y) is of the form o-ii(t, y) = y(t)), where the Borel-

measurable functions d: [0, co) x are bounded on compact sets.

PROOF. We have already established the equivalence of (A) and (B.) If P is asolution to the martingale problem and f e C2(Rd) does not necessarily havecompact support, we can define for every integer k > 1 the stopping time

(4.19) Sk A inf.{ t 0: Y(011 > k }.

Let gke CAW) agree with f on Ix e Rd; 11x11 < k }. Under P, each Mrk is amartingale which agrees with Mir for t < Sk. It follows that Mf e dff'"; thus(C) (B). Under (A.1), Problem 4.3 shows (A) (C); under (A.2), the argumentfor this implication is given in Proposition 4.2.

4.12 Remark. It is not always necessary to verify the martingale property ofM-1. under P for every f e CARd), in order to conclude that P solves themartingale problem. To wit, consider fi(x) A xi and fii(x) A xixi for 1j < d, and choose sequences I 10,5)11°_, of functions in Co(Rd) suchthat g1 (x) = fi(x), g1 (x) = fy(x) for 114 k. If Me) and /1/19`,k; are mar-tingales for 1 < i, j < d and k > 1, then M-1; and M-fq are local martingales.According to Proposition 4.6, there is a weak solution to (3.15), and Corollary4.8 now implies that P solves the local martingale problem. Under either ofthe assumptions (A.1) or (A.2), P must also solve the martingale problem.

It is also instructive to note that in the Definitions 3.1 and 3.14 of weaksolution to a stochastic differential equation, the Brownian motion W appearsonly as a "nuisance parameter." The Stroock & Varadhan formulation elimi-nates this "parametric" process completely. Indeed, the essence of the implica-tion (B) (A) proved in Proposition 4.6 is the construction of this process.

In keeping with our practice of working with filtrations which satisfy theusual conditions, we have constructed from ,4,(C[0, oo )d) such a filtra-tion {A} for the Definitions 4.5 and 4.10. Later in this section, we shall insteadwant to deal with {A} itself, because this filtration does not depend on theprobability measure under consideration and each is countably determined.Toward this end, we need the following result.

4.13 Problem. Assume either

(A.1)' b (t, + KT; 0 5 t < T, y e C[0, 05)d, for every0 < T < co, where KT is a constant depending on T, or else

(A.2)' b,(t, y) and 0-0(t, y) are of the form b,(t, y) = (t, y(t)),y) = er(t,y(t)), where the Borel-measurable functions bi, au: [0, oc) xFl are bounded on compact sets.

Let P be a probability measure on (C[0, oo)d, AC[0, °or)); {.f,} be as inDefinition 4.5; and f E ce(ll 0). Show that if {Mif, A; 0 < t < oo} is a martin-gale, then so is {Mir, ,97,; 0 < t < co}.

C. Well-Posedness and the Strong Markov Property

We pause here in our development of the martingale problem to discuss thestrong Markov property for solutions of stochastic differential equations.Consistent with our discussion of Markov families in Chapter 2, we shedthe trappings of time-dependence. Thus, we have time-homogeneous andBorel-measurable drift and dispersion coefficients bi: Rd R, R;

1 < i < d,1 < j < r, and we shall study the time-homogeneous version of (2.1),written here in integral form as

(4.20) X, = x + f b(Xs) ds + f a(X5) diVs; 0 < t < co.

This model actually does allow for time-dependence because time can beappended to the state variable; i.e., we may take Xr1) = t, b, (x) = 1,crd+i,J(x) = 1 :54 :5_ r. We adopt the time-homogeneous assumption pri-marily for simplicity of exposition. Note, however, that some results (e.g.,Remark 4.30 and Refinements 4.32) require the nondegeneracy of the diffusioncoefficient, which is not valid in such an augmented model.

4.14 Definition. The stochastic integral equation (4.20) is said to be well posedif, for every initial condition X E Rd, it admits a weak solution which is uniquein the sense of probability law.

We know, for instance, that (4.20) is well posed if b and a satisfy Lipschitzand linear growth conditions (Theorems 2.5, 2.9). If a is the (d x d) identitymatrix and b is uniformly bounded, (4.20) is again well posed (Proposition 3.6and Corollary 3.11). Later in this section, we shall obtain well-posedness undereven less restrictive conditions on a (Theorems 4.22 and 4.28, Corollary 4.29).

If (4.20) is well posed, then the solution X with initial condition X0 = xinduces a measure Fix on (C[0, co)", a(c[o,cor)). One can then ask whetherthe coordinate mapping process on this canonical space, the filtration {M, },and the family of probability measures {Px},ce R. constitute a strong Markovfamily. We shall see that if b and a are bounded on compact subsets of Rd,the answer to this question is essentially affirmative. Our analysis proceedsvia the martingale problem, which we now specialize to the case at hand. Wedenote by

d d 02f(x) d Of(x)(1.2) (df)(x) =

1

- E E a,(x) E2 1=1 k=1 CX OXk 1=1 OX/

the time-homogeneous version of the operator in (4.1).

4.15 Definition. Assume that R and oii: Rd R; 1 < i < d, 1 < j 5 rare bounded on compact subsets of Rd. A probability measure P on (Q, a.) A(C[0, oo)d, ((C[0, oo)d)) under which

(4.21) E[f(y(t)) - f(y(s)) - (szif)(y(u))du Ad= 0, a.s. P

holds for every 0 < s < t < oo, f e CARd), is called a solution to the time-homogeneous martingale problem. We denote by 13' any solution for which

(4.22) 13' [y e C[0, oo)d; y(0) = x] = 1.

We say that the time-homogeneous martingale problem is well posed if, forevery x e Rd, there is exactly one such measure Px.

4.16 Remark. The replacement of ..97, by Ms in (4.21) is justified by Problem4.13.

4.17 Remark. Under the conditions of Definition 4.15, well-posedness of thetime-homogeneous martingale problem is equivalent to well-posedness of thestochastic integral equation (4.20) (Corollaries 4.8 and 4.9 and Proposition4.11).

4.18 Lemma. For every bounded stopping time T of the filtration {A}, we have

= (431(t A T); 0 t < oo).

PROOF. The MT-measurability of each y(t n T) follows directly from Definition1.2.12 and Proposition 1.2.18. It remains to show MT g o-(y(t n T); 0 < t < oo).

Let cpt: C[0, oo)d C[0, oo)d be given by (cpty)(s) = y(t n s); 0 < s < oo, forarbitrary t > 0. Problem 1.2.2 shows that T(y) = T(pT(y)(y)) holds for everyy e C[0, oo)d and so, with A e ,IT and t A T(y), we have

ye A ye[A n 5 t1]-4* cpty e [A n 5 tn.:* (Aye A.

The second of these equivalences is a consequence of the facts An{ T< t} e Mand y(s) = (cp,(y))(s); 0 < s < t. We conclude that

A = {ye C[0, oo)d; ch(y)(y)e Al = e C[0, oo)d; y( n T)e Al.

For the next lemma, we recall the discussion of regular conditional prob-abilities in Subsection 3.C, as well as the formula (2.5.15) for the shift operators:(Osy)(t) = y(s + t); 0 < t < oo for s > 0 and y e C[0, oo)d.

4.19 Lemma. Let T be a bounded stopping time of {.4,} and 5 a countablydetermined sub-a-field of MT such that y(T) is 5-measurable. Suppose that band Q are bounded on compact subsets of Rd, and that the probability measureP on (SI, = (C[0, co)°, .4(C [0, cx)r), solves the time-homogeneous martingaleproblem of Definition 4.15. We denote by Q,(F) = Q(a); F): S2 x P4 -+ [0,1]the regular conditional probability for .4 given 1.

There exists then a P-null event N e 5 such that, for every w N, the probabilitymeasure

P. 4 Q,° 13i.1

solves the martingale problem (4.21), (4.22) with x = w(T(w)).

PROOF. We notice first that, thanks to the assumptions imposed on /,Theorem 3.18 (iv)' implies the existence of a P-null event N e 5, such that

Q(0); {yen; Y(T(Y)) = w(T(w)) }) = 1,

and therefore also

PAY e f); .Y(0) = (0(7.(0))] = Q.Ey e n; Y(T(Y)) = w(T(w))] = 1

hold for every w 0 N. Thus (4.22) is satisfied with x = w(T(w)).In order to establish (4.21), we choose 0 < s < t < co, Gel, Fe.4

f e q(Rd); define

Z(Y) f(Y(t)) - f(Y(s)) - I (sin1Y1un du;

and observe that

(4.23)

ye C[0, co)d,

Z(y)1F(Y)13.(dY) = Z(07-0).01F(07.61Y)0(0;dY)1-1

= E[loiF(Z 0 OT)ig](a)

= E[E(Z 0 oTiaT+3). lei.,F11] (03)

= 0, P-a.e. co.

We have used in the last step the martingale property (4.21) for P and theoptional sampling theorem (Problem 1.3.23 (i)).

Let us observe that, because of our assumptions, the random variable Zis bounded; relation (4.23) shows that the /-measurable random variablew Z(y)P,(dy) is zero except on a P-null event depending on s, t, f, and F.Consider a countable subcollection g of as and a P-null event N(s, t, f) e ,such that

Z(y)Pc,(dy) = 0; V co N(s, t, f), V F e g.

Then the finite measures v,±-(F) A IF Z± (y)P,(dy); F e agree on 6', and sinceis countably determined, the subcollection g can be chosen so as to permit

the conclusion

f,Z(y)P,(dy) = 0; V ru N(s,t, f), V F e

We may set now N(f) = lj Li e Q N(s, t, f), and use the boundedness and0 s<t<ao

continuity (in s, t) of Z to conclude that

Mir f(y(t)) - f(y(0)) - ft (sif )(y(u)) du, .4t; 0 < t < co

is a martingale under P., for every w N(f). Finally, we see that there existsa P-null event N a under which M-r is a P.; martingale for all w N andcountably many f e CROld); because of Remark 4.12, Pc, solves the time-homogeneous martingale problem for all a) 0 N.

4.20 Theorem. Suppose that the coefficients b, a are bounded on compact subsetsof Rd, and that the time-homogeneous martingale problem of Definition 4.15(or equivalently, the stochastic integral equation (4.20)) is well posed. Then forevery stopping time T of {A}, F e .4(C [0, co )d) and x e , we have the strongMarkov property

(4.24) Px[0,-1 F1.4,]((o) = P'm[F], Px-a.s. on {T < col.

PROOF. If the stopping time T is bounded, we let in Lemma 4.19 be .4T, whichis countably determined by Lemma 4.18. Using the notation of Lemma 4.19,we may write then, for every F e a/(C[0, co)d):

Px F I .431,] (w) = Q(o);OV F) = P,(F) = Pc'm"[F],

for Px-a.e. w en. The last identity is a consequence of the uniqueness ofsolution to the time-homogeneous martingale problem with initial conditionx = co(T(w)).

Unbounded stopping times are handled as in Problem 2.6.9 (iii).

4.21 Remark. The strong Markov property of (4.24) is the same as condition(e") of Theorem 2.6.10, except that we have succeeded in proving it only forstopping (rather than optional) times.

Condition (a) of Definition 2.6.3 is satisfied under the assumptions ofTheorem 4.20. Indeed, well-posedness implies that the mapping x H Px(F)is Borel-measurable for every F e .4(C[0, co )d), but the proof of this state-ment requires a rather extensive set-theoretic development (see Stroock &Varadhan (1979), Exercise 6.7.4, and Parthasarathy (1967), Corollary 3.3,p. 22). This result is of rather limited interest, however, because when a proofof well-posedness is given, it typically provides a constructive demonstrationof the measurability of the mapping x Px(F).

D. Questions of Existence

It is time now to use the martingale problem in order to establish the funda-mental existence result for weak solutions of stochastic differential equationswith bounded, continuous coefficients.

4.22 Theorem (Skorohod (1965), Stroock & Varadhan (1969)). Consider thestochastic differential equation

(4.25) d X , = b(X,) dt + o- (X ,) dW

where the coefficients bi, are bounded and continuous functions.Corresponding to every initial distribution kt on 4(91°) with

fRaIIx1I2mµ(dx) < co, for some m > 1,

there exists a weak solution of (4.25).

PROOF. For integers j > 0, n > 1 we consider the dyadic rationals 4") = jr"and introduce the functions i(i(t) = t11); t e [tj"), 01_1). We define the newcoefficients

(4.26) b(n)(t, b(Y(0.(t))), (7(")(t, Y) g 6(Y(IP.(0);

0 < t < co, y e C[0, oo)d,

which are progressively measurable functionals.Now let us consider on some probability space (0,F, P) an r-dimensional

Brownian motion W = {W, Fr; 0 < t < col and an independent randomvector with the given initial distribution and let us construct the filtration1,,1 as in (2.3). For each n > 1, we define the continuous process V") =Wm), 0 < t < co} by setting X,(,,") = and then recursively:

X") = 41!) + b(X(n)))(t - t(p) + o-PC:!))(1,V, - K.)); j 0, tj") < t < .

Then V") solves the functional stochastic integral equation

(4.27) X:") = + f b(")(s, X("))ds + ft o-(")(s, X("))dWs; 0 < t < co.

Fix 0 < T < co. From Problem 3.15 we obtain

sup E II X:") - X!")1121" C(1 +FII 112"1)(t - s)m; 0 ,s<t.,T,n>1

where C is a constant depending only on m, T, the dimension d, and the boundon 11b112 + 110112. Let P(") c P(X("))-1; n > 1 be the sequence of probabilitymeasures induced on (C[0, oo)d,a(C[0, oo)d)) by these processes; it followsfrom Problem 2.4.11 and Remark 2.4.13 that this sequence is tight. We maythen assert by the Prohorov Theorem 2.4.7, relabeling indices if necessary,

that the sequence 1/34")};°_, converges weakly to a probability measure P* onthis canonical space.

According to Proposition 4.11 and Problem 4.13, it suffices to show

(4.28) P*[y e C[0, cot y(0) e I-1 = p(F); I- e A(Rd),

(4.29) E*C./(Y(0) - .AY(s)) -1t

(sif)(Y (14)) du IA] = 0, a.s. P*,s

for every 0 < s < t < oo, fe Cj,(Rd). For every fE Cb(Old), the weak convergenceof {/34")},°,'_, to P* gives

E*f(y(0)) = lim E(n)f(y(0)) = i f(x)p(dx),n-.co Rd

and (4.28) follows. In order to establish (4.29), let us recall from (4.27) andProposition 4.11 that for every f e oRd) and n > 1,

f(y(t)) -f (y(0)) -f (s1,;")f )(y) du, ,4,; 0 t < coo

is a martingale under P("), where

(sit(n).n(Y) A i bln)(t, y) af (At)) + 1 is i 0-1;*, y)011)(t, y)a2f (Y(t)).i=1 axi 2 1=1 k=l J=I oXiaKk

Therefore, with 0 < s < t < co and g: C[0, co)d -> II a bounded, continuous,AS measurable function, we have

(4.30) E4")[{f(y(t)) - f(y(s)) -f 1 (sii;")f )(y) du} g(y)1= 0.

We shall show that for fixed 0 < s < t < co, the expression

Fn(Y) A f(Y(t)) - f(y(s)) f (clin)f)(Y) dus

converges uniformly on compact subsets of C[0, co)d to

F(Y) A f(Y(t)) - f(y(s)) f (slf )(Y (u)) du.

Then Problem 2.4.12 and Remark 2.4.13 will imply that we may let n -> co in(4.30) to obtain E*[F(y)g(y)] = 0 for any function g as described, and (4.29)will follow. Let K c C[0, co)d be compact, so that (Theorem 2.4.9 and (2.4.3)):

M A sup 11)7(41 < GC, lim sup trit(y,2-") = 0.yEK n-co yeX

o5u5t

Because b and a are uniformly continuous on {x E Rd; II x II < M }, we can findfor every s > 0 an integer n(s) such that

sup (11b(")(s, y) - b(Y(s))II + II a(")(s,y) - cr(Y(s))11) 5 v; n > n(E).0<styeK

The uniform convergence on K of Fn to F follows.

4.23 Remark. It is not difficult to modify the proof of Theorem 4.22 to allowb and a to be bounded, continuous functions of (t, x) e [0, oo) x Rd, or even toallow them to be bounded, continuous, progressively measurable functionals.

E. Questions of Uniqueness

Finally, we take up the issue of uniqueness in the martingale problem.

4.24 Definition. A collection 9 of Borel-measurable functions cc,: Rd -4 R iscalled a determining class on Rd if, for any two finite measures it, and it2 onR(Rd), the identity

P(x)Iii(dx) =J

49(x)112(dx), V 9Qga 68a

implies it =

4.25 Problem. The collection C83(Rd) is a determining class on Rd.

4.26 Lemma. Suppose that for every f e Cg)(Rd), the Cauchy problem

(4.31)au

at= siu; in (0, oo) x Rd,

(4.32) u(0, ) = f; in Rd,

has a solution uf e C([0, oo) x Rd) n C12((0,00) x Rd) which is bounded oneach strip of the form [0, T] x Rd. Then, if Px and 17:" are any two solutions ofthe time-homogeneous martingale problem with initial condition x E Rd, theirone-dimensional marginal distributions agree; i.e., for every 0 < t < co:

(4.33) Px[y(t) e V] = Px[y(t) e v r e R(Rd).

PROOF. For a fixed finite T > 0, the function g(t, x) A uf(T - t, x); 0 < t <x e W is of class Cb([0, T] x Rd) r C1 ,2(0, x Rd) and satisfies

-ag + = 0; in (0, T) x Rd,at

T,

g(T, ) = f; in Rd.

Under either F" or /3', the coordinate mapping process X, = y(t) on C[0, 0o)dis a solution to the stochastic integral equation (4.20) (with respect to some

Brownian motion, on some possibly extended probability space; see Proposi-tion 4.11). According to Proposition 4.2, the process {g(t, y(t)), .1t; 0 5 t 5 T}is a local martingale under both Px and Px; being bounded and continuous,the process is in fact a martingale. Therefore,

Exf(y(T)) = Ex g(T, y(T)) = g(0, x) = Exg(T,y(T)) = Exf(y(T))

Since f can be any function in the determining class Co (Rd), we conclude that(4.33) holds.

We are witnessing here a remarkable duality: the existence of a solutionto the Cauchy problem (4.31), (4.32) implies the uniqueness, at least for one-dimensional marginal distributions, of solutions to the martingale problem.In order to proceed beyond the uniqueness of the one-dimensional marginalsto that of all finite-dimensional distributions, we utilize the Markov-likeproperty obtained in Lemma 4.19. (We cannot, of course, use the Markovproperty contained in Theorem 4.20, since uniqueness is assumed in thatresult.)

4.27 Proposition. Suppose that for every x e Rd, any two solutions Px and Px ofthe time-homogeneous martingale problem with initial condition x have the sameone-dimensional marginal distributions; i.e., (4.33) holds. Then, for every xe Rd,the solution to the time-homogeneous martingale problem with initial conditionx is unique.

PROOF. Since a measure on C[0, corl is determined by its finite-dimensionaldistributions, it suffices to fix 0 < t, < t2 < < t < a) and show that Pxand Px agree on A o(y(t,),..., y(t)). We proceed by induction onn. We have assumed the truth of this assertion for n = 1. Suppose now that Pxand Px agree on q(t11. t, _1), and let Qy be a regular conditional probabilityfor M(C[0, oc)°) given W(tI, tn_i), corresponding to Px. According toLemma 4.19, there exists a Px-null set N e q(tI, to _1) such that for y 0 N,the measure Py A Qy o et lI solves the time-homogeneous martingale problemwith initial condition y(tn_i). Likewise, there is a regular conditional prob-ability Oy corresponding to Px and a Px-null set e 5(t , , t_,), such thatP

Y0

Y0-1 solves this martingale problem whenever y R. For y not in theIn-I

null (under both Px, P) set N u R, we know that Py and Py have the sameone-dimensional marginals. Thus, with A e .4(Rd(n-1)) and B e .4(Rd), we have

Px[(y(t,),... , y(t,,_,))e A, y(t) e B]

= Ex[110,40,...,yon_inE A}Pylco co(tn - tn_i)EB }]

= Ex [1 {(Ati), IDE Al PY {W (.0(tn -tn_i)EB

}]

= x[ 1]1(0,00,...,yon D,EA}Pylw e w(tn - tn_i )e B

= Px[(y(ti),..., y(tn_i))e A, y(tn)e 13],

where we have used not only the equality of Py {co ef2; co(t. - tn_i )e B} andPy{w e S2; w(t - t_, )e B}, but also their W(t1,... , t_1)-measurability. It isnow clear that P and Pr agree on g(t1,..., t).

We can now put the various results together.

4.28 Theorem (Stroock & Varadhan (1969)). Suppose that the coefficientsb(x) and a(x) in Definition 4.15 are such that, for every f e Co (RI), the Cauchyproblem (4.31), (4.32) has a solution uf e C([0, oo) x Rd) n Cl '2((0, co) x lild)which is bounded on each strip of the form [0, T] x g8 °. Then, for every x ell?,there exists at most one solution to the time-homogeneous martingale problem.

4.29 Corollary. Let the coefficients b(x), a(x) be bounded and continuous andsatisfy the assumptions of Theorem 4.28. Then the time-homogeneous martingaleproblem is well posed.

4.30 Remark. A sufficient condition for the solvability of the Cauchy problem(4.31), (4.32) in the way required by Theorem 4.28 is that the coefficients bi(x),aik(x); 1 < j, k < d be bounded and Holder-continuous on Rd, and the matrixa(x) be uniformly positive definite; i.e.,

d d

(4.34) E E a ik(x)M >_ A11112; V x, e Rd and some A > 0.i=1 k=1

We refer to Friedman (1964), Chapter 1, and Friedman (1975), §6.4, §6.5, orStroock & Varadhan (1979), Theorem 3.2.1, for such results; see also Remark7.8 later in this chapter.

4.31 Remark. If aike C2(Rd) for 1 < i, k < d, then a(x) has a locally Lipschitz-continuous square root, i.e., a (d x d)-matrix-valued function a' (x) ={iiii(x)} 1 , i, j, such that a ik(x) = El=iiiii(x)a,;(x); xe Rd (Friedman (1975),Theorem 6.1.2). We do not necessarily have a(x) = Q(x) (indeed, one interest-ing case is the one-dimensional problem with a(x) = sgn(x) and a(x) = 1;x e R). If, in addition, bi e C 1 (Rd); 1 < i 5 d, then according to Theorem 2.5,Remark 3.3, Proposition 3.20, Corollary 4.9, and Proposition 4.11, for everyx e I there exists at most one solution to the time-homogeneous martingaleproblem. This result imposes no condition analogous to (4.34) and is thusespecially helpful in the study of degenerate diffusions.

4.32 Refinements. It can be shown that if the coefficients b(x), a(x) are boundedand Borel-measurable, and the matrix a(x) is uniformly positive definiteon compact subsets of Rd, then the time-homogeneous martingale problemadmits a solution. In the cases d = 1 and d = 2, this solution is unique. SeeStroock & Varadhan (1979), Exercises 7.3.2-7.3.4, and Krylov (1969), (1974).

F. Supplementary Exercises

4.33 Exercise. Assume that the coefficients bi: IR, o-ij: IRd IR; 1 < i < d,1 < j < r are measurable and bounded on compact subsets of Rd, and let albe the associated operator (1.2). Let X = { X .f.,; 0 < t < co} be a continuousprocess on some probability space P) and assume that {.F,} satisfiesthe usual conditions. With

('f

e C2(Rd) and a e IR, introduce the processes

Mt g f (X,) - f(X0) - J df(Xs)ds, .F,; 0 < t < co,

A, -A e-x`f(X,) - f(X0) + J e-"(ocf(Xs) - alf(X))ds, 0 < t < oo,

and show M e dr' a A loc. If f is bounded away from zero on compactsets and

N, g f(X,)expdf(X,)

o .f(XS) dsf (x .97,; 0 < 00,

then these two conditions are also equivalent to: N e .1r1". (Hint: Recall fromProblem 3.3.12 that if M e dr' and C is a continuous process of boundedvariation, then C,M, - f o MS dC, = Ito CS dM, is in dr 1".)

4.34 Exercise. Let (X, W), P), 1,9;1 be a weak solution to the functionalstochastic differential equation (3.15), where condition (A.1)' of Problem 4.13holds. For any continuous function f: [0, co) x IRd FR of class C'2((0, co) xRd) and any progressively measurable process Ik .97,; 0 < t < col, show that

,

At f(t, X,)e-fr'ku" - f(0, X0) - + als'f - ksf)e-P'ku" ds, Ft;o as

0 < t < oo

is in dr'. If, furthermore, f and its indicated derivatives are bounded andk is bounded from below, then A is a martingale.

4.35 Exercise. Let the coefficients b, r be bounded on compact subsets of Fe,and assume that for each x e Fe, the time-homogeneous martingale problemof Definition 4.15 has a solution Px satisfying (4.22). Suppose that there existsa function f: [0, oc) of class C2(11?) such that

df(x) + Af(x) < c,

holds for some A > 0, c > 0. Then

Exf(y(t)) f(x)e-At + ;.(1 - e-1`); 0 < t < co, x e

5.5. A Study of the One-Dimensional Case 329

5.5. A Study of the One-Dimensional Case

This section presents the definitive results of Engelbert and Schmidt con-cerning weak solutions of the one-dimensional, time-homogeneous stochasticdifferential equation

(5.1) dX, = b(Xi)dt + a(X,) dW,

with Borel-measurable coefficients b: R R and a: R -+ R. These authorsprovide simple necessary and sufficient conditions for existence and unique-ness when b --- 0, and sharp sufficient conditions in the case of general driftcoefficients. The principal tools required for this analysis are local time, thegeneralized Ito rule, the Engelbert-Schmidt zero-one law (see Subsection 3.6.E),and the notion of time-change.

Solutions of equation (5.1) may not exist globally, but rather only up toan "explosion time" S; see, for example, Remark 2.8. We formalize the idea ofexplosion.

5.1 Definition. A weak solution up to an explosion time of equation (5.1) is atriple (X, W), (S2, F, 11, {,;}, where

(i) (SI, ,F, P) is a probability space, and {..F,} is a filtration of sub-a-fields ofY., satisfying the usual conditions,

(ii) X = IX ,f",,; 0 < t < col is a continuous, adapted, 11 u 1 + co 1-valuedprocess with 1X01 < co a.s., and { 147 gt; 0 < t < co} is a standard, one-dimensional Brownian motion,

(iii) with

(5.2) Sn A- inf{t _>._ 0; 1X,1 n},

we have

,AS(5.3) P[f{1b(Xs)I + e(Xs)} ds < col = 1; V0<t<oo

o

and(iv)

(5A) PEX,s. = X0 + b(X01{,s.}dso

+J0-(X.011ssId3; V 0 t < co] = 1

o

valid for every n > 1.We refer to

(5.5) S = liM Sn->co

as the explosion time for X. The assumption of continuity of X in theextended real numbers implies that

(5.6) S = inf{t > 0: X, st RI and Xs = +co a.s. on {S < co}.

We stipulate that X, = Xs; S < t < co.

The assumption of finiteness of X0 gives P[S > 0] = 1. We do not assumethat lim, X, exists on {S = col, so Xs may not be defined on this event. IfP[S = co] = 1, then Definition 5.1 reduces to Definition 3.1 for a weaksolution to (5.1).

We begin with a discussion of the time-change which will be employed inboth the existence and uniqueness proofs.

A. The Method of Time-Change

Suppose we have defined on a probability space a standard, one-dimensionalBrownian motion B = {Bs, FB; 0 < s < co} and an independent randomvariable with distribution p. Let {s) be a filtration satisfying the usualconditions, relative to which B is still a Brownian motion, and such that

is q0-measurable (a filtration with these properties was constructed forDefinition 2.1). We introduce

(5.7) T Afs+ U

du

2g Bu); < < CC,0

a nondecreasing, extended real-valued process which is continuous in thetopology of [0, co] except for a possible jump to infinity at a finite time s.From Problem 3.6.30 we have

(5.8) To A lim T, = co a.s.s t co

We define the "inverse" of Ts by

(5.9) A, -4 inf{s > 0; T > t}; 0 < t < co and A3, lim A,.

Whether or not Ts reaches infinity in finite time, we have a.s.

(5.10) Ao = 0, A, < co; 0 t < co and Acc, = inf {s 0: T, = co}.

Because Ts is continuous and strictly increasing on [0, IL), A, is also continuouson [0, co) and strictly increasing on [0, TA,,_). Note, however, that if T, jumpsto infinity (at s = A), then

(5.11) A, = Az°, V t > TA._ ;

if not, then TA._ = TA.= co, and (5.11) is vacuously valid. The identities

(5.12) TA, = t; 0 5 t < T/c_,

(5.13) AT= = s; 0 < s < Aco,

hold almost surely. From these considerations we deduce that

(5.14) 'Ts = inflt 0; A, > sl; 0 < s < oo, a.s.

In other words, A, and Ts are related as in Problem 3.4.5.Let us consider now the closed set

E dy(5.15) /(a) = { x E R; f_e = co, V e > 01,anddefine

(5.16) R A inf{s z 0; + Bs e 40)}.

5.2 Lemma. We have R = Acc, a.s. In particular,c R+ du

(5.17) = co, a.s.v2( + B.)

PROOF. Define a sequence of stopping times

1R,, A inf s 0; p( + B 1(a)) -1; n > 1,

where

(5.18) p(x, 1(a)) A inf{ x - yl; ye/(a)}.

Because 1(a) is closed, we have limn, R = R, a.s. (recall Solution 1.2.7). Forn > 1, set

a(x) =

1

(x); p(x,1(a)) zn,

p(x, 1(a)) <1

We have cr,;-2(x) dx < co for each compact set K c R, and the Engelbert-Schmidt zero-one law (Proposition 3.6.27 (iii) Problem 3.6.29 (v)) gives a.s.

[R." du R^^s du s du

J 0 a + R 1 0 + Bu ) ± B.) < co; 0 s < oo.ff

For P-a.e. w e f and s chosen to satisfy s < R (co), we can let n oo to conclude

cs du

o cr2g(w) + B((0)) < c).

It follows that Ts < co on {s < R}, and thus A. 5 R a.s. (see (5.10)).For the opposite inequality, observe first that 1(a) = 0 implies R =

and in this case there is nothing to prove. If 1(a) 0 0 then R < co a.s. and

for every s > 0,R+s du

>s du

Jo cr2( ± B.) fo Q2( + BR + Wu)'

where W --4- -BR+u -BR is a standard Brownian motion, independent of BR(Theorem 2.6.16). Because + BR e 1(a), Lemma 3.6.26 shows that the latterintegral is infinite. It follows that TR = CO, so from (5.10), fl, < R a.s.

We take up first the stochastic differential equation (5.1) when the drift isidentically zero. One important feature of the resulting equation

(5.19) dX, = a(X,)dW,

is that solutions cannot explode. We leave the verification of this claim to thereader.

5.3 Problem. Suppose (X, W), (0,F,P), {,,} is a weak solution ofequation (5.19) up to an explosion time S. Show that S = co, a.s. (Hint: RecallProblem 3.4.11.)

We also need to introduce the set

(5.20) Z(a) = Ix E R; a(x) = 0 }.

The fundamental existence result for the stochastic differential equation (5.19)is the following.

5.4 Theorem (Engelbert & Schmidt (1984)). Equation (5.19) has a non-exploding weak solution for every initial distribution a if and only if

(E)

i.e., if

1(6) g Z(6);

re dy

j_, a2(x + y)=co' V e > 0 Cr(X) = O.

5.5 Remark. Every continuous function a satisfies (E), but so do many dis-continuous functions, e.g., a(x) = sgn(x). The function o(x) = 1{0}(x) does notsatisfy (E).

PROOF OF THEOREM 5.4. Let us assume first that (E) holds and let { + B A;0 < s < oo 1 be a Brownian motion with initial distribution ii, as described atthe beginning of this subsection. Using the notation of (5.7)-(5.16), we canverify from Problem 3.4.5 (v) that each A, is a stopping time for {A}. We set

(5.21) M, = B, X, = + M g" =WA; 0 5 t < oo.Because A is continuous, I.F,1 inherits the usual conditions from { '.,} (cf.Problem 1.2.23). From the optional sampling theorem (Problem 1.3.23 (i)) and

the identity A, T, = A, n s (Problem 3.4.5 (ii), (v)), we have for 0 < t1 < t2 < coand n > 1:

E[Mt, A = E[BA, A nIA, ] = BA, An = Mt! A Tn, a.s.

Since lim, T = co a.s., we conclude that M e Jr'. Furthermore,Mt2A T - At A T = nn -(A, A n) is in ,fic for each n > 1, so

(5.22) <M>, = A,; 0 < t < co, a.s.

As the next step, we that the process of (5.9) is given by

(5.23) A, = t 62 (X )dv; 0 < t < co, a.s.0

Toward this end, fix w e {R = 4,0}. For s < A,(w), (5.7) and (5.10) show thatthe function 7'(w) restricted to u e [0, s] is absolutely continuous. Thechange of variable v = 7;,(w) is equivalent to Ay(w) = u (see (5.12), (5.13)) andleads to the formula

At(m)

(5.24) A,(w) = 62(((o) + B.(0)dr,,((o) = a2 (X (w)) dv,

valid as long as A,(w) < 21,0(w), i.e., t < T(w), where

(5.25) T TA._ = inf { u > 0; A = A. }.

If t(w) = co, we are done. If not, letting t T t(w) in (5.24) and using thecontinuity of A(w), we obtain (5.23) for 0 < t < t(o)). On the interval [-r(w), co],A,(w) A00(co) = A.(w) = R(w). If R(w) < co, then

A',40(co) c(w) + BR(W)(w) E 1(a) Z(o-),

and soo-(X,(w)) = o-(X too(w)) = 0; t(w) < t < co.

Thus, for t > t(co), equation (5.23) holds with both sides equal to 240.)(co).From (5.22), (5.23), and the finiteness of A, (see (5.10)), we conclude that

<M> is a.s. absolutely continuous. Theorem 3.4.2 asserts the existence ofa Brownian motion 4V = 1147 0 < t < col and a measurable, adaptedprocess p = Ip At; 0 < t < col on a possibly extended probability space

F), such that

M, = f dITT, <M>, = f py2 dv; 0 t < co , P a.s.

In particular,

We may setP[pi2 = 62(Xj) for Lebesgue a.e. t > 0] = 1.

W=ft sgn(p09-(XJ)dIR; 0 < t < co;0

observe that W is itself a Brownian motion (Theorem 3.3.16); and write

X,= + M, = + f t a(X)dW; 0 < t < co, P-a.s.0

Thus, (X, W) is a weak solution to (5.19) with initial distribution ft.To prove the necessity of (E), we suppose that for every x e 64E, (5.19) has

a nonexploding weak solution (X, W) with X0 = x a.s. Here W = {W, ",;0 < t < co} is a Brownian motion with W0 = 0 a.s. Then

(5.26)

is in de.'" and

M, = X, - X0, ..17;,; 0 < t < co

(5.27) <M>, = I: a2(Xv)dv < co; 0 < t < oo, a.s.o

According to Problem 3.4.7, there is a Brownian motion B = 14, W's; 0s < co} on a possibly extended probability space, such that

(5.28) M, = Boot; 0 < t < co, a.s.

Let Ts = inf{t _. 0; <M>, > s }. Then s A <M>co = <M> T., (Problem 3.4.5 (ii)),so using the change of variable u = <M> (ibid. (vi)) and the fact that d<M>assigns zero measure to the set {v > 0; a2(X) = 0 }, we may write

5 A <MX° du 1<m)T. du ii; d<mx,(5.29)

a2(X0 + B.) j0 a2(X0 + B.) j0 a2(X,,)

d<M>v= f T,

l {,,,2(x d>, 0}

(x

= J.0 1 {2(x 0") dv _<_ T.

Let us choose the initial condition X0 = x in Z(o)`. Then a(x) 0 0, so asolution to (5.19) with such an initial condition cannot be almost surelyconstant. Moreover P[<M>co > 0] > 0, and thus for sufficiently small positives, P[T, < co, <MX° > 0] > 0. Now apply Lemma 3.6.26 with

T(wA <M>.(w); if Ts(co) < oo, <M>.(w) > 0,

s; otherwise.

T,

We conclude from this lemma and (5.29) that x e 1 (o) .̀ It follows that/(u) g Z(a).

5.6 Remark. The solution (5.21) constructed in the proof of Theorem 5.4 isnonconstant up until the time

(5.30) t A inf.{ t > 0; X, e /(a)}.

(This definition agrees with (5.25).) In particular, if X0 = x e 1(o)`, then X is

not identically constant. On the other hand, if x e Z(a), then Y = x alsosolves (5.19). Thus there can be no uniqueness in the sense of probability law ifZ(a) is strictly larger than 1(a). This is the case in the Girsanov Example 2.15;if a(x) = jxIa and 0 < a < (1/2), then 1(a) = 0 and Z(a) = {0}.

Engelbert & Schmidt (1985) show that if a: IR -* IR is any Borel-measurablefunction satisfying 1(a) = 0, then all solutions of (5.19) can be obtained fromthe one constructed in Theorem 5.4 by "delaying" it when it is in Z(a). Anidentically constant solution corresponds to infinite delay, but other, lessdrastic, delays are also possible.

5.7 Theorem. (Engelbert & Schmidt (1984)). For every initial distribution p,the stochastic differential equation (5.19) has a solution which is unique in thesense of probability law if and only if

(E + U) 1(a) = Z(cr).

PROOF. The inclusion 1(a) g Z(a), i.e., condition (E), is necessary for existence(Theorem 5.4); in the presence of (E), the reverse inclusion is necessaryfor uniqueness (Remark 5.6). Condition (E) is also sufficient for existence(Theorem 5.4), so it remains only to show that the equality 1(a) = Z(a) issufficient for uniqueness. We shall show, in fact, that the inclusion 1(a) Z(0)is sufficient for uniqueness.

Assume 1(a) Z(a) and let (X, W), (n,F,p), {,,} be a weak solutionof (5.19) with initial distribution It. We define M as in (5.26) and obtain(5.27)-(5.29), where B is a standard Brownian motion. We also introducevia (5.30). By assumption,

(5.31) a2(X,) > 0; 0 t < T, a.s.,

so <M> is strictly increasing on [0, t]. In particular,

(5.32) R A inf {s 0; X0 + Bse 1(a)} = <M>t, a.s.

We set

(5.33) T, A int.{ t 0; <M>, >

so T is nondecreasing, right-continuous, and

(5.34)

We claim that

{TS < = Is < <M>,}.

(5.35) 0 < s < <MX°, a.s.JS

a2

du

(X0 + Bu) -Ts ;

In light of (5.29), (5.31), to verify this claim it suffices to show

(5.36) R <M>., a.s.

Indeed, on the set {R < <M>,,,} we have by the definition (5.33) that TR < CO;

letting sIR in (5.29), we obtain thenTR+ du

o 6.2(X0 + B)

Comparing this with (5.17), we see that PER < <M>] = 0, and (5.35) isestablished.

We next argue that

< T R < c o on {R < <M> 1.

Ts =

s+ du(5.37) ; 0 < s < co, a.s.

.1.0 6 (X0 + B)

For s < <M>, we have Ts < co and so (5.37) follows immediately from (5.35).For s = <M> = R, it is clear from (5.33) that Ts = oo, and (5.37) follows from(5.17). It is then apparent that (5.37) also holds for s > <M> with both sidesequal to infinity.

Comparing (5.37) and (5.7), we conclude that the discussion precedingLemma 5.2 can be brought to bear. The process A defined in (5.9) coincidesthen with <M>, which is thus seen to be continuous and real-valued and tosatisfy

(5.38) <M>, = inf Is 0; }s+ du

fo u2(X0 + B)> t ; 0 t < co.

We now prove uniqueness in the sense of probability law by showing thatthe distribution induced by the solution process X on the canonical space(C[0, co), ,l(C[0, co))) is completely determined by the distribution it of Xo.The distribution induced on this space by the process X0 + B is completelydetermined by it because B is a standard Brownian motion independent of X0(Remark 3.4.10). For CO E C [0, Go), define 9.(w) to be the right-continuousprocess

tp,(w) A inf Is 0;iso+ du

o-2(w(u))> t}; 0 < t < co

with values in [0, co]. Because of its right-continuity, 9 is 4( [0, co)) ®.4(C[0, co))-measurable (Remark 1.1.14). Let 7C: [0, CO) x C [0, CO) -+ R be themeasurable projection mapping tr,(co) = w(t). Consider the P. co)) ®M(C [0, co))-measurable process Co)) A co(0) + tr ,,,,(,,)(a)). We have

(5.39) X, = X0 + Boo, = 11/,(X0 + B.), 0 t < co,

and so the law of X is completely determined by that of X0 + B..

5.8 Remark. It is apparent from the proof of Theorem 5.7 that under theassumption 1(a) 2 Z(r), any solution to (5.19) remains constant after arrivingin 1(o). See, in particular, the representation (5.39) and recall that <M>, isconstant for t < t < CO.

Using local time for continuous semimartingales, it is possible to give asimple but powerful sufficient condition for pathwise uniqueness of the solutionto equation (5.19).

5.9 Theorem. Suppose that there exist functions f: L -> [0, co) and h: [0, co) ->[0, co) such that:

(i) at every x el(a)t, the quotient (f/a)2 is locally integrable; i.e., there existse > 0 (depending on x) such that

EE(f(Y))2 dy<oo,

J . 0(Y)

(ii) the function h is strictly increasing and satisfies h(0) = 0 and (2.25);(iii) there exists a constant a > 0 such that

+ y) - a(x)I f(x)klYD; V x e y e [ - a, a].

Then pathwise uniqueness holds for the equation (5.19).

PROOF. Observe first of all that if a(x) = 0, then (ii) and (iii) imply that

fe

cf2(x +

dy> 2f -2(x) h'(y)dy = oo, V e > O.

0

Thus, Z(a) 1(a).Suppose now that X(`) = {XI°, .F.,; 0 < t < co}; i = 1, 2, are solutions to

(5.19) relative to the same Brownian motion W = {W, 0 < t < co} onsome probability space (SI, .F, P), with P[XP = XVI = 1. Defining -c(l) =inf{t > 0; XP) e /(a)}, we recall from Remark 5.8 that XP) = 'CPL.; 0 < t < co,a.s., so it suffices to prove that

(5.40) xp) )02); 0 < t < T(1),

We set A, = XF) - ForFor each integer k > (1/a), there exists sk > 0for which 1,11c h-2 ) d = k. Using the continuity of the local time for A(Remark 3.7.8), (3.7.3), and assumption (iii), we may write for every randomtime r:

(5.41) A°(0) = lim - h-2(x)M(x)dxk a0

1 lik

k

t= lim - h-2(A)1 (As )(a(X(2)) - a(XP)))2 ds1 fsk 0

1 ft< lim -,, f 2 (XV)) ds, a.s. P.

k-,,,, PC 0

Now M, A XI" - 4 ) is in jr.loc, and so M admits the representationM, = B<m>, where B is a Brownian motion (Problem 3.4.7). Furthermore,

<M>, = focr2(X!,1))dv,

and 62(X,")) > 0 for 0 < v < -r"), so a change of variable results in the relation

<m>,

t f 2 (XV)) dv = f2041) B.)0.-2(A-,i)o + B,Jdu; 0 < t < -r").fo o

We take

T. A infIt > 0; p(X:1),I(o-)) -1n},

R A <M>,. = inf {s > 0; p(X0 + B5,1(0)) < ;11},

where p is given by (5.18). We alter f and a near 1(6) by setting

f(x) = o-(x) = 1; if p(x, /(o-)) <1

f(x) = f(x), o-(x) = o-(x); if p(x,l(u)) .._. 1,n

so f20-;2 is locally integrable at every point in 11. From the implication(iii) =. (v) in Proposition 3.6.27 and Problem 3.6.29, it follows that

r R,,

f 2 (XP)) dv = 1. f 2 (XW ) + B,Jo-- 2 (X W ) + B) du1o o

TR fn2(xW) + B.)0;2, ,I1)klio + Buidu < co, a.s.0

From (5.41) we see now that Azt(0) = 0; n > 1, a.s. Since t T -c" ) as n co, weconclude from the continuity of ANO) that A'.:`0)(0) = 0, a.s. P. Remark 3.7.8shows that E I X).,,i, - X:1,),i)1 = 0; 0 < t < oo, and (5.40) follows.

We recall from Corollary 3.23 that the existence of a weak solution andpathwise uniqueness imply strong existence. This leads to the following result.

5.10 Corollary. Under condition (E) of Theorem 5.4 and conditions (i)-(iii) ofTheorem 5.9, the equation (5.19) possesses a unique strong solution for everyinitial distribution p.

5.11 Remark. If the function f is locally bounded, condition (i) of Theorem 5.9follows directly from the definition of 1(6). We may take h(y) = y"; a > (1/2),and (iii) becomes the condition that a be locally Holder-continuous withexponent at least (1/2); see Examples 2.14 and 2.15.

B. The Method of Removal of Drift

It is time to return to the stochastic differential equation (5.1), in which a driftterm appears. We transform this equation so as to remove the drift and therebyreduce it to the case already studied. This reduction requires assumptions ofnondegeneracy and local integrability:

(ND)

(LI) V x e R, 3

a2 (x) > 0; V

> 0 such that

x e R,

x+(AI dy' b< co.

Under these assumptions, we fix a number c e R and define the scale function

(5.42) P(x) x exp { -2 14 b() 61} g x e R.a2g)

The function p has a continuous, strictly positive derivative, and p" existsalmost everywhere and satisfies

(5.43) p"(x) -2b(x)0.2(x) p'(x).

Henceforth, whenever we write p", we shall mean the (locally integrable)function defined by (5.43) on the entire of R; this definition is possible becauseof (ND).

The function p maps R onto (p( -co), p(co)) and has a continuously differ-entiable inverse q: (p( -co), p(oo)) -> R. This latter function has derivativeqf(y)= (1/ p'(q(y))), which is actually absolutely continuous, with

P(q(Y))" y))2

p'(q(Y))

2b(q(y)) 1a.e. in (p( -co), p(oo)).a2(q(y)) (q( y))) 2

We extend p to [-co, co] and q to [p(- co), p(co)] so that the resultingfunctions are continuous in the topology on the extended real number system.

(5.44)

5.12 Problem. Although not explicitly indicated by the notation, p(x) definedby (5.42) depends on the number c e R. Let us for the moment display thisdependence by writing pc(x). Show that

(5.45) Pa(x) = pa(c) + PL(c)Pc(x).

In particular, the finiteness (or nonfiniteness) of pc(+ co) does not depend onthe choice of c.

5.13 Proposition. Assume (ND) and (LI). A process X = {X Ft; 0 < t < oo}is a weak (or strong) solution of equation (5.1) if and only if the process

Y = Y p(X,), r#,; 0 < t < col is a weak (or strong) solution of

(5.46) = + f d(Ys)dWs; 0 < t < oo,

where

(5.47) p(-00) < Yo < p(00) a.s.,

{p'(q(y))6.(q(y)); p( -co) < y < p(co),(5.48) (y) =

0; otherwise.

The process X may explode in finite time, but the process Y does not.

PROOF. Let X satisfy (5.1) up to the explosion time S, define Y, = p(X,), andrecall S from (5.2); we obtain from the generalized Ito rule (Problem 3.7.3)and (5.43):

t 5,,

(5.49) LAS = P(X0) 6(Ys)dWs; 0 < t < co, n > 1.

Because X is a continuous, extended-real-valued process defined for all 0 <t < co (Definition 5.1) and p: [-co, co] [p(-oo),p(co)] is continuous, Y isalso continuous for all 0 < t < co. Hence, as n co, the left-hand side of(5.49) converges to Y n s, and the right-hand side must also have a limit. Inlight of Problem 3.4.11, this means that st0A s d2(Ys) ds < co; 0 < t < co, a.s.Because Xs, and hence Ys, is constant for S < s < co, we must in fact haveSto ei2(Ys)ds < co; 0 Lc. t < co, a.s. It follows that Po 6(Ys)d147, is defined for all0 < t < cc and is a continuous process. From (5.49) we see that (5.46) holdsfor 0 _5 t < S. This equality must also hold for 0 < t S on the set {S < co},because of continuity. The integrand a(Ys) vanishes for S < s < co, and so(5.46) in fact holds for all 0 < t < co. The solution of this equation cannotexplode in finite time; see Problem 5.3.

For the converse, let us begin with a process Y satisfying (5.46), (5.47) whichtakes values in [p( -co), p(oo)]. We can always arrange for Y to be constantafter reaching one of the endpoints of this interval, because the integrand in(5.46) vanishes when Ys (p( -co), p(oo)). Define X, A q(Y); 0 5 t < co, and letS be given by (5.2). Then the generalized Ito rule of Problem 3.7.3 gives, inconjunction with the properties of the function q (in particular, (5.44)) and(5.48),

1

Xt S,, = X0 ± (107,$)dY.s ±o

(1"117.0d<Y>s

tAS 1t A s- b(Xs) 1

ei(Ys)dW, + ii2(Y )ds= X°

+PVCs) fo 472(Xs) (P'1Xs02 3

tAs,, tAs,,= X0 + b(Xs) ds + cr(Xs)dWs; 0<t<co

almost surely, for every n 1.

5.14 Exercise. Show by example that if (LI) holds but (ND) fails, then (5.46)can have a solution Y which is not of the form p(X), for some solution X of(5.1 )

5.15 Theorem. Assume that a-2 is locally integrable at every point in R, andconditions (ND) and (LI) hold. Then for every initial distribution II, the equation(5.1) has a weak solution up to an explosion time, and this solution is unique inthe sense of probability law.

PROOF. Let 6 be defined by (5.48). According to Theorem 5.7 and Proposition5.13, it suffices to prove that 1(6) = 43). Now Z(6) = (p( -co), p(+ co)) ,̀ and1(6) contains this set. We must show that 6 -2 is locally integrable at everypoint yo e (p(-co), p(co)). At such a point, choose E > 0 so that

P( -Do) < Yo - E < yo + 6 <P(c30),

and write

fYo' dy 9(Y°+') dx

Y o 62(Y) - q(yo-e) P'(072(X)

The second integral is finite, because p' is bounded away from zero on finiteintervals and a-2 is locally integrable.

5.16 Corollary. Assume that a-2 is locally integrable at every point in R, andthat conditions (ND), (LI), and

I b(x) - b(y)I < K ix - yl; (x, y) e 082

I o-(x) - a(y)I -. h( I x - ye); (x, y) e R2

hold, where K is a positive constant and h: [0, co) - [0, co) is a strictly increasingfunction for which h(0) = 0 and (2.25) hold. Then, for every initial condition

independent of the driving Brownian motion W = {W, .Ft; 0 < t < co}, theequation (5.1) has a unique strong solution (possibly up to an explosion time).

(5.50)

PROOF. Weak existence (Theorem 5.15) and pathwise uniqueness (Proposition2.13 and Remark 3.3) imply strong existence (Corollary 3.23). These resultscan easily be localized to deal with the case of possible explosion of thesolution.

5.17 Proposition. Assume that b: R R is bounded and a: R -- R is Lipschitz-continuous with a2 bounded away from zero on every compact subset of 01.Then, for every initial condition independent of the driving Brownian motionW = {Hi, ,,; 0 5 t < co}, equation (5.1) has a nonexploding, unique strongsolution.

PROOF. We show first that the boundedness of b prevents the explosion of anysolution X to (5.1). We fix t e (0, co) and let n -, co in (5.4); the Lebesgue integral

Ito b(XN)1{,,s.} ds converges to .1:)^ s b(Xs)ds, a finite expression because bis bounded. On the other hand, the stochastic integral ro a(Xs)1{s<s} dWsconverges to ro^ s a(X.,)d W., on the event A ijbA S a 2 (Xs)ds < col, and doesA ,

not have a limit on A`; cf. Problem 3.4.11. It develops that the limit of theright-hand side of (5.4) exists and is finite a.s. on A, and does not exist on /1`.On the left-hand side of (5.4) we have lim, X, , s. = X, , s, which is defineda.s. and is equal to +co on {S < t }. It follows that P[S < t] = 0 holds forevery t > 0, so P[S = co] = 1.

We turn now to the questions of existence of a weak solution, and ofpathwise uniqueness, for equation (5.1). The assumptions on b and a imply(ND), (LI), and the local integrability of a', so weak existence follows fromTheorem 5.15. According to Proposition 5.13, the pathwise uniqueness of (5.1)is equivalent to that of (5.46). Because p" -(2b/o-2)p' is locally bounded,p' is locally Lipschitz. It follows that ii(y) defined by (5.48) is locally Lipschitzat every point y E(p( -oo), p(oo)). We have shown that any solution X to (5.1)does not explode, so any solution Y to (5.46), (5.47) must remain in the interval(p( -co), p(co)). Under these conditions, the proof given for Theorem 2.5 showsthat pathwise uniqueness holds for (5.46). We appeal to Corollary 3.23 in orderto conclude the argument.

Proposition 5.13 raises the interesting issue of determining necessary andsufficient conditions for explosion of the solution X to (5.1). Since Y givenby (5.46) does not explode, and Y, = p(X,), it is clear that the conditionp(+cc) = +oo guarantees that X is also nonexploding; however, this sufficientcondition is unfortunately not necessary (see Remark 5.18). We develop thenecessary and sufficient condition known as Feller's test for explosions inTheorem 5.29.

5.18 Remark. Consider the case of b(x) = sgn(x), a(x) a > 0. The scalefunction p of (5.42) is bounded, and according to Proposition 5.17 the equation(5.1) has a nonexploding, unique strong solution for any initial distribution.

5.19 Remark. The linear growth condition

I b(x)I + I a(x)I < K(1 + Ix I ); VX E R

is sufficient for P[S = co] = 1; cf. Problem 3.15.

C. Feller's Test for Explosions

We begin here a systematic discussion of explosions. Rather than workingexclusively with processes taking values on the entire real line, we start withan interval

1 = (1",r); -oo < i < r < co

and assume that the coefficients a: I R, b: I -4 F satisfy

(ND)' Q2(x) > 0; V x e l,

ix", + I b(y)I

J.-. a2(Y)

We define the scale function p by (5.42), where now the number c must bein I. We also introduce the speed measure

(LI)' V x e /, 3 e> 0 such that dy < co.

(5.51)2 dx

m(dx) 4p'(x)a 2 (X) '

xe I

and the Green's function

(5.52) Ga,b(x, .Y) =(p(x A y) - p(a))(p(b) - p(x v y))

; x, y E [a, b] g_ I.p(b) - p(a)

In terms of these two objects, a solution to the equation

(5.53) b(x)M'(x) + lo-2(x)M"(x) = - 1; a < x < b,

(5.54) M(a) = M(b) = 0

is given byto

(5.55) M,(x) A- i G,,,(x, y)m(dy)a

=

-1.

(P(x) - P(.0)m(dY) +P(x) P(a) f b (PO) - P(A)m(dY)a p(b) - p(a) a

As was the case with p", the second derivative Ma",6 exists except possibly ona set of Lebesgue measure zero; we define M,". at every point in (a, b) by using(5.53). Note that G., be , ), and hence Ma.b(), is nonnegative.

5.20 Definition. A weak solution in the interval I = (1', r) of equation (5.1) isa triple (X, W), (Si, ,F, P), {..-,}, where

(i)' condition (i) of Definition 5.1 holds,(ii)' X = {X ..*--,; 0 < t < co} is a continuous, adapted, [t, r]-valued process

with X0 e / a.s., and { W A; 0 < t < col is a standard, one-dimensionalBrownian motion,

(iii)' with {e}n°°_,, and {r},7,, strictly monotone sequences satisfying t < to <r < r, lim,,,, f = t, iim 00 rn = r, and

(5.56) S A inf{t z 0: X, it (e, r)}; n > 1,

the equations (5.3) and (5.4) hold.

We refer to

(5.57) S = inf { t _ _ 0: X, 0 (e , 0} = tun sn-co

as the exit time from I. The assumption X0 E I a.s. guarantees that P[S > 0] = 1.If = -co and r = +co, Definition 5.20 reduces to Definition 5.1, once westipulate that Xi = Xs; S < t < cc.

Let (X, W) be a weak solution in 1 of equation (5.1) with X0 = X E (a, b) I,and set

t = inf { t ._ 0: o-2(Xs)ds> n ; n = 1, 2, ...,o

Tab= inf {t >0;Xt (a, b)}; e < a < < r.We may apply the generalized Ito rule (Problem 3.7.3) to M,,,b(X,) and obtain

t A T A T.,b

Ma,b(Xt A T A T,,,b) = Ma,b(X) (t A Tr, A 1:7,0 ± J Nra,b(X.00-(Xs)do

Taking expectations and then letting n co, we see that

(5.58) E(t A Tad)) = Ma,b(X) Eltla,b(XtA T b) 5 M a b(X) < 00,,

and then letting t co we obtain ET,b < Ma.b(x) < co.In other words, X exits from every compact subinterval of ((,r) in finite

expected time. Armed with this observation, we may return to (5.58), observefrom (5.54) that limi, EM,,,b(Xt A Tab) = 0, and conclude

(5.59) ET,b = Ma,b(x); a < x < b.

On the other hand, the generalized Ito rule applied in the same way to p(X,)gives p(x) = Ep(X, A Tab), whence

(5.60) p(x) = Ep(XT.,,b)= p(a)P[XTo.b = a] + p(b)P[XT..b = b],

upon letting t co. The two probabilities in (5.60) add up to one, and thus

p(b) - p(x) p(x)- p(a)(5.61) P[XT = a] =

p(b)- p(a)'P[XT = b] =

a.13 p(b) - p(a)

These expressions will help us obtain information about the behavior of Xnear the endpoints of the interval (t,r) from the corresponding behavior ofthe scale function. Problem 5.12 shows that the expressions on the right-handsides of the relations in (5.61) do not depend on the choice of c in the definitionof p.

5.21 Remark. For Brownian motion Won I = (-co, co), we have (with c = 0in (5.42)) p(x) = x, m(dx) = 2 dx. For a process Y satisfying

= + d(K)diVs,0

we again have p(R) = g, but m(dx) = (2 dife12(50). Now Y is a Brownianmotion run "according to a different clock" (Theorem 3.4.6), and the speedmeasure simply records how this change of clock affects the expected value of

exit times:

(5.62) ET'a,s(5e) = f b ((x A - (5-c v y)) 2 di)

b - a (5:2

Once drift is introduced, the formulas become a bit more complicated, but theidea remains the same. Indeed, suppose that we begin with X satisfying (5.1),compute the scale function p and speed measure m for X by (5.42) and (5.51),and adopt the notation g = p(x), y = p(y), a = p(a), b = p(b); then (5.55), (5.59)show that ET,,,b(x) is still given by the right-hand side of (5.62), where now ifis the dispersion coefficient (5.48) of the process Y, A p(Xi). We say Y, = p(X,)is in the natural scale because it satisfies a stochastic differential equationwithout drift and thus has the identity as its scale function.

5.22 Proposition. Assume that (ND)', (LI)' hold, and let X be a weak solutionof (5.1) in I, with nonrandom initial condition X0 = x e I. Let p be given by (5.42)and S by (5.57). We distinguish four cases:

(a) pv +) = -co, p(r -) = co. Then

P[S = co] = P[ sup X, = = P[ inf X, = el= t.ot<co

In particular, the process X is recurrent: for every ye I, we have

P[X, = y; some 0 t < co] = 1.

(b) pv +)> -co, p(r -) = co. Then

P[lim X, = P[ sup X, < 7.1= 1.its

(c) p(1 +) = -co, p(r-) < co. Then

P[lim X, = P[ inf X, > 1.its 0<t<s

(d) p(e +) > -co, p(r -) < co. Then

P[lim X = (1=1 - P[lim X = r] =p(r -) - p(x)

its its P(r -) - p(e +).

5.23 Remark. In cases (b), (c), and (d), we make no claim concerning thefiniteness of S. Even in case (d), we may have P[S = co] = 1, as demonstratedin Remark 5.18.

PROOF OF PROPOSITION 5.22. For case (a), we have from (5.61) for 1' < a <x < b < r:

(5.63) p[ inf X1 ,,P[XT = = 1 - (p(x) /p(b))

o5.t<s

Letting b T r, we obtain P [info <,,s X, < a] = 1 for every a e /. Now we let a J eto get P[info ,,,s X, = = 1. A dual argument shows that P[supo <,<s X, =r] = 1. Suppose now that P[S < co] > 0; then the event {limas X, exists andis equal to e or r} has positive probability, and so {supo <,<5 X, = r} and{info <,<s X, = el cannot both have probability one. This contradiction showsthat P[S < oo] = 0.

For case (b), we first observe that (5.63) still implies P [info <,<5 X, = = 1.If, however, we recall P[X7...b = b] = (p(x) - p(a))/(p(b) - p(a)) from (5.61)and let a le, we see that

P[X, = b; some 0 < t < S] = P(x) - P(e+)P(b) - p(e +)

Letting now b T r, we conclude that P[suPo<i<s X, = r] = 0. We have thusshown

P[ inf X , . =-el= P[ sup Xt < ri = 1.ot<s 0<t<s

It remains only to show that limits X, = info <,<5 X and for this it suffices toestablish that the limit exists, almost surely. With S as in (5.56), the processYt(n) A P(XtAs,,) - +); 0 < t < co is for each n > 1 a nonnegative localmartingale (see (5.49)); letting n oo and using Fatou's lemma, we see thatY A p(X,,$) - p(e +); 0 < t < oo is a nonnegative supermartingale. As such,it converges almost surely as t oo (Problem 1.3.16). Because p: [e ,r)has a continuous inverse, lim,, X, AAS must exist.

Case (c) is dual to (b), and case (d) is obtained easily, by taking limits in(5.61). 0

5.24 Example. For the Brownian motion X, = ut + trig, on / = ( -co, co) withdrift u > 0 and variance a2 > 0, we have (setting c = 0 in (5.42)) that p(x) =(1 - e-Px)/13 and m(dx) = (2efix /0-2) dx, where /3 = 41/(72. We are in case (c).Compare this result with Exercise 3.5.9.

5.25 Example. For the Bessel process with dimension d > 2 (Proposition3.3.21), we have I = (0, co), b(x) = (d - 1)/2x, and a2(x) = 1. With c = 1, weobtain

(i) for d = 2: p(x) = log x, m(dx) = 2x dx (case (a)),(ii) for d > 3: p(x) = (1 - x')/(d - 2), m(dx) = 2x" dx (case (c)).

Compare these results with Problem 3.3.23.

Proposition 5.17, Remark 5.19, and part (a) of Proposition 5.22 providesufficient conditions for nonexplosion of the process X in (5.1), i.e., forP[S = co] = 1. In our search for conditions which are both necessary andsufficient, we shall need the following result about an ordinary differentialequation.

We define, by recursion, the sequence fun }`°_,) of real-valued functions on /,by setting u, ar 1 and

x Y

(5.64) u,,(x) = fc p'(y) i u_1(z)m(dz)dy; X E 1, n > 1c

where, as before, c is a fixed number in I. In particular we set for x e /:

TY

p ( )o-

2 dz(5.65) v(x) A u1 (x) = .1 x p'(y)

(z)dy = f x

(p(x) - p(y))m(dy), z ,

5.26 Lemma. Assume that (ND)' and (LI)' hold. The seriesCO

(5.66) u(x) = > un(x); x e In=0

converges uniformly on compact subsets of I and defines a differentiable functionwith absolutely continuous derivative on I. Furthermore, u is strictly increasing(decreasing) in the interval (c, r) (respectively, (e, c)) and satisfies

(5.67)

(5.68)

as well as

(5.69)

Icr2(x)u"(x) + b(x)u'(x) = u(x); a.e. x el,

u(c) = 1, u'(c) = 0,

1 + v(x) < u(x) < ev(x); x e /.

PROOF. It is verified easily that the functions funI,T_I in (5.64) are nonnegative,are strictly increasing (decreasing) on (c, r) (respectively, v, 0 , and satisfy

(5.70) lo-2(x)14(x) + b(x)u'(x) = un_1(x), a.e. xe I.

We show by induction that

v"(x)(5.71) u(x) -_ ; n = 0, 1, 2, ...

n!

Indeed, (5.71) is valid for n = 0; assuming it is true for n = k - 1 and notingthat

x

(5.72) uk(x) = p'(x) f uk_1(z)m(dz); x e I,c

we obtain for c < x < r:x

(k -

y vk-1 2)1 fx

13Uk(X) i '01) M(dZ)dy(k - 1)! ,

p'(y)vk-1(y)m((c, Ody

=(k 1)!

1 .1.x 0 (y)dv(y) =vk(kx)

.- , !

A similar inequality holds for e < x 5 c. This proves (5.71), and from (5.72)

348

we have also

5. Stochastic Differential Equations

v"-1(x)lu.(x)1

(n - 1)!; n = 1, 2, ... .

It follows that the series in (5.66), as well as I,T=c, u,,'(x), converges absolutelyon I, uniformly on compact subsets. Solving (5.70) for u;; (x), we see thatEnTh u;; (x) also converges absolutely, at each point x e /, to an integrablefunction. Term-by-term integration of this sum shows that E`°_, u;; (x) isalmost everywhere the second derivative of u in (5.66), and that u'(x) =

u'(x) holds for every x e I. The other claims follow readily.

5.27 Problem. Prove the implications

(5.73) p(r -) = co v(r -) = co,

(5.74) p(( +) = -co .v(e+) = co.

5.28 Problem. In the spirit of Problem 5.12, we could display the dependenceof v(x) on c by writing

x v 2 dz(5.75) yc(x) -4' PjY) dy.

P:(z)o- 2 (Z)

Show that for a, c e 1:

(5.76) va(x) = v(c) + vL(c)p(x) + vc(x); x e I.

In particular, the finiteness or nonfiniteness of vc(r -), vc(i + ) does not dependon c.

5.29 Theorem (Feller's (1952) Test for Explosions). Assume that (ND)' and(LI)' hold, and let (X, W), be a weak solution in I = (e,r) of(5.1) with nonrandom initial condition X0 = zeI. Then P[S = co] = 1 orP[S = co] < 1, according to whether

(5.77) v(( +) = v(r-) = co

or not.

PROOF. Set r A inf {t > 0: yo o-2(Xs)ds> n} and Z") A u(X, A sn n). Accordingto the generalized Ito rule (Problem 3.7.3) and relation (5.67), Z(") has therepresentation

(X0ds +4") = Z,(3") + u u'(Xs)o-(X0dWs.To

Consequently, MP) A e-t A S" 2" 4") has the representationtA5, Ar,,

Min) = Mg e-su'(Xs)o-(Xs)dIgs0

as a nonnegative local martingale. Fatou's lemma shows that any nonnegativelocal martingale is a supermartingale, and that this property is also enjoyedby the process M, A lim, MP) = e-tAsu(X,); 0 5 t < cc. Therefore,

(5.78) Mx = lim e-t^su(X, As)

exists and is finite, almost surely (Problem 1.3.16).Let us now suppose that (5.77) holds. From (5.69) we see that u(f + ) =

u(r-) = co, and (5.78) shows that Mx = oe a.s. on the event {S < }. Itfollows that P[S < oo] = 0.

For the converse, assume that (5.77) fails; for instance, suppose that v(r -) <cc. Then (5.69) yields u(r-) < oo. In light of Problem 5.28, we may assumewithout loss of generality that c < x < r, and set = inf{t > 0; X, = c }. Thecontinuous process

ME A T.,. =e-(t A A Tc)I4(X

t ASA Tc); < t < GO

is a bounded local martingale, hence a bounded martingale, which thereforeconverges almost surely (and in 1,1) as t co; cf. Problem 1.3.20. It developsthat

-U(X) = Ee-S A T`U(XS AT) = u(r-)Ee- s _1 {s<Tc} + u(c)Ee Tc 1 {T<s}.

If P[S = co] = 1, the preceding identity gives u(x) = u(c)Ee-T` < u(c), con-tradicting the fact that u is strictly increasing on [c, x]. It follows thatP[S co] < 1.

5.30 Example. For Brownian motion W on 1 = (-oo, co), we have alreadycomputed p(x) = x, m(dx) = 2 dx (with c = 0). Consequently, v(x) = x2 andv(+oo) = oo.

5.24 Example (continued). For Brownian motion with drift it > 0 and vari-ance 62 > 0, it turns out that v(x) = 2(fix - 1 + e- tot)332a2, with /3 _ (2/1/62)

and c = 0. Again, v(+oo) = co.

5.25 Example (continued). For the Bessel process with dimension d > 2and c = 1, we have v(x) = [x2 - 1 - 2p(x)]/d; 0 < x < co, and v(0 + ) =v(oo) = oo.

5.31 Exercise. Consider the geometric Brownian motion X, which satisfies thestochastic integral equation

(5.79) X,=x+ttf Xsds + v Xsdl/K,

where x > 0. Use Theorem 5.29 and Proposition 5.22 to show that X, e (0, oo)for all t, and

(i) if it < v2/2, then X, = 0, suPo < X, < oo, a.s.;(ii) if it > v2/2, then info.<,,, X, > 0, X, = co, a.s.;(iii) if it= v2/2, then info <,<, X, = 0, supo <t. X, = (x), a.s.

(The solution to (5.79) is given by

X, = xexpt(it - iv2)t + vW,I,

and so (i)-(iii) can also be deduced from the properties of Brownian motionwith drift (see, e.g., Problem 2.9.3 and Proposition 2.9.23).)

Let us suppose now that (5.77) is violated, so that P[S < co] is positive.Under what additional conditions can we guarantee that this probability isactually equal to one, i.e., that explosion occurs almost surely?

5.32 Proposition. Assume that (ND)' and (LI)' hold. We have P[S < co] = 1if and only if one of the following conditions holds:

(i) v(r -) < co and v(e +) < co,(ii) v(r -) < co and p(e +) = -co, or

(iii) vv +) < co and p(r-) = co.

In the first case, we actually have ES < co.

5.33 Remark. If (i) prevails, then we also have finiteness of p(r-) and p(e +)(Problem 5.27), and we can define by analogy with (5.52) the Green's function

G(x, y) = [P1x A y) - p(e +)][P(r -) - P1x v .0]; (x, y)e I2

I* -) - XI' +)for the entire interval I = (C, r). We also define the counterpart of (5.55):

M(x) = f,

G(x, y)m(dy); x e(e,r).e

This function satisfies M(e+) = M(r-) = 0, has an absolutely continuousderivative on I, and satisfies the equation (5.53) there. The same procedurethat led to (5.59) now gives ES = M(x) < co, justifying the last claim inProposition 5.32.

PROOF OF SUFFICIENCY IN PROPOSITION 5.32. We just dealt with (i). Supposethat (ii) holds; then with {e},T=, and {r};,°=, as in Definition 5.20 (iii)', weintroduce the stopping times

R g inftt 0; X, = enl; n 1,

T, . g inf tt 0; X, = /1, 'Te -A inftt 0; X, = el = iim Rn.n-,co

Because v(r -) < oo, v( e ) < co, we obtain as in Remark 5.33 that E(R,, A Tr) <co; n 1. On the other hand, (5.73) gives p(r -) < co, so we are in the case (c)of Proposition 5.22. Therefore, for P-a.e. o.) e D, R(a)) = co for sufficientlylarge n (depending on w), and thus S(w) = 'T,.(a.)) < co.

Condition (iii) is dual to (ii). 0

PROOF OF NECESSITY IN PROPOSITION 5.32. Assume that P[S < co] = 1; fromTheorem 5.29 we conclude that either v(( +) < co or v(r -) < co. Let ussuppose v(( + ) < co, and that none of (i), (ii), (iii) holds. Then necessarilyp(r -) < oo = v(r -) and p(( + ) > -co (remember (5.74)), so we are in case(d) of Proposition 5.22, and thus A, g {limo., X, = r} has positive probability.But now we recall from the proof of Theorem 5.29 that M, = e-` ^su(X,)is a nonnegative supermartingale with M° of (5.78) an almost surely finiterandom variable. According to (5.69), u(r -) = co, and so S = co on A,. Thisshows that P[S < co] < 1, contradicting our initial assumption. It followsthat at least one of (i), (ii), (iii) must hold, if P[S < co] = 1 does. 1=1

D. Supplementary Exercises

5.34 Exercise. Take o(x) 1 and I = (-co, co).

(a) If b(x) = lx2, show that P[S < co] = 1.(b) If b(x) = 2x3, show that ES < oo.

5.35 Exercise. Take I = (0, oo) and b(x) = k, o(x) = where k and e arepositive constants.

(i) Show that we are in case (a), (b), or (c) of Proposition 5.22 according asit A (2k/e) is equal to, less than, or greater than one, respectively.

(ii) In the first and third cases, the solution is nonexploding; in the secondcase the origin is reached in finite time with probability one.

(iii) The cases e = 2, k = 2, 3, ... should be familiar; can you relate the solutionto a known process?

(iv) Solve the stochastic differential equation explicitly in the case e2 = 4k.(Hint: Recall Proposition 2.21.)

5.36 Exercise. Show that the solution of the equation

7IdX, = (1 + X,)(1 + X,2)dt + (1 + X,2)dW,; X0e(- -2' )'

explodes (to +co) in finite expected time.

5.37 Exercise. Discuss the possibility of explosion in the cases:

(i) b(x) - x, o(x) = + x2), I = R;(ii) b(x) = o-(x) = Sx, / = (0, co), and e R, > 0;(iii) b(x) = 1 + o-(x) = 6x, I = (0, oo), and ye R, S > 0.

(Hint for (iii): Recall Exercise 3.3.33.)

5.38 Exercise. Let the function b: R R be locally square-integrable, i.e.,V x e R, 3e > 0 such that P,`,!:b2(y)dy < co.

(0 Show that, corresponding to every initial distribution /4. on (FR, R(R)), theequation

dX, = b(X,) dt + dW,

has a weak solution up to an explosion time S, and this solution is uniquein the sense of probability law.

(ii) Prove that for every finite T > 0, this solution obeys the generalizedGirsanov formula:

T

P[X e S > T] = E[exp (f b(W) digs --2

b2(1Vds)- 1{w.e r}1

for every I" earT(c[o, oon, the a-field of (3.19) with m = 1. Here, W is aone-dimensional Brownian motion with P(Wo e B) = p(B); B ER(R).(Hint: Try to emulate the proof of Proposition 3.10, and recall theEngelbert-Schmidt zero-one law of Section 3.6.)

(iii) In particular, conclude that with W as in (ii), the nonnegative super-martingale

1 fZ, = exp b(Ws) dills -- b2(W)ds}; 0 t < co,

2 0

is a martingale if and only if P[S = co] = 1; this is equivalent tocp( + co) = co, where

cp(x) = fx 1 Y Yexp { -2 I b(u) du} dz dy.o o

5.39 Exercise. In the setting of Subsection C, show that for every bounded,piecewise continuous function h: I -> [0, co) and x e (a, b) 1, we have

To. I,

Ex h(X,) dt = Ga,b(x, y)h(y)m(dy)o a

Here and in the following exercises we denote by a superscript x on probabilitiesand/or expectations the initial condition X0 = x. (Hint: Proceed by analogywith (5.55), (5.59).)

5.40 Exercise (Pollack & Siegmund (1985)). In the setting of Subsection Cwith b and a bounded on compact subintervals of I and with the scale functionp(x) and the speed measure m(dx) satisfying

p(e+) = -co, p(r-) = oo, m(I) < co,

the solution X to (5.1) starting at x e I never exits I (Proposition 5.22). Weassume that

(5.80) Px(X, = z) = 0; V x, z e /, t > 0.

Show in the following steps that the normalized speed measure (m(dx)/m(1))is the limiting distribution of X:

((i, z ))(5.81) lirn Px(X, < z) =

m; x, z e I.

m(/)

(i) Introduce the stopping times T, = inf {t 0; X, = a}, Sod, = inf { t Ta;

X, = b} for a, b e I. Deduce from the assumptions and (5.59) the positiverecurrence properties, with e<a<x<u<b< r:

Ex T = -f (p(x) - p(y))m(dy) + (p(x) - p(a))- m((a, r)) <a

Ex Tb = (P(Y) P(x))m(dY) + (P(b) - P(x))' m(((, b)) < co,x

ExS,x = Ex T + E"Tx = (p(u) - p(x))- m(I) < co.

(ii) Show that the same methodology as in (i) allows us to conclude, with thehelp of Exercise 5.39:

s,Ex 1 1 (e, z)(X f)dt = (p(u) - p(x)) m((e , z)); V z e I.

Jo

(iii) With the aid of assumption (5.80), show that for e<x<u<r and z e I,z x, the function

a(t) Px(X, < z, S,x > t); 0 t < co,

is continuous and D,°-1 max.-1 s ,< a(t) < co. This last condition impliesdirect Riemann integrability of a() (see Feller (1971), Chapter XI,Section 1).

(iv) Establish a renewal-type equation

F(t) = a(t) + ft F(t - s)v(ds); 0 < t < co

for the function F(t) s Px(X, < z), with appropriate measure v(dt) on[0, co). Show that the renewal theorem (Feller (1971), Chapter XI) implies(5.81).

5.41 Exercise (Le Gall (1983)). Provide a proof of Proposition 2.13 based onsemimartingale local time.

(Hint: Apply the Tanaka-Meyer formula (3.7.9) to the difference X ±4. X(1) -X(2' between two solutions X(1), X(2) of (2.1). Use Exercise 3.7.12 to show thatthe local time at the origin for X is identically zero, almost surely.)

5.42 Exercise (Le Gall (1983)). Suppose that uniqueness in the sense ofprobability law holds for (2.1), and that for any two solutions X(1), X(2) on thesame probability space and with respect to the same Brownian motion andinitial condition, the local time of X = X(1) - X(2) at the origin is identicallyequal to zero, almost surely. Then pathwise uniqueness holds for (2.1).

(Hint: Use a Tanaka-Meyer formula to show that X(1) v X(2) also solves(2.1).)

5.6. Linear Equations

In this section we consider d-dimensional stochastic differential equations inwhich the solution process enters linearly. Such processes arise in estimationand control of linear systems, in economics (see Section 5.8), and in variousother fields. As we shall see, one can provide a fairly explicit representationfor the solution of a linear stochastic differential equation.

For most of this section, we study the equation

(6.1) dX, = [A(t)X, + a(t)] dt + a(t)dW, 0 < t < oo,

X, =

where W is an r-dimensional Brownian motion independent of the d-dimensional initial vector and the (d x d), (d x 1) and (d x r) matrices A(t),a(t), and a(t) are nonrandom, measurable, and locally bounded. In Problem6.15 we generalize (6.1) for one-dimensional equations by allowing the solutionX also to appear in the dispersion coefficient.

The deterministic equation corresponding to (6.1) is

(6.2) (t) = A(t)l;(t) + a(t); (0) =

Standard existence and uniqueness results (Hale (1969), Section 1.5) imply thatfor every initial condition E Rd, (6.2) has an absolutely continuous solutionW) defined for 0 < t < oo. Likewise, the matrix differential equation

(6.3) 4(t) = A(t)1(t), 0(0) = I

has a unique (absolutely continuous) solution defined for 0 < t < co. (Here /is the (d x d) identity matrix.) This matrix function b is called the fundamentalsolution to the homogeneous equation

(6.4) (t) = A(t)c(t).

For each t > 0, the matrix C(t) is nonsingular, for otherwise there would bea to > 0 and a nonzero vector Ae Old such that (1)(t0)). = 0. But t(t)A is asolution to (6.4), and since the identically zero function is the unique solutionwhich vanishes at to, we must have 4:I)(t)). = 0 for all t. This would contradictthe initial condition OM = I.

In terms of (I), the solution of the deterministic equation (6.2) is simply

(6.5) (t) = (1)(t) (0) + J (I)-1 (s)a(s)ds].0

(See Hale (1969), Chapter 3, for additional information.)A pleasant fact is that the solution of (6.1) has a representation similar to

(6.5). Indeed, it is easily verified by Ito's rule that 7

(6.6) X, A 4)(t) [X, + j'' 4I (s)a(s) ds + (V' (s)6(s)d141,1; 0 < t < co

solves (6.1). Pathwise uniqueness for equation (6.1) follows from Theorem 2.5.

5.6. Linear Equations 355

6.1 Problem. Suppose that EIRC0112 < co, and introduce the mean vector andcovariance matrix functions

(6.7) m(t) A EX

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

(6.9) V(t) A- p(t, t).

Show that

(6.10) m(t) = OW [m(0) + J i (1)-1(s)a(s)dsi,o

SAt

(6.11) p(s, t) = col- V(0) + 01)'(u)o-(u)(0-1(u)o-(u))T du]f:DT (t),L w0

hold for every 0 < s, t < co. In particular, m(t) and V(t) solve the linearequations

(6.12) rh(t) = A(t)m(t) + a(t),

(6.13) V(t) = A(t)V(t) + V(t)AT(t) + a(t)aT(t).

6.2 Problem. Show that if X0 has a d-variate normal distribution, then X isa Gaussian process (Definition 2.9.1).

A. Gauss-Markov Processes

If X0 is normally distributed, then the finite-dimensional distributions of theGaussian process X in (6.6) are completely determined by the mean andcovariance functions. In this case, we would like to know under what addi-tional conditions we can guarantee the nondegeneracy of the distribution ofX i.e., the positive definiteness of the matrix

(6.14) V(t) = OW [V(0) + f t (I)-1(u)o-(u)(4)-1(u)a(u))T dui (DT (t),o

for every t > 0. In order to settle this question, we shall introduce the conceptof controllability from linear system theory.

6.3 Definition. The pair of matrix-valued, measurable, locally bounded func-tions (A, a) is called controllable on [0, T] if for every pair x, y E Rd, there existsa measurable, bounded function v: [0, T] -> Rr, such that

(6.15) Y(t) = x + ft A(s)Y (s) ds + ..1: cr(s)v(s) ds; 0 t < T,o o

satisfies Y(T) = y. In other words, for every pair x, y e Rd, there exists a controlfunction v) which steers the linear system (6.15) from Y(0) = x to Y(T) = y.

6.4 Proposition. The pair of functions (A, a) is controllable on [0, T] if and onlyif the matrix

(6.16) M(T) A f 41)-'(t)o-(t)(0-1(t)o-(t))T dt

is nonsingular.

PROOF. Let G(t) = (1)-1(t)o-(t). From (6.5), the solution to (6.15) is

Y(t) = (I)(t)[x + J G(s)v(s)dsi,I)

and because of the nonsingularity of (1)(T), controllability is equivalent to thecondition that G(s) v(s) ds range over all of OrI as v ranges over the bounded,measurable functions from [0, T] to Or.

Choose an arbitrary z e Old. Under the assumption of nonsingularity ofM(T), we may set v(s) = GT (s)M-1 (T)z, and then we have z = G(s)v(s) ds.On the other hand, if M(T) is singular, then there exists a nonzero z EIR° suchthat zT M(T)z = 0, i.e., SI; z TG(s)GT(s)z ds = 0, which shows that zT G(s) = 0for Lebesgue-almost every se [0, T]. Consequently, 2T g G(s)v(s) ds = 0 forany bounded, measurable v, which contradicts controllability.

We see from its definition that V(0) is positive semidefinite, so the non-singularity (and hence the positive-definiteness) of M(T) implies the sameproperty for V(T). We obtain thereby the following result: if the pair (A, a) iscontrollable on [0, T], then the matrix V(T) of (6.14) is nonsingular. A bit ofreflection shows that the two conditions are actually equivalent, providedV(0) = 0, i.e., X0 in (6.1) is nonrandom.

When the matrices A and a appearing in (6.1) are constant, this result takesa more explicit form. In this case, the fundamental solution to (6.4) is

(6.17)tn(Dm etA A E An

n=0 n!

and controllability reduces to the following rank condition.

6.5 Proposition. The pair of constant matrices (A, a) is controllable (on anyinterval [0, T]) if and only if the (d x d) controllability matrix C A [a, Ao-,A2 Ad-10] has rank d.

PROOF. Let us first assume that rank(C) < d, and show that M(T) givenby (6.16) is singular for every 0 < T < co. The assumption rank(C) < d isequivalent to the existence of a nonzero vector z a l for which

(6.18) ZT = ZT Acr = ZT A20- = = Z T Ad-1 Cr = O.

By the Hamilton-Cayley theorem, A satisfies its characteristic equation; thus,

every positive, integral power of A can be written as a linear combination of/, A, It follows from (6.18) that z TAna = 0; n > 0, and thereby

z7-0-1 zT e-tA E (-trzTA"o-= 0; t > 0.

n=0 n!

The singularity of M(T) follows from zT M(T)z = 0.

Let us now assume that for some T > 0, M(T) is singular and show thatrank (C) < d. The singularity of M( T)enables us to find a nonzero vector z E Rd

such that

0 = zT M(T)z = f lIzTe-mo-112 dt.

From this we see that f(t) zT e-"`a is identically zero on [0, T], andevaluating f(0), f'(0),...,f(d-1)(0) we obtain (6.18).

When A and a are constant, equations (6.13) and (6.14) take the simplifiedform

(6.13)' V(t) = AV(t) + V(t)AT aaT,

ll(6.14)' V(t) = e" [V(0) + --sAcro.re-sAT ds etAT.

One could hope that by proper choice of V(0), it would be possible to obtaina constant solution to these equations. Under the assumption that all theeigenvalues of A have negative real parts, so that the integral

(6.19) V A esito.o.resAT ds

converges, one can verify that V(t) V does indeed solve (6.13)', (6.14)'. Weleave this verification as a problem for the reader.

6.6 Problem. Show that if V(0) in (6.14)' is given by (6.19), then V(t) = V(0).In particular, V of (6.19) satisfies the algebraic matrix equation

(6.20) AV + VAT = --o-o-T.

We have established the following result.

6.7 Theorem. Suppose in the stochastic differential equation (6.1) that a(t) a a,a(t) 0, all the eigenvalues of A(t) A have negative real parts, and the initialrandom vector = X0 has a d-variate normal distribution with mean m(0) = 0and covariance V = E(X0.11) as in (6.19). Then the solution X is a stationary,zero-mean Gaussian process, with covariance function

(e("-"V;0 < t < s < op

(6.21) p(s, =Ve(`-"T; 0 5 s < t < co.

PROOF. We have already seen that V(t) = V; (6.21) follows from (6.14)' and(6.11).

6.8 Example (The Ornstein-Uhlenbeck Process). In the case d = r = 1, a(t) .m-0, A(t) = -a < 0, and a(t) a > 0, (6.1) gives the oldest example

(6.22) dX, = -aXidt + a dW,

of a stochastic differential equation (Uhlenbeck & Ornstein (1930), Doob(1942), Wang & Uhlenbeck (1945)). This corresponds to the Langevin (1908)equation for the Brownian motion of a particle with friction. According to(6.6), the solution of this equation is

ftX, = X 0e-Ga + o- e-2"-s) dWs; 0 < t < co.o

If EX,!, < co, the expectation, variance, and covariance functions in (6.7)-(6.9)become

m(t) A- EX, = m(0)e',0.2 0.2

V(t) A Var(X,) = + (V(0) - -2a) e-2',

0.2

p(s,t) A Cov(X Xt) = [V(0) + -2a

(e2'(` ^ s) - Me-2(f').

If the initial random variable X0 has a normal distribution with mean zeroand variance (a2/2a), then X is a stationary, zero-mean Gaussian process withcovariance function p(s, t) = (a2/2a)e-'1".

B. Brownian Bridge

Let us consider now the one-dimensional equation

T X,(6.23) dX, =

Tdt + dWi; 0 < t < T, and Xo = a,

t

for given real numbers a, b and T > 0. This is of the form (6.1) with A(t) =-11(T - t), a(t) = b /(T - t) and a(t) ._ 1, whence OW = 1 - (t /T). From (6.6)we have

t ) t dig'T.X,=a(1--t)+b-T +(T -Of

0' Ts; 0<t<

6.9 Lemma. The process

{(T - t)(6.24) X =

0;

' digs

f Cl-t<T"t = T,

T s;

is continuous, zero-mean, and Gaussian, with covariance function

(6.25) p(s, t) = (s A t) 0 < S, t < T

PROOF. The process M, f o (T - s)-1 dWs; 0 < t < T is a continuous, square-integrable martingale with quadratic variation

o ds 1 1

<M>t 0 (T - s)2 T - t T

According to Theorem 3.4.6, there exists a standard, one-dimensionalBrownian motion B such that M, = Boot; 0 < t < T It follows from thestrong law of large numbers for Brownian motion (Problem 2.9.3) that Y, -4B0011(<M>, + T-1) converges almost surely to zero as t T T The process Yof (6.24) is thus seen to be almost surely continuous on [0, T], and to be azero-mean Gaussian process with

tAs du stE(Y,Y,) = (T - t)(T - s) f

(T - u)2= (s t)--T,

o

provided 0 < s,t<TIfsv t = T, the preceding expectation is trivially zero,in agreement with (6.25).

6.10 Corollary. The process

(6.26) (I -T-t ) + b-t

T+ (T - t) fo T

0- <t<T-ss '

' dW

b; t = T,

is Gaussian with a.s. continuous paths, expectation function

(6.27) m(t) A E(X,) = a 1 - -t + bT-t

'0 < t < T

and covariance function p(s, t) given by (6.25). This process is the pathwise uniquesolution of equation (6.23) on [0, T).

6.11 Problem. Show that the finite-dimensional distributions for the processX in (6.26) are given by

(6.28) P[X,e dx,, , X,. e dxn]

p(T - t; xn, b)= p(ti - ti-i; x,- xi) dx, dx,,,

t=i p(T; a, b)

where 0 = to < t, < < t < T, x, = a, (x,, , xn)e ER", and p(t; x, y) is theGaussian kernel (2.2.6). (Hint: The normal random variables Zr Xi,/(T - ti) - X, i/(T - 4_1); i = 1, . . . ,n, are independent.)

6.12 Definition. A Brownian bridge from a to b on [0, T] is any almost surelycontinuous process defined on [0, T], with finite dimensional distributionsspecified by (6.28).

It is apparent that any continuous, Gaussian process on [0, T] with meanand covariance functions specified by (6.27) and (6.25), respectively, is aBrownian bridge from a to b. Besides the representation of Brownian bridgeas the solution to (6.23), there are two other ways of thinking about thisprocess; they appear in the next two problems.

6.13 Problem. Let {l4',.; 0 < t < cm}, (S2,,), {Pa}ac n be a one-dimensionalBrownian family. Show that for 0 = to < t, < < t,, < T, x0 = a, and(x ... , x.) e IV% the conditional finite-dimensional distributions I [W,, edx"..., W. e dx,,IWT = b] are given by the right-hand side of (6.28), forLebesgue-almost every be P. In other words, Brownian bridge from a to b on[0, T] is Brownian motion started at a and conditioned to arrive at b at time T.

6.14 Problem. Let W be a standard, one-dimensional Brownian motion anddefine

(6.29) Brb A a 1( t t t0 < t < 'T.

Then Barb is a Brownian bridge from a to b on [0, T].

C. The General, One-Dimensional Linear Equation

Let us consider the one-dimensional (d = 1, r > 1) stochastic differentialequation

(6.30) dX, = (t) X, + a(t)] dt + [S;(t)X, + o-j(t)]i =i

where W = {141, = (Him, , W,(6), 0 < t < co} is an r-dimensionalBrownian motion, and the coefficients A, a, Si, oi are measurable, {. }-

adapted, almost surely locally bounded processes. We set

(6.31)ct

ift

1 r

S(u)dW°)"

- -J=1 o 2 j=1

Zt 4- exp [ft A(u) du +

ToS.12(u)du,

6.15 Problem. Show that the unique solution of equation (6.30) is

Wol(6.32) X, = Z, [X0 + f 1

- la(u) - S;(u)o-,(u)} du +0 1=1 .1=1 f af(u)d

Zu "

In particular, the solution of the equation

(6.33) dX, = A(t)X,dt + S;(t)X,dW,u)J=1

is given by

1 ,(6.34) X, = Xoexp[f {A(u) - Si (u)} au 2, Si(u) dW,,u)].

.1= f0

In the case of constant coefficients A(t) = A, S;(t) S; with 2A < E;=, s.,2 in(6.34), show that lim X, = 0 a.s., for arbitrary initial condition Xo.

D. Supplementary Exercises

6.16 Exercise. Write down the stochastic differential equation satisfied byY = X1`; 0 < t < co, with k > 1 arbitrary but fixed and X the solution ofequation (5.79). Use your equation to compute E(X,k).

6.17 Exercise. Define the d-dimensional Brownian bridge from a to b on [0, T](a, b e Rd) to be any almost surely continuous process defined on [0, T], withfinite dimensional distributions specified by (6.28), where now

p(t; x, y) = (27rt)-(d/2) expiix Yil 2

2t Jx, y e t > O.

(i) Prove that the processes X given by (6.26) and BO given by (6.29),where W is a d-dimensional Brownian motion with Wo = 0 a.s., ared-dimensional Brownian bridges from a to b on [0, T].

(ii) Prove that the d-dimensional processes fig'; 0 < t < T} and { Br: ;0 < t < T} have the same law.

(iii) Show that for any bounded, measurable function F: C[0, T]' R, wehave

(6.35) EF(a + W.) = f EF(B!')p(T; a, b)db.ud

6.18 Exercise. Let (I): R be of class C2 with bounded second partialderivatives and bounded gradient V(I), and consider the Smoluchowski equa-tion

(6.36) dX, = V(1)(X,)dt + dWi; 0 < t < co,

where W is a standard, Rd-valued Brownian motion. According to Theorems2.5, 2.9 and Problem 2.12, this equation admits a unique strong solution forevery initial distribution on Xo. Show that the measure

(6.37) u(dx) = e2G(x)dx on a(Rd)

is invariant for (6.36); i.e., if X(') is the unique strong solution of (6.36) withinitial condition Xr = a e Rd, then

(6.38) µ(A) = f P(X:")e A) p(da); V A e M(0')

holds for every 0 < t <(Hint: From Corollary 3.11 and the Ito rule, we have

(6.39) Ef Ma)) = E[f(a + exp Istro(a + - (1)(a)

1 f- (AO + IIV(1)112)(a + ds}1

for every f e C43(01°). Now use Exercise 6.17 (ii), (iii) and Problem 4.25.)

6.19 Remark. If d = 1 in Exercise 6.18, the speed measure of the process X isgiven by m(dx) = 2 exp{ - 20(c)}tt(dx) and is therefore invariant. RecallExercise 5.40.

6.20 Exercise (The Brownian Oscillator). Consider the Langevin system

d X, = Y, dt

dY, = - X, dt - aY, dt + o- dW

where W is a standard, one-dimensional Brownian motion and /3, a, and a arepositive constants.

(i) Solve this system explicitly.(ii) Show that if (X0, Yo) has an appropriate Gaussian distribution, then

(X 171) is a stationary Gaussian process.(iii) Compute the covariance function of this stationary Gaussian process.

6.21 Exercise. Consider the one-dimensional equation (6.1) with a(t) = 0,a(t) a > 0, A(t) < -a < 0, V 0 t < cc, and = x e R. Show that

0.2 0.2

E X,2 <2a

x22a

e-2t; Vt > O.

6.22 Exercise. Let W = = (W,(1), Wt(r)),,t; 0 < t < co} be an r-dimensional Brownian motion, and let A(t), S(P)(t); p = 1, . . . , r, be adapted,bounded, (d x d) matrix-valued processes on [0, T]. Then the matrix stochas-

5.7. Connections with Partial Differential Equations 363

tic integral equation

(6.40) X(t) = I + f A(s) X (s)ds + J t S(P)(s)X (s)dW,(P)p=1 o

has a unique, strong solution (Theorems 2.5, 2.9). The componentwise for-mulation of (6.40) is

d

Xki(t) = oki + Et

A ke(s)X o(s) ds + E Ed

snsp,;(s)dws(P)C =1 p=11=1 0

.

Show that X(t) has an inverse, which satisfies

(6.41) X-1(t) = I + I X -1(s)[ (S(P)(s))2 - A(s)idso P -1

-E f x-i(s)s(P)(s) d HIP)P =1 0

5.7. Connections with Partial Differential Equations

The connections between Brownian motion on one hand, and the Dirichletand Cauchy problems (for the Poisson and heat equations, respectively) onthe other, were explored at some length in Chapter 4. In this section wedocument analogous connections between solutions of stochastic differentialequations, and the Dirichlet and Cauchy problems for the associated, moregeneral elliptic and parabolic equations. Such connections have already beenpresaged in Section 4 of this chapter, in the prominent role played there bythe differential operators a', and (a /at) + si as well as in the relevance ofthe Cauchy problem to the question of uniqueness in the martingale problem(Theorem 4.28).

In Chapter 4 we employed probabilistic arguments to establish the exis-tence and uniqueness of solutions to the Dirichlet and Cauchy problemsconsidered there. The stochastic representations of solutions, which were souseful for uniqueness, will carry over to the generality of this section. As faras existence is concerned, however, the mean-value property for harmonicfunctions and the explicit form of the fundamental solution for the heatequation will no longer be available to us. We shall content ourselves, there-fore, with representation and uniqueness results, and fall back on standardreferences in the theory of partial differential equations when an existenceresult is needed. The reader is referred to the notes for a brief discussion ofprobabilistic methods for proving existence.

Throughout this section, we shall be considering a solution to the stochasticintegral equation


364

(7.1) X!") = x+s

f 6(0, XV))d0 +i

5. Stochastic Differential Equations

o-(0, Xl,"))dW,; t s s< cc,

under the standing assumptions that

(7.2)

(7.3)

the coefficients bi(t, x), 6,,(t, x): [0, co) x Rd R arecontinuous and satisfy the linear growth condition (2.13);

the equation (7.1) has a weak solution (X("), W),(SI, .F, P), {cr's} for every pair (t, x); and

(7.4) this solution is unique in the sense of probability law.

We frequently suppress the superscripts (t, x) in X(), and write E" to indicatethe expectation computed under these initial conditions. Associated with (7.1)is the second-order differential operator At, of (4.1). When b and o- do notdepend on t, we write Al (as in (1.2)) instead of si, and EX instead of E".

A. The Dirichlet Problem

Let D be an open subset of Rd, and assume that b and a do not depend on t.

7.1 Definition. The operator .sal of (1.2) is called elliptic at the point x e l if

d d

E Eaik(xgi4 > 0; ;/ E Old \ {0}.1=1 k=1

If Ai is elliptic at every point of D, we say that .91 is elliptic in D; if there existsa number 6 > 0 such that

d d

E E aik(xgik 611a2; VxeD,eRd,1=1 k=1

we say that sal is uniformly elliptic in D.

Let .sal be elliptic in the open, bounded domain D, and consider the con-tinuous functions k: D [0, co), g: D R, and f: ap R. The Dirichletproblem is to find a continuous function u: D P such that u is of class C2(D)and satisfies the elliptic equation

(7.5) - ku = -g; in D

as well as the boundary condition

(7.6) u = f; on ap.

7.2 Proposition. Let u be a solution of the Dirichlet problem (7.5), (7.6) in theopen, bounded domain D, and let tD -4 inf It > 0; X,It DI. If

(7.7) Ex D < co; VxeD,

then under the assumptions (7.2)-(7.4) we have

(7.8)rift tb

U(X) = Ex[f(X,D)exp { - i k(Xs)ds}+ f g(X,)exp { - J k(Xs)ds} dt]o o o

for every x ED.

7.3 Problem. Prove Proposition 7.2.

When should the condition (7.7) be expected to hold? Intuitively speaking, ifthere is enough "diffusion" to guarantee that, in at least one component, Xbehaves like a Brownian motion, then (7.7) is valid. We render this idea precisein the following lemma.

7.4 Lemma. Suppose that for the open, bounded domain D, we have for some1 < 1 < d:

(7.9)

Then (7.7) holds.

min ciee(x) > 0.xeD

PROOF (Friedman (1975), p. 145). With b A mazE 51b(x)I, a A minx 5 ciee(x),

q A mine 5 Xe, and v > (2b/a), we consider the function h(x) = -fiexp(vxj);x = (x1,...,xd)eD, where the constant ti > 0 will be determined later. Thisfunction is of class C' (D) and satisfies

1 2b-(.91h)(x) = ye"i{iv2aei(x) + vbj(x)} -

2'Iva e" v - - ; xeD.

a

Choosing p sufficiently large, we can guarantee that .cdh < -1 holds in D.Now the function h and its derivatives are bounded on D, so by Ito's rule wehave for every x ED, t > 0:

ET A ID) _.._ h(x) - Ex h(X, A,D) s 2 max ih(y)1 < 00.yD

Let t -> c0 to obtain (7.7). El

75 Remark. Condition (7.9) is stronger than ellipticity but weaker thanuniform ellipticity in D. Now suppose that in the open bounded domain D,we have that

(i) sad is uniformly elliptic,(ii) the coefficients au, b,, k, g are Holder-continuous, and

(iii) every point a e OD has the exterior sphere property; i.e., there exists a ballB(a) such that B(a) n D = 0, B(a) n 8D = {a}.

We also retain the assumption that f is continuous on OD. Then there exists

a function u of class C(D) n CZ(D) (in fact, with Holder-continuous secondpartial derivatives in D), which solves the Dirichlet problem (7.5), (7.6); seeGilbarg & Trudinger (1977), p. 101, Friedman (1964), p. 87, or Friedman(1975), p. 134. By virtue of Proposition 7.2, such a function is unique and isgiven by (7.8).

B. The Cauchy Problem and a Feynman-Kac Representation

With an arbitrary but fixed T > 0 and appropriate constants L > 0, 2 > 1,we consider functions f (x): Rd -, R, g(t, x): [0, T] x08° -, 01 and k(t, x):[0, T] x08° -.10, co) which are continuous and satisfy

(7.10) (i) 1./(x)1 < L(1 + 11421)

as well as

(7.11) (i) Ig(t,x)I L(1 + 11421)

Or (ii) f(x) > 0; V xe Rd

or (ii) g(t, x) > 0; V 0 < t < T, x e Rd.

We recall also the operator .21, of (4.1), and formulate the analogue of theFeynman-Kac Theorem 4.4.2:

7.6 Theorem. Under the preceding assumptions and (7.2)-(7.4), suppose thatv(t, x): [0, T] x08° - 08° is continuous, is of class CL2([0,T) x08°) (Remark4.1), and satisfies the Cauchy problem

av(7.12) --at + kv = At) + g; in [0,T) x08°,

(7.13) v(T, x) . f(x); x e01°,

as well as the polynomial growth condition

(7.14) max Iv(t, x)1 M(1 + II x112P); x e Rd,0<t<T

for some M > 0, p _. 1. Then v(t, x) admits the stochastic representation

T

(7.15) v(t, x) = E"[f (X 7.) exp { -i k(0, XOrli

s

+ 1T

g(s, Xs) exp { -i k(0,X0)0} ds]

on [0, T] x Rd; in particular, such a solution is unique.

PROOF. Proceeding as in the proof of Theorem 4.4.2, we apply the Ito rule tothe process v(s, Xs)exp{ -If k(0, X 8)0}; s e [t, T], and obtain, with Ti, ''inf{s > t; IIX sll n},

T

(7.16) v(t, x) = E"[J. g(s, XS) exp -J k(0, X8) dO} ds]

+ E" [v(t, exp {- k(0, X0) dei

+ E'x[f (X T) exp {- k(0, X0) de} iltLet us recall from (3.17) the estimate

(7.17) Et' max 11X9112m C(1 0012m)eC(s-t); t < < T,i<c4<s

valid for every m > 1 and some C = C(m, K, T, d) > 0. Now the first term onthe right-hand side of (7.16) converges as n -p cc to

J'g(s, Xs) exp - k(0, X0)0} ds,

by either the dominated convergence theorem (thanks to (7.11) (i) and (7.17))or the monotone convergence theorem (if (7.11) (ii) prevails). The second termis bounded in absolute value by

(7.18) E"[I ver, M(1 + n2")Pfx[T,, < T].

However, this last probability can be written as

pt,x[tn x[r max n ri-2'"E" max II X 0112inr<OT

cn-2m11

by virtue of (7.17) and the Ceby§ev inequality. Simply selecting m > u, we seethat the right-hand side of (7.18) converges to zero as n cc. Finally, the lastterm in (7.16) converges to

Et.x[f(X T) exp j -f k(0, X0) d0}1,

either by the dominated or by the monotone convergence theorem.

7.7 Problem. In the case of bounded coefficients, i.e.,

(7.19) I b,(t, x) I + E 0-4(t,x) < p; 0 < t < co, x 1 0 and 0 < u < (1/18p Td). (Hint: Use Problem 3.4.12.)

7.8 Remark. For conditions under which the Cauchy problem (7.12), (7.13)has a solution satisfying the exponential growth condition (7.20), one shouldconsult Friedman (1964), Chapter I. A set of conditions sufficient for theexistence of a solution v satisfying the polynomial growth condition (7.14) is:

(i) Uniform ellipticity: There exists a positive constant S such that

d d

E E a,(t, x)M _., 451a21=1 k=1

holds for every e Rd and (t, x) e [0, co) x Rd;(ii) Boundedness: The functions aik(t, x), bi(t, x), k(t, x) are bounded in

[0, 7] x Rd;(iii) Holder continuity: The functions aik(t, x), bi(t, x), k(t, x), and g(t, x) are

uniformly Holder-continuous in [0, T] x Rd;(iv) Polynomial growth: The functions f(x) and g(t, x) satisfy (7.10) (i) and (7.11)

(i), respectively.

Conditions (i), (ii), and (iii) can be relaxed somewhat by making them localrequirements. We refer the reader to Friedman (1975), p. 147, for preciseformulations.

7.9 Definition. A fundamental solution of the second-order partial differentialequation

(7.21)a-tu + ku = .21,u

is a nonnegative function G(t, x; t, defined for 0 < t < r < T, x e Rd, e Rd,

with the property that for every f e Co(Rd), t e (0, T], the function

(7.22) u(t, x) 4=='- G(t, X; T, (0 cg; 0 < t < T, x eRid

is bounded, of class C1'2, satisfies (7.21), and

(7.23) lim u(t, x) = f(x); x e Rd.ttt

Under conditions (i)-(iii) of Remark 7.8 imposed on the coefficients bi(t, x),aik(t, x), and k(t, x), a fundamental solution of (7.21) exists (see Friedman (1975),pp. 141, 148 and Friedman (1964), Chapter I). For fixed (r, e (0, T] x Rd, thefunction

co(t, x) g G(t, x; t,

is of class C1'2([0, r) x Rd) and satisfies the backward Kolmogorov equation(7.21) in the backward variables (t, x). If, in addition, the functions (a/ext)bat, x),(0/axi)aik(t, x), (a2 /ax; axk)aik(t, x) are bounded and Holder-continuous, thenfor fixed (t, x) e [0, T) x l the function

thr, -4 G(t, x; t,

is of class C12((t, T] x Rd) and satisfies the forward Kolmogorov equation

0 1d d 32

(7.24)ar t/(T,

2 1=1 k=1= E E [a.k(T, )tl(T,

- [bAT , 00-r, 01 - k(-r, 00T, 0i=1

in the forward variables (r,Returning to the Cauchy problem (7.21), (7.23) with k = 0, we recall from

Theorem 7.6 that its solution is given by

(7.25) u(t, x) = Ef(n.x)); f e Co(Rd).

A comparison of (7.22), (7.25), in conjunction with Problem 4.25, leads to theconclusion that any fundamental solution G(t, x; r, is also the transitionprobability density for the process )0) determined by (7.1); i.e.,

(7.26) P[Xx) e A] = G(t, x; r, g; A e .4 0 t < TA

In particular, under the conditions (7.2)-(7.4), this fundamental solution isunique, and the representation (7.15) of the solution to the Cauchy problem

av(7.27) --at = sit!) + g; in [0, T) x Rd,

(7.28)

now takes the form

v(T, x) = f(x); x e Rd,

(7.29) v(t, x) = J G(t, x; T, Of(0 do + J

T

J G(t, x; T, g(r, )(g dr.Rd t Rd

C. Supplementary Exercises

7.10 Exercise. The Cauchy problem (7.27), (7.28) does not include the potentialterm kv appearing in (7.12). The case of nonzero k corresponds to a diffusionwith killing. In particular, let X() be the solution to (7.1); let Y be anindependent, exponentially distributed random variable with mean 1; anddefine the lifetime

p(t.X) inf > t; k(9, XS")) d0 >

The killed diffusion process is

(i.x) A Txp-); t < s < p("),A; s y po-x),

where A is a cemetery state isolated from Rd. Assume the conditions ofTheorem 7.6 and let G(t, x; r, be a fundamental solution of (7.21). Then wehave

(7.30) P[X!" e A] = J G(t, x; r, g; A e .4(GV), 0 < t < r < T,A

and the solution (7.15) of the Cauchy problem (7.12), (7.13) takes the form(7.29).

7.11 Exercise. Suppose that bi(t, x); 1 < i < d, are uniformly bounded on[0, T] x R' and that f(x) and g(t, x) satisfy (7.10) and (7.11), respectively. Ifv(t, x) is a solution to the Cauchy problem

-at1

= Av + (b, Vv) + g; in [0, T) x Rd

v(T, x) = f(x); x e

and (7.20) holds, thenT-t

v(t, x) -= Ex[f(W7-_,) exp 1. (b(t + 0, WO, dWo)o

t + 0, W0)112 d0}o

T-t8g(t + s, Ws) exp f (b(t + 0, Wo), d Wo)

Il b(t- -12

+ Wo)112 d0} ds],0

where {W .F,; 0 < t < T }, (CI, Px 1.E Re is a d-dimensional Brownianfamily.

7.12 Exercise. Write down the Kolmogorov forward and backward equationswith k 0 for one-dimensional Brownian motion with constant drift it, andverify that the transition probability density of this process satisfies theseequations in the appropriate variables.

7.13 Exercise. Let the coefficients b, a in (7.1) be independent of t, and assumethat condition (7.9) holds for every open, bounded domain D c Rd. Supposealso that there exists a function f : R° \ {0} R of class C2, which satisfies

(7.31) df(x) 5 0 on Rd\ {0}

and is such that F(r) is strictly increasing with F(r) = co.

(i) Show that we have the recurrence property

5.8. Applications to Economics 371

(7.32) Px(T, < co) = 1; V x e \B,

for every r > 0, where B, = {X G Rd; II X II < r} and T,. = inf It 0; X, E B, 1.(ii) Verify that (7.32) holds in the case

(7.33) (x, b(x)) + 1 > aii(x)(x, a(x)x);

1x112V x E Rd\{0}.

(iii) If (7.31) is strengthened to sif(x) < -1; V x e Rd\ {0}, then we have thepositive recurrence property

(7.34) ExT, < 00 V X E \ fir.

5.8. Applications to Economics

In this section we apply the theory of stochastic calculus and differentialequations to two related problems in financial economics. The first of theseis option pricing, where we derive the celebrated Black & Scholes (1973) optionpricing formula. The second application is the optimal consumption/investmentproblem formulated by Merton (1971). These problems are unified by theirreliance on the theory of stochastic differential equations to model the tradingof risky securities in continuous time. In the second problem, this theoryallows us to characterize the value function and optimal consumption processin a context more general than considered heretofore. We subsequently spe-cialize the model to the case of constant coefficients, so as to illustrate the useof the Hamilton-Jacobi-Bellman equation in stochastic control.

A. Portfolio and Consumption Processes

Let us consider a market in which d + 1 assets (or "securities") are tradedcontinuously. We assume throughout this section that there is a fixed timehorizon 0 < T < oc. One of the assets, called the bond, has a price Po(t) whichevolves according to the differential equation

(8.1) dPo(t) = r(t)P0(t)dt, P0(0) = po; 0 < t < T.

The remaining d assets, called stocks, are "risky"; their prices are modeled bythe linear stochastic differential equation for i = 1, .. , d:

(8.2) dPi(t) = bi(t)Pi(t)dt + PM) Pi(0) = pi; 0 < t < TJ =1

The process W = {W = (W,"),..., W,49)T,,F,; 0 < t < T} is a d-dimensionalBrownian motion on a probability space (f2, P), and the filtration 1.F, isthe augmentation under P of the filtration {",""} generated by W. The interestrate process {r(t), .9v,; 0 5 t < T}, as well as the vector of mean rates of return

{b(t) = (131(t),...,bd(t))T,,,: 0 S t 5 T} and the dispersion matrix {O(t) =(a1 i(t))1< i,j<d, Wit; 0 < t < T }, are assumed to be measurable, adapted, andbounded uniformly in (t, co)e [0, T] x f2. We set a(t) A a(t)aT(t) and assumethat for some number e > 0,

(8.3) .ra(t) Eg112; e[Fad, 0 < t < T,a.s.

8.1 Problem. Under assumption (8.3), a T(t) has an inverse and

(8.4)11(aT(01-1- , - a; V e Rd, 0 < t < T, a.s.

Moreover, with d(t) A 6T(t)a(t), we have

(8.5) Ta(cg gl12; v a.s.

so a(t) also has an inverse and

(8.6) 11(010)-111 II; V e 0_ <t <T, a.s.

We imagine now an investor who starts with some initial endowment x > 0and invests it in the d + 1 assets described previously. Let NM denote thenumber of shares of asset i owned by the investor at time t. Then X0 a x =El=c, N,(0)p,, and the investor's wealth at time t is

(8.7) X, = E Ni(t)P,(t).i=o

If trading of shares takes place at discrete time points, say at t and t + h, andthere is no infusion or withdrawal of funds, then

(8.8) Xt+h - X, = INI,(t)[P,(t + h) - PM].i=0

If, on the other hand, the investor chooses at time t + h to consume an amounthCH.,, and reduce the wealth accordingly, then (8.8) should be replaced by

d

(8.9) Xt+h - Xt = Ni(t)[P,(t + h) - P,(t)] - hC,+h.i=o

The continuous-time analogue of (8.9) is

dX, = Ni(t)d13,(t) - cdt.i=o

Taking (8.1), (8.2), (8.7) into account and denoting by tr,(t) A N,(t)P,(t) theamount invested in the i-th stock, 1 < i < d, we may write this as

(8.10)d d d

dX, = (r(t)X, - C,)dt + E (kW - r(t))n,(t)dt + E E ni(t)a,i(t)dW,(11.1=1 1=1 j =1

8.2 Definition. A portfolio process it = {n(t) = (n 1(t), ,nd(t))T,..97,; 0 5 tis a measurable, adapted process for which

d T

(8.11) E ir(t)dt < co, a.s.i=1 o

A consumption process C = {C,Ft; 0 < t < T} is a measurable, adaptedprocess with values in [0, co) and

(8.12) C, dt < co, a.s.fo

83 Remark. We note that any component of n(t) may become negative, whichis to be interpreted as short-selling a stock. The amount invested in the bond,

d

no(t) X t - E nt(t),i=1

may also be negative, and this amounts to borrowing at the interest rate r(t).

8.4 Remark. Conditions (8.11), (8.12) ensure that the stochastic differentialequation (8.10) has a unique strong solution. Indeed, formal application ofProblem 6.15 leads to the formula

(8.13) X, = e.V. r(s) as x + to e-P,roo du ur(s)T(b(s) - r(s)1) - C5] ds{

+ft CS° r(u)d" irT(s)o-(s)dWs}; 0 < t < T,

where 1 is the d-dimensional vector with every component equal to 1. Allvectors are column vectors, and transposition is denoted by the superscriptT The verification that under (8.11), (8.12), the process X given by (8.13) solves(8.10) is straightforward.

8.5 Definition. A pair (7c, C) of portfolio and consumption processes is said tobe admissible for the initial endowment x > 0 if the wealth process (8.13)satisfies

(8.14) X > 0; 0 < t < T, a.s.

If b(t) = r(t)1 for 0 S t < T, then the discount factor CP. os) ds exactly offsetsthe rate of growth of all assets and (8.13) shows that

(8.15) M, -A X te- r(s) ds x + r(u) du Cs ds

is a stochastic integral. In other words, the process consisting of current wealthplus cumulative consumption, both properly discounted, is a local martingale.When b(t) r(t)1, (8.15) is no longer a local martingale under P, but becomes

one under a new measure P which removes the drift term n(t)T(b(t) - r(t)1)in (8.10). More specifically, recall from Problem 8.1 that the process

(8.16)

is bounded, and set

(8.17)

0(t) (6(t))-1(b(t) - r(t)1)

Z = exp {- Ei=1

0(s)dWP1 f 110(012 ds}.2 0

Then Z = {Z 0 < t < T} is a martingale Corollary 3.5.13); the newprobability measure

(8.18) P(A) A E[ZT1A]; A E .FT

is such that P and P are mutually absolutely continuous on .FT, and theprocess

(8.19) PfrAW,+fi 0(s)ds; 0< t< T,

is a d-dimensional Brownian motion under P (Theorem 3.5.1). In terms of IV,(8.13) may be rewritten as

(8.20) Xie-f; r(s)ds e-po r(u) du ds = x + Ir e-p,r(u)du ITT (00.(s) dais;

0 JO

0 < t < T, a.s.

For an admissible pair (it, C) the left-hand side of (8.20) is nonnegative andthe right-hand side is a P-local martingale. It follows that the left-hand side,and hence also Xie-A r(s)ds is a nonnegative supermartingale under P (Problem1.5.19). Let

(8.21) To = T A inf It e [0, T]; X, = 0 }.

According to Problem 1.3.29,

X, = 0; To < t < T holds a.s. on {To <

If To < T, we say that bankruptcy occurs at time To.From the supermartingale property in (8.20) we obtain

(8.22) E[XTe-gr(s)ds Cte- r(s)ds dt < x,

whence the following necessary condition for admissibility:

(8.23) E e-Jo r(s)ds C, dt < x.

This condition is also sufficient for admissibility, in the sense of the followingproposition.

8.6 Proposition. Suppose x > 0 and a consumption process C are given so that(8.23) is satisfied. Then there exists a portfolio process it such that the pair (it, C)is admissible for the initial endowment x.

PROOF. Let D 4 g, ce-ro 'Is)" dt, and define the nonnegative process

E [ Cse-T

ll

-4 Pr(u)" ds ..F; + (x - ED) el° r(s)ds,

so that

(8.24)

where

e50 r(s) ds Ix + - ft Cse- if, r(u) du dsSt

o

m,=° E[DI,F,] - ED =E[DZTI ]

E(DZT)Zi

from the Bayes rule of Lemma 3.5.3. Thanks to Theorem 1.3.13, we mayassume that P-a.e. path of the martingale

Ni A E(DZTIA), ..Ft; 0 <t _<_ T,

is RCLL, and so, by Problem 3.4.16, there exists a measurable, {.F,}- adapted,Rd-valued process Y with

(8.25)fT

II 17 (t)112 dt < oo0

and

(8.26) 1%4= E(DZT) + i ft y(s)dWsu); 0 t T,t---1 o

valid a.s. P. Now mi = u(/V Zi) - E(DZT), where u(x, y) = (x/y), and from Ito'srule we obtain with (1)4) A (Y(t) + Alt0(t))/Zi:

d

'(8.27) m, = J.(pi(s)dtillm; 0 < t < Tj =1 o

We have used the relations dZi = -Z,OT(t)dWt and (8.19). Now define

(8.28) ir(t) A el0'")ds(aT(t))-1 (1)4),

so that (8.24) becomes (8.20) when we make the identification = X. Condi-tion (8.11) follows from (8.4), (8.25), the boundedness of 0, and the pathcontinuity of Z and N (the latter being a consequence of (8.26)).

Remark. The representation (8.27) cannot be obtained from a direct applica-tion of Problem 3.4.16 to the P-martingale fm A; 0 t < T }. The reason isthat the filtration {A} is the augmentation (under P or P) of {Aw}, not of{Awl.

B. Option Pricing

In the context of the previous subsection, suppose that at time t = 0 we signa contract which gives us the option to buy, at a specified time T (calledmaturity or expiration date), one share of stock 1 at a specified price q, calledthe exercise price. At maturity, if the price PP ) of stock 1 is below the exerciseprice, the contract is worthless to us; on the other hand, if PP) > q, we canexercise our option (i.e., buy one share at the preassigned price q) and thensell the share immediately in the market for PP. This contract, which is calledan option, is thus equivalent to a payment of (PV - q)+ dollars at maturity.Sometimes the term European option is used to describe this financial instru-ment, in contrast to an American option, which can be exercised at any timebetween t = 0 and maturity.

The following definition provides a generalization of the concept of option.

8.7 Definition. A contingent claim is a financial instrument consisting of:

(i) a payoff rate g = Ig Ft; 0 < t < T }, and(ii) a terminal payoff fT at maturity.

Here g is a nonnegative, measurable, and adapted process, fT is a nonnegative,3,7T-measurable random variable, and for some pi > 1 we have

T

(8.29) E[fT + gt dt < oo.

8.8 Remark. An option is a special case of a contingent claim with g 0 andfT = (PP) - qr.

8.9 Definition. Let x > 0 be given, and let (ir, C) be a portfolio/consumptionprocess pair which is admissible for the initial endowment x. The pair (it, C)is called a hedging strategy against the contingent claim (g, fT), provided

(i) C, = gt; 0 t T, and(ii) XT = fT

hold a.s., where X is the wealth process associated with the pair (n, C) andwith the initial condition X0 = x.

The concept of hedging strategy is introduced in order to allow the solutionof the contingent claim valuation problem: What is a fair price to pay at timet = 0 for a contingent claim? If there exists a hedging strategy which isadmissible for an initial endowment X0 = x, then an agent who buys at timet = 0 the contingent claim (g, fT) for the price x could instead have investedthe wealth in such a way as to duplicate the payoff of the contingent claim.Consequently, the price of the claim should not be greater than x. Could onebegin with an initial wealth strictly smaller than x and again duplicate the

payoff of the contingent claim? The answer to this question may be affirmative,as shown by the following exercise.

8.10 Exercise. Consider the case r = 0, d = 1, bl = 0 and a 1. Let thecontingent claim g 0 and fT = 0 be given, so obviously there exists a hedgingstrategy with x = 0, C 0, and it = 0. Show that for each x > 0, there is ahedging strategy with X0 = x.

The fair price for a contingent claim is the smallest number x > 0 whichallows the construction of a hedging strategy with initial wealth x. We shallshow that under the condition (8.3) and the assumptions preceding it, everycontingent claim has a fair price; we shall also derive the explicit Black &Scholes (1973) formula for the fair price of an option.

8.11 Lemma. Let the contingent claim (g, fT) be given and define

(8.30) Q e-p; r(u)dufT r(u) du J-Y., C16*

0

Then EQ is finite and is a lower bound on the fair price of (g, fT).

PROOF. Recalling that r is uniformly bounded in t and w, we may writeQ < L( fT + f o gs ds), where L is some nonrandom constant. From (8.17) wehave for every v > 1:

ZT = exp {-T

vOi(s)dWP - - IlvO(s)112 ds}i=1 2 0

a

x expv(v - 1) f

2 0110(s)112 ds},t

and because 11011 is bounded by some constant K, it follows that

v(v - 1) K27,}2

EZ'4. < exp

With p as in (8.29) and v given by (1/v) + (1/p) = 1, the Holder inequalityimplies

EQ LE[ZT fT + I gsds)1

E fT +T

gsds < CC.

Now suppose that (it, C) is a hedging strategy against the contingent claim(9,M, and the corresponding wealth process is X with initial conditionX0 = x. Recalling the Definition 8.9 and (8.30), we rewrite (8.22) as EQ 5 x.

8.12 Theorem. Under condition (8.3) and the assumptions preceding it, the fairprice of a contingent claim (g, fT) is EQ. Moreover, there exists a hedgingstrategy with initial wealth x = EQ.

PROOF. Define

(8.31) ey, r(s)ds[EQ m, f e-pc, r(u) du ds],0

where m, = E EQ1 - EQ. Proceeding exactly as in the proof of Proposition8.6 with D replaced by Q, we define it by (8.28) and C g, so that (8.31)becomes (8.20) with the identifications X = x = EQ. But then (8.31) can alsobe cast as

[T

(8.32) X, = E e- 1 7- r(u) du fT e- -1; r(u) dugs ds

i

whence X, > 0; 0 < t < T and = fT are seen to hold almost surely.

,F,i; 0 < t < T,

1=1

8.13 Exercise. Show that the hedging strategy constructed in the proof ofTheorem 8.12 is essentially (in the sense of meas x P-a.e. equivalence) the onlyhedging strategy corresponding to initial wealth x = EQ. In particular, theprocess X of (8.32) gives the unique wealth process corresponding to the fairprice; it is called the valuation process of the contingent claim.

8.14 Example (Black & Scholes (1973) option valuation formula). In thesetting of Remark 8.8 with d = 1 and constant coefficients r(t) r > 0,611(t) a- o- > 0, the price of the bond is

Po(t) = poe"; 0 < t < 'T,

and the price of the stock obeys

dP, (t) = 61 (t)P, (t) dt + o- Pi(t)(114/, = r P,(t) dt + o- (t)

For the option to buy one share of the stock at time T at the price q, we havefrom (8.32) the valuation process

(8.33) X, E[e-rci--0(131(T) - ig) + 1 Al 0 T.

In order to write (8.33) in a more explicit form, let us observe that the function

(8.34)

(T - t, x)) - qe-r(T-1)(1)(p_(T - t, x)); 0 < t < T, x > 0,v(t, x)

(x - q)+; t= T, x > 0

with1 0-2 x

p+ (t, x) = [log x + t (7- +2)], 4:1)(x) = f e 12 dz,

o-.1i ,./27r -op

satisfies the Cauchy problem2all 0

+ ry =1 a2 x2 + rxO-x v; on [0, T) x (0, oc,)

Ot 2 ex e

v(T, x) = (x - q)+ ; x 0,

as well as the conditions of Theorem 7.6. We conclude from that theorem andthe Markov property applied to (8.33) that

(8.36) X, = v(t,P,(t)); 0 < t T, a.s.

We thus have an explicit formula for the value of the option at time t in termsof the current stock price P, (t), the time-to-maturity T - t, and the exerciseprice q.

(8.35)

8.15 Exercise. In the setting of Example 8.14 but with fT = h(P, (T)), whereh: [0, co) [0, co) is a convex, piecewise C2 function with h(0) = h'(0) = 0,show that the valuation process for the contingent claim (0,fT) is given by

(8.37) X, = E[cr(T-')h(P,(T))1.97,1 = i h"(9)1,,,T(t, P, (t)) dq.

We denote here by ty,,,T(t, x) the function of (8.34).

C. Optimal Consumption and Investment (General Theory)

In this subsection we pose and solve a stochastic optimal control problem forthe economics model of Subsection A. Suppose that, in addition to the datagiven there, we have a measurable, adapted, uniformly bounded discountprocess 13 = {Ns), Fs; 0 < s < T} and a strictly increasing, strictly concave,continuously differentiable utility function U: [0, co) - [0, co) for whichU(0) = 0 and U'( co) --' lime U'(c) = 0. We allow the possibility that[r(0) A limy, 0 U'(c) = co. Given an initial endowment x > 0, an investorwishes to choose an admissible pair (ir, C) of portfolio and consumptionprocesses, so as to maximize

T

V,,,e(X) A E 1 e-ilk")"U(COds.o

We define the value function for this problem to be

(8.38) V(x) = sup V,c(x),(n,c)

where the supremum is over all pairs (n, C) admissible for x. From theadmissibility condition (8.23) it is clear that V(0) = 0.

Recall from Proposition 8.6 that for a given consumption process C, (8.23)is satisfied if and only if there exists a portfolio it such that (n, C) is admissiblefor x. Let us define 2(x) to be the class of consumption processes C for which

(8.39)T

E.1e-P.'")dsC,dt = x.

o

It turns out that in the maximization indicated in (8.38) we may ignore theportfolio process it and we need only consider C e 9(x).

8.16 Proposition. For every x > 0 we haveT

V(x) = sup E e-f 0 t3(s)ds U(C,) dt.C e fd(x) 0

PROOF. Suppose (it, C) is admissible for x > 0, and set

i T

y '' E e-J°'(."' Ctdt < x.0

If y > 0, we may define C, = (x/y)C, so that 0 e g(x). There exists then aportfolio process A such that (A, 0) is admissible for x, and

(8.40) V,c(x) < Vi,e(x).

If y = 0, then: C, = 0; a.e. t e [0, T], almost surely, and we can find a constantc > 0 such that 0, .--- c satisfies (8.39). Again, (8.40) holds for some it chosenso that (A, 0) is admissible for x. 1=1

Because U': [0, co] '''' [0, U'(0)] is strictly decreasing, it has a strictlydecreasing inverse function 1: [0, U'(0)] --4onto [0, co]. We extend I by setting1(y) = 0 for y > U'(0). Note that /(0) = co and /(x)) = 0. It is easily verifiedthat

(8.41) U(I(y)) - yl(y) U(c) - yc; 0 < c < op, 0 < y < oo.

Define a function X: [0, co] - [0, co] byT

E e-1,. r(u) du(8.42) I (yZsef°(fl(")-1( "))")ds,0

and assume that

(8.43) X(y) < co; 0 < y < Do.

We shall have more to say about this assumption in the next subsection, wherewe specialize the model to the case of constant coefficients. Let us definey 4- sup{ y 0; .1. is strictly decreasing on [0, y]}.

8.17 Problem. Under condition (8.43), .i" is continuous and strictly decreasingon [0, y] with X(0) = co and gly) = 0.

toLet : [0, co] --o-4 [0, y] be the inverse of .?C. For a given initial endow-ment x > 0, define the processes


(8.44)

(8.45)

r/: g V(x)zsePo (poo-rom du

C: A join.

The definition of -N implies C* eg(x). We show now that C* is an optimalconsumption process.

8.18 Theorem. Let x > 0 be given and assume that (8.43) holds. Then theconsumption process given by (8.45) is optimal:

(8.46) V(x) = E CP° P(s)ds U(Ct*)dt.0

PROOF. It suffices to compare C* to an arbitrary C e g(x). For such a C, wehave

E e-S°P(s)ds(U(Ct*) - U(Ct))dt

= E CS° P(s)'[(U(I(e)) - ri7 Re)) - (U(C,) - ri7Ct)]dt

V(X)E e-f`o r(s)ds (ct)I, Ct)dt.

The first expectation on the right-hand side is nonnegative because of (8.41);the second vanishes because both C* and C are in (x).

Having thus determined the value function and the optimal consumptionprocess, we appeal to the construction in the proof of Proposition 8.6 for thedetermination of a corresponding optimal portfolio process 7E*. This does notprovide us with a very useful representation for n *, but one can specialize themodel in various ways so as to obtain V, C* and ir* more explicitly. We dothis in the next subsection.

D. Optimal Consumption and Investment(Constant Coefficients)

We consider here a case somewhat more general than that originally studiedby Merton (1971) and reported succinctly by Fleming & Rishel (1975),pp. 160-161. In particular, we shall assume that U is three times continuouslydifferentiable and that the model data are constant:

(8.47) fl(t) 13, r(t) r, b(t) b, Q(t) = a,

where b ege and a is a nonsingular, (d x d) matrix. We introduce the linear,second-order partial differential operator given by


IAP(t,Y) - (Pt(t, y) + i3P(t,.11) - 113 - 031(Py(t, y) - 111011 2.Y2(P(t,37),

where B = o-' (b - r1) in accordance with (8.16). Our standing assumptionthroughout this subsection is that B is different from zero and there exist C' '3functions G: [0, T] x (0, co) [0, oo ) and S: [0, T] x (0, co) [0, co) such that

(8.48) LG(t, y) = U (1 (y)); 0 < t < 'T, y > 0

(8.49) G(T, y) = 0; y > 0

and

(8.50) LS(t, y) = y1 (y); 0 < t < T, y > 0

(8.51) S('T, y) = 0; y > O.

Here we mean that G,(t, y), Gty(t, y), Gy(t, y), Gyy(t, y), and Gyyy(t, y) exist for all0 < t < T, y > 0, and these functions are jointly continuous in (t, y). The sameis true for S. We assume, furthermore, that G, Gy, S, and Sy all satisfy poly-nomial growth conditions of the form

(8.52) max H(t, y) < M(1 + y -'' + y A); y > 01) T

for some M > 0 and A > O.

8.19 Problem. Let H: [0, T] x (0, co) -, [0, co) be of class C I 2 on its domainand satisfy (8.52). Let g: [0, T] x (0, co) [0, co) be continuous, and assumethat H solves the Cauchy problem

LH (t, y) = g(t, y); 0 < t < 7; y > 0

H('T, y) = 0; y > O.

Then H admits the stochastic representationT

H(t, y) = E e-fl(s-`)g(s,17,(`'''))ds,t

Iwhere, with t < s < T:

(8.53) ys(1,y) A ye(fl -rgs-t) Zt

(8.54) 4 4 exp{ -0T(Ws - Wt) - 1110112(s - t)}.

(Hint: Consider the change of variable e = log y.)

From Problem 8.19 we derive the stochastic representation formulasT

(8.55) G(t, y) = E i e-134-`)U (1(17,("))ds,t

T

(8.56) S(t, y) = yE f e-r(s-`)Z's I (V' Y)) ds.t

It is useful to consider the consumption/investment problem with initialtimes other than zero. Thus, for 0 < t 5 T fixed and x > 0, we define the valuefunction

(8.57) V(t, x) = sup E f fisU (Cs) ds,0, ,c) r

where (it, C) must be admissible for (t, x), which means that the wealth processdetermined by the equation

(8.58) Xs = x + (r X - C) du + E (b1 - r)tri(u) dufd d

+ E E dW,P); t s < T,r

remains nonnegative. Corresponding to a consumption process C, a portfo-lio process it for which (7r, C) is admissible for (t, x) exists if and only if(cf. Proposition 8.6)

(8.59) f$

T

E e-r(s-`)Zst Cs ds < x.

For 0 < t < T, define a function X(t, ): [0, oo] [0, co] by

(8.60) X(t, y) 9 E J T e-r(s-`)41(Y,"Y))ds.

Comparison of (8.56) and (8.60) shows that

(8.61) y) = S(t, y) < co; 0 < y < oo.

Now y(t) g sup It > 0; X(t, ) is strictly decreasing on [0, y] = co, and wehave just as in Problem 8.17 that for 0 < t < T, -) is strictly decreasingon [0, co] with .X(t, 0) = co and (t, co) = 0. We denote by ): [0, co] -°'[0, co] the inverse of X(t, ):

(8.62) W(t, y)) = y; 0 y co, 0 < t < T

If we now define for t < s < T:

(8.63)

(8.64)

then

171") 4 V(t,x)Zst e(11-04-0,

C: 9. /(n(st,x)),

(8.65) V(t, x) = E f CI" U (C,*)ds.

This claim is proved as in Theorem 8.18; the new feature here is that fory = x), one has ii(") = Y("), and consequently

(8.66) V(t, x) = e-fl`G(t,(t,x)); 0 < t < T, x > 0.

Thus, if we can solve the Cauchy problems (8.48), (8.49) and (8.50), (8.51), thenwe can express V(t, x) in closed form.

8.20 Exercise. Let U(c) = cb, where 0 < 6 < 1. Show that if

1

r61

II 0112k A 1 - 2 1 -

is nonzero, then

GO, = -1(1 - e-ku-_,))(yro-t)

S(t, y) = 6G(t, y),

e-k( -6V(t, = e-13` )5.

k

If k = 0, then G(t, y) = (T - t)(1)°1(", S(t, y) = 45G(t, y), andV(t,x) = e-NT - -6 X b.

Although we have the representation (8.66) for the value function in ourconsumption/investment problem, we have not as yet derived representationsfor the optimal consumption and portfolio processes in feedback form, i.e., asfunctions of the optimal wealth process. In order to obtain such representa-tions, we introduce the Hamilton-Jacobi-Bellman (HJB) equation for thismodel. This nonlinear, second-order, partial differential equation offers acharacterization of the value function and is the usual technique by whichstochastic control problems are attacked. Because of its nonlinear nature, thisequation is typically quite difficult to solve. In the present problem, we havealready seen how to circumvent the HJB equation by solving instead the twolinear equations (8.48) and (8.50).

8.21 Lemma (Verification Result for the HJB Equation). Suppose Q: [0, T] x[0, co) -> [0, co) is continuous, is of class C1'2([0, T) x (0, co)), and solves theHJB equation

(8.67)

Then

(8.68)

Qt(t, + max f[rx -c + (b - r1)T n] Qx(t,1

+ Mal' 702 Q.(t ,c>oneRd

+ e-fitU(c)} =0; 0 t T, 0 < x < co.

V(t, x) < Q(t, x); 0 < t < T, 0 5 x < co.

PROOF. For any initial condition (t, x) e [0, T) x (0, co) and pair (ir, C) admis-sible at (t, x), let {Xs; t 5 s < T} denote the wealth process determined by(8.58). With

//

n

l 1

Tn A T - - A id s e [t, T]; Xs > n or Xs - or 1114412 du = n ,

we have E 1;"Qx(s, X On'. (s)o- OK = 0. Therefore, It6's rule implies, in con-junction with (8.58) and (8.67),

0 < EQ(T., X,,,)rn

= Q(t,x) + E 1 (Qt(s, Xs) + [rX3 - Cs + (b - rpr ir(s)]Qx(s, Xs)

1r

+ - II 0" T n (Ai 2 Qxx(s, Xs)} ds 5 Q(t, x) -E f2

e-PsU(Cs)ds.

Letting n -+ co and using the monotone convergence theorem, we obtainE if CP' U(Cs) ds < Q(t, x). Maximization of the left-hand side over admissiblepairs (n, C) gives the desired result.

A solution to the HJB equation may not be unique, even if we specify theboundary conditions

(8.69) Q(t,0) = 0; 0 < t < T and Q(7; x) = 0; 0 < x < co.

This is because different rates of growth of Q(t, x) are possible as x approachesinfinity. One expects the value function to satisfy the HJB equation, and, inlight of (8.68), to be distinguished by being the smallest nonnegative solutionof this equation.

8.22 Proposition. Under the conditions set forth at the beginning of this sub-section, the value function V: [0, T] x [0, co) -* [0, co) is continuous, is of classC1,2(IV,,,,, T) x (0, co)), and satisfies the HJB equation (8.67) as well as theboundary conditions (8.69).

PROOF. If 0 < y < U'(0), then

(8.70) U(I(y)) = U'(I(y))1'(y) = yr(y);

if y > U'(0), then 1(y) = l'(y) = 0 and (8.70) still holds. Because of our as-sumption that G and S are of class C, we may differentiate (8.48), (8.50)with respect to y and observe that (pi (t, A A -YGy(t9Y) and (P2(t, A A-y2(0/0y)(S(t, y)/y) both satisfy

1.4p,(t, y) = - y2 l' (y); 0 5 t 5 T, y > 0,

Sai(T, Y) = 0; y > O.

In particular, I' is continuous at y = U'(0), i.e., a necessary condition for ourassumptions is U"(0) = oo. Problem 8.19 implies cp, = (p2, because both func-tions have the same stochastic representation. It follows that

(8.71) Gy(t, y) = y-0(-1

S(t, y)) = ydy(t, y)aY

and from (8.66), (8.62) we have

(8.72) Vx(t, x) = CP' V(t, x),

(8.73) Vt(t, gr(t, y)) = - Vx(t, lit, y)) gr,(t, y).

Finally, (8.50) and (8.61) imply that

(8.74)

-1;(t, y) + r.T(t, y) - (f - r + Heir )Y .Ty(t, Y) - 2 110112 Y2 (t, Y) = 1(y);

0 < y < oo, 0 t < 'T.

We want to check now that the function V(t, x) of (8.66) satisfies the HJBequation (8.67). With Q = V, the left-hand side of this equation becomes CP'times

(8.75) Gt(t, V(t, x)) - /3G(t, V(t, x)) + Gy(t,V(t, x)) Vt(t, x)

+ max[((rx - c) + (b - rpr n)V(t, x) + ll o-Tn112Vx(t, x) + U(c)].c> 0

The maximization over c is accomplished by setting

(8.76) c = I(V(t, x)).

Because of the negativity of Vx, the maximization over it is accomplished bysetting

(8.77) n = -(craT)-` (b r1)V(t,x)

V ,c(t, x)

Upon substitution of (8.76), (8.77) into (8.75), the latter becomes

(8.78) Gt(t,V(t, x)) - I1G(t,V(t, x)) + Gy(t,V(t, x))V,(t, x)

1 V2(t x+ rx?J(t, x) - V(t, x)I(V(t, x)) - -2 1102 Vt,

,

x)

)+ U(I(Mt, x))).

We may change variables in (8.78), taking y = '(t, x) and using (8.71), (8.73),(8.48) to write this expression as

GM, y) - K(t, Y) - A(t, y) + ry.lit, Y) -Y 1(Y) - +110112y2xy(t, y) + U (I (Y))

= A Thirt(t, y) + rajt, y) - (13 - r + 110112)Y.gry(t, Y)

- 1110112Y2Aryy(t, Y) - 4.01

which vanishes because of (8.74). This completes the proof that V satisfies theHJB equation (8.67). The boundary conditions (8.69) are satisfied by V byvirtue of its definition (8.57) and the admissibility condition (8.59) appliedwhen x = O. 1=1

In conclusion, we have already shown that for fixed but arbitrary (t, x) e[0, T) x (0, oo), there is an optimal pair (n *, C*) of portfolio/consumptionprocesses. Let {X:; t < s S T} denote the corresponding wealth process. Ifwe now repeat the proof of Lemma 8.21, replacing (it, C) by (tr*, C*) and Q byV, we can derive the inequality

(8.79)T

0 V(t, x) + E ft {V,(s, X:) + [rX: -C: + (b - r1)T tr*(s)] Vx(s, X:)

+ -211 6 T it* (s)112 Vx(s, X:)} ds V(t, x) -ET

e- Ps U (Cnds.t

1

We have used the monotone convergence theorem and the inequality

(8.80) Vi(s, X:`) + [r X: -C: + (b - r DT tr*(s)] Vx(s, X:)

+ Dar n*(s)112 Vxx(s, X:) - - CP' [1 (Cn 0; t < S < T,

which follows from the HJB equation for V. But (8.65) holds, so equalityprevails in (8.79) and hence also in the first inequality of (8.80), at least formeas x P-almost every (s, w) in [t, T] x n. Equality in (8.80) occurs if andonly if le: and C: maximize the expression

[rX: -c + (b - r1)T it] Vx(s, X:)

i.e. (cf. (8.76), (8.77)),

(8.81)

(8.82)

C: = .1((s, X:)),

+ ilia TnIi2 vx.(s,xs) + e-Pt um;

(s, X:)ir: (0.0.T r 1 0 r1),c(s, X:) '

where again both identities hold for meas x P-almost every (s, w) E [t, T] x aThe expressions (8.81), (8.82) provide the optimal consumption and portfolioprocesses in feedback form.

8.23 Exercise. Show that in the context of Exercise 8.20, the optimal con-sumption and portfolio processes are linear functions of the wealth processX *. Solve for the latter and show that XI = 0 a.s.

5.9. Solutions to Selected Problems2.7. We have from (2.10)

-d(e-flt f ' g(s)ds)=(g(t)- 13 f g(s)ds)

dte-flt < a(t)e-flt,

o o

whence f u g(s) ds _5 e°' f u a(s)e - as ds. Substituting this estimate back into (2.10),we obtain (2.11).

2.10. We first check that each V) is defined for all t > 0. In particular, we must showthat for k > 0,

(II b(s, XI') )11 + 110(s, X?))112)ds < oo; 0 < t < cc, a.s.Jo

In light of (2.13), this will follow from

(9.1) sup EIP0k)112 < co; 0 T < co,

a fact which we prove by induction. For k = 0, (9.1) is obvious. Assume (9.1) forsome value of k. Proceeding similarly to the proof of Theorem 2.5, we obtain thebound for 0 5 t < T:

(9.2) EI1V+"112 < 9E11 112 + 9(T + 1)K2 (1 + E11X?)112)ds,

which gives us (9.1) for k + 1. From (9.2) we also have

EMk+i)112 < C(1 + E11112) + C f E11X?)112ds; 0 < t < T,o

where C depends only on K and T Iteration of this inequality gives

2r +1)112 < C(1 + Eg112)[1 + Ct + ("2)(CO'(k + 1)!

and (2.17) follows.

2.11. We will obtain (2.4) by letting k -+ co in (2.16), once we show that the two integralson the right-hand side of (2.16) converge to the proper quantities. With T > 0,(2.21) gives maxo,,,T1,1X,(co) - )0k)(o))11 < 2', V k > N(co). Consequently,

b(s, X?))ds - ft b(s, X5) ds2

< K2T f X s12 ds0

converges to zero a.s. for 0 < t < T, as k In order to deal with the stochasticintegral, we observe from (2.19) that for fixed 0 < t < T, the sequence of randomvariables {.)0},T_I is Cauchy in L2(52, P), and since V) -+ X, a.s., we must haveEll r) - X,I12 -0 as k- co. Moreover, (2.17) shows that Ell r)II 2 is uniformlybounded for 0 < t < T and k > 0, and from Fatou's lemma we conclude thatE11X,112 <_ limk E11V)112 is uniformly bounded as well. From (2.12) and thebounded convergence theorem, we have

1E2

1cr(s, Vsk))dWs -j o-(s, Xs) dli/s = E .1110-(s,.V))fo- a(s, X s)II2 ds

J5 K2 E E II X?) - XsII2 ds -0 as k --+ oo; 0 5 t 5 T.o

2.12. For each positive integer k, define the stopping time for {,F,}:

{0 if 1111 > k,Tk =

if II 115 k,ao

and set 6, = 1{,4 <k}. We consider the unique (continuous, square-integrable)strong solution V) of the equation

= t b(s, V') ds + ft o-(s, V)) dgis; 0 < t < x.

For t > k, we have that (r) - r))11, 4 <10 is equal to

j. f:^ Tk {b(s, V)) - b(s, Xr)} ds + t '' Tk { 0(S, V)) - CYCS, Xr)} d Wso

= {b(s, V)) - b(s, Xr)}1{s< Ts} ds + {Os, V)) - a(s, V))}1{s< Tki dWs.

By repeating the argument which led to (2.18), we obtain for 0 < t < T:

E[max II - X?)(12 E max 11V) - V42 1{ 4 <k} dt,1{1141 <k} Lo.ct 0 0..s5t

and Gronwall's inequality (Problem 2.7) now yields

max II Xsrl - X?) I1 = 0, a.s. on {II II 5 14.0<s<T

We may thus define a process {X,; 0 < t < co } by setting X Jo)) = V)(co), wherek is chosen larger that II (c0)11 Then

X,1{1141,<k} - XP')1{114'11<k}

tnTk EAT,

= f Ns, V)) ds + f V)) dWs

tnrk Tk

= b(s, X s) ds + cr(s, X s)dWs

+ b(s, X s)ds + ft cr(s, Xs) dWdl 1,14,1 kl.

Since II 5 = 1, we see that X satisfies (2.4) almost surely.

3.15. We use the inequality for p > 0;

(9.3) I ailP + + la IP < n(lail + + aI)P < nP41(la1 IP + + laIP).

We shall denote by C(m,d) a positive constant depending on m and d, notnecessarily the same throughout the solution. From (iv) and (9.3) we have, almostsurely:

t

2m t 2m

IIX,112m C(m, d)[11X0112m + j' b(u, X) du (U, X) dW.

and the Holder inequality provides the bound

fob(u X) du

2m

= E b,(u, X) du)2T r[..1 1119(u, X)!! 2 dd

t2m-1 b(u, X)112'" du.

The stopping times Sk A inf {t 0; IIX, k} tend almost surely to infinity ask co, and truncating at them yields

A Sk

max II X 5112' < C(m,d) [II X 0112' + t2m-1 b(u, X)112' duOSSStASk 0

+ maxfsA Sk

0(14, X) OK a.s.

Remark 3.3.30 and Holder's inequality provide a bound for the last term, to wit,

E[ max f5 s A

a(u, X)d147.o

tA sk

C(m,d) E X)II2 dur

I AS,

< C(m, d)tm- E f a(u, X)112'n du.

Therefore, upon taking expectations, we obtain

E( max IIX5112m)otAsk

C(m, d)[E1IX0112m + C(T) E f (111,(4, X)II 2m + 110(u, X)112m)dui

for every 0 < t < T, where C(T) is a constant depending on T, m, and d. But nowwe employ the linear growth condition (3.16) to obtain finally,

tnSk

OM A E( max UXkl12") < C[1 + EIIXollzm + f tpk(u) du];0..5.s..5tASk 0

0 < t < T,and from the Gronwall inequality (Problem 2.7):

Ok(t) C(1 + EllX0112m)ect; 0 T.

Now (3.17) follows from Fatou's lemma. On the other hand, if we fix s < t in[0, T] and start from the inequality IRC, - Xs112'" < C(m,d)(11.g b(u, X) dull2m +.R 0(u, X) dW.II2m), we may proceed much as before, to obtain

Ell - Xsem C(t - s)"' -' f t [1 + E( max 11X0112m)idus

< C(1 + CecT)(1 + EllX0112m)(t - s)m.

3.21. We fix F .4,(C [0, oor). The class of sets G satisfying (3.25) is a Dynkin system.The collection of sets of the form (3.26) is closed under pairwise intersection andgenerates the a-field .4(08') .4(C[0, coy). By the Dynkin System Theorem, itsuffices to prove (3.25) for G of the form (3.26). For such a G, we have

oqi(x,w;F)/.1(dx)P,(dw)

=E[.F)1.1(dx) ,G; P{o-,-1G31.4,(C[0, CO)')}

GI

= Es [J Q;(x, ; F)p(dx) 1 ,Gz] P:[al G3]

= PAGI x 9,- G2 x F] Ps[a,' G3].

The crucial observation here is that, under Wiener measure Ps, Crt-1 G3 is inde-pendent of A(C[0, con From (3.21), we have

Ps[a,1 G3] = pi[(x, w, y) E 0; ajw E G3] = Vi[47, Wth E G3],

PAGI x G2 x F] = vi[XP E G , 9,Wu) E G2, ri) E Pl.

Therefore,

T.Q;(x, w; F)p(dx)13,(dw) = vi[XY)E G,,(p,WthE G2, Yth E F] vi[aiW(i) E G3]

= vi[X cji) E GI , W(i)E G2, CT, VV(i) E G3, y(j) F]

= vi[(XV, W(J)) E G, E F] = PJ[G x F],

because (XeE G, , (p,W(i)E G2, YU) E F) belongs to the a-field andWu) E G3 is independent of it.

3.22. The essential assertion here is that Qi(x, w; ) assigns full measure to a singleton,which is the same for j = 1, 2. To see this, fix (x, w) E x C[0, Do)' anddefine the measure Q(x, w; dy,,dy2) A Qi(x,w; dy 1)Q2(x, w, dye) on (S, 9') A(C[0, co)" x C[0, co)", Af(C[0, co)") 0 At(C[0, co)d). We have from (3.23)

(9.4) P(G x B) = f Q(x,w; B)i(dx)Ps(dw); Be G E .gi(Rd) 0 .4(C [0, °or).

With the choice B = {(YI,Y2)ES; Y = Y2} and G = [Rd x C[0, ooY, relations(3.24) and (9.4) yield the existence of a set N E d(Rd) .2(C[0, cor ) with (p x P.)(N) = 0, such that Q(x, w; B) = 1 for all (x, w) ct N. But then from Fubini,

1 = Q(x, w; B) = f Qi(x,w; {y}) Q2(x,w; dy); (x, w)0 N,cto..)d

which can occur only if for some y, call it k(x, w), we have Qi(x, w; {k(x, w) } ) = 1;j = 1, 2. This gives us (3.27). For (x, w) 0 N, and any BE AC [0, oa)d), we havek(x, w)E B <r> Q i(x , w; B) = 1. The A WI ) 0 Af(C [0, con/I( [0, 1])-measurabilityof (x, w)i- Qi(x, w; B) implies the A Rd) 0 .4(C[0, co yvalc[o, co)")- measurabilityof k. The .4,/.4,(C[0, co) ")- measurability of k follows from a similar argument,which makes use of Problem 3.21. Equation (3.28) is a direct consequence of(3.27) and the definition (3.23) of P.

4.7. For (p, a) E D, let A = pp' = o-o-T . Since A is symmetric and positive semidefinite,A

1 0

0

0there is an orthogonal matrix Q such that QAQT = - -1 - - , where A is a (k x k)[diagonal matrix whose diagonal elements are the nonzero eigenvalues of A. Since

QAQT = (QP)(Q P)T = (12a)(120)T,

rQp must have the form Qp =[-1- , where y, YiT = A. Likewise, Qa = Z1

0 01

where Z1ZT = A. We can compute an orthonormal basis for the (d - k)-dimensional subspace orthogonal to the span of the k rows of Y1. Let Y2 be the(d - k) x d matrix consisting of the orthonormal row vectors of this basis, and

A ' 0 .,. A 0set Y= EY-11. Then (Qp)yr yy, where / is the (d - k) x

Y2 0 I 0 0,

(d - k) iden ity matrix. In the same way, define Z = -[Z- - so that (Qa)ZT =Z2

(Qp)112., ZZT = YYT. Then a = pYr(ZT)', so we set R = YT(ZT)-1. It is easilyverified that RRT = 1. All of the steps necessary to compute R from p and a canbe accomplished by Borel-measurable transformations.

4.13. Let us fix 0 < s < t < co. For any integer n > (t - s)-' and every A e Ws_ft,/), wecan find B e A+(0,) such that P(A A B) = 0 (Problem 2.7.3). The martingaleproperty for {Mt,f, gk; 0 < u < co} implies that

(9.5) E [{ f(y(t)) - f(y(s +n) ) -J

± 1 1 , )(s1 J)

1A]

I t

= E[ff( y(t)) -f(y (s + -)) -..1 (sit,: f )( y) du} ,,d =0.n s+(tin)

If follows that the expectation in (9.5) is equal to zero for every A e.f.s = Ws+ .We can then let n -oo and obtain the martingale property E[(Mir - MS )1A] =0 from the bounded convergence theorem.

4.25. Suppose jud (P(x)Pi(dx) = 1Rd 9(x)/12(dx) for every ys, Cr (Rd), and take ik eCo(111°). Let p E Co (Rd) satisfy p > 0, cRd p(x) dx = 1, and set (p(x) A juld tk(x +(y /n))p(y) dy = n IRdt1/(z)p(nz - nx) dz. Then cp e q°(Rd) and 9(x) --1//(x) forevery x e Old. It follows from the bounded convergence theorem that.fRa ii/(x)111 (dx) = jRa ti/(x)/12(dx), for every 0 a Co(Rd). Now suppose G c Rd isopen and bounded. Let 0(x) = 1 A infy# G nlly - xII. Then On E Co( ) for all n,and ifr 11,. It follows from the monotone convergence theorem that it,(G) =i12(G) for every bounded open set G. The collection of sets Cif = {BE At(Rd);PI(B) = 142(B)} form a Dynkin system, and since every bounded, open set is in', the Dynkin System Theorem 2.1.3 shows that elf = g(ge).

5.3. For t 0, let Et =we can show that

{Po' a2(Xs) ds = co}. Using the method of Solution 3.4.11,

lim AS. = XO

lim XrAS = X0 +n-,co

lim a(Xs)dRis = o o, a.s. on En-.c0

ftAS

lim ct(Xs)dW = - co, as. on E,..-.0 0

But X, is continuous in the topology of the extended real numbers, so P(E,) = 0.Consequently, X, ,,s = X0 roA 0.(xs) a'His is real-valued a.s., for every t 0,so S = co a.s.

5.27. For s > 0 and c+s<x<r, we have2 dz dz

v(x) = p'(y) dy [p(x) p(c + e)] rp'(z)a2(z) jc p'(z2)a2(z)'

and (5.73) follows. A similar argument works for (5.74).

6.2. It suffices to show that if Q(t) is a locally bounded, measurable, (d x r)-matrix-valued function of t, then l', = cc,Q(s)dW, is a Gaussian process. This iscertainly true if the components of Q are simple, and the general Q can beapproximated by simple matrix functions Q(") so that for each fixed t, we have

Elifo [V")(s) - Q(s)] 6VA' = 0. Because the L2 limit of normal randomvectors must be normal, we have the desired result.

6.11. The jointly normal random variables Z,, Z2, Z are uncorrelated, as one cansee from (6.25) or by observing from (6.26) that

b(t, - t,_,) digsZ;= = 1, n.

(T - td(T - t,_,) T - s

It is also apparent that

b(tiVar

t1_,) t,- ti_,EZ, =

(T - 4)(T - ti_1)'(Z,) - (T - td(T - t,_,)

Now write down the joint density of (Z,, , Z") and make the change of variableszi = x; /(T - ti) - x;_1 /(T - ti_1) to obtain the desired result.

6.15. Observe that

dZ, = Z,[A(t)dt + S.,(t)dWP1,J=1

and apply ItO's rule to the right-hand side of (6.32) to see that X defined by thisformula satisfies (6.30). Uniqueness follows from Theorem 2.5.

In the case of constant coefficients for (6.33), the solution becomes X, =exp( Y,), where l', at + aB a 'A .,./E;=, SI, p = A - a2 /2 and B, A

(1/0E'i=1 SWU) is Brownian motion (by the P. Levy characterization, Theo-rem 3.3.16). The strong law of large numbers (Problem 2.9.3) shows that

(1',/t) = p, and hence also lim X, = 0, a.s.

7.3. Proceeding as in Solution 4.2.25, we show that

t A to

M, A u(X, tdexp - k(Xdds}

A tD

g(Xs) exp -f k(X8)(10}ds; 0 < t < op

is a uniformly integrable martingale under Px. The identity PM° = Pitt, is(7.8).

8.1. For a (d x d) matrix r, define the operator norm lin! = sup,o (11r-11/1111). Weshow that Ilrll = !IrTil. According to the Cauchy-Schwarz inequality,

Ilt1TrT11 = II ri111 gillIrt111 < 111-11RII 11n11.

Now set n = F% to obtain FT and thus by definition ilFT11 11F11

We obtain the opposite inequality by reversing the roles of F and FT.Now (8.3) implies that aT(t) is nonsingular, for otherwise we could find a

nonzero vector which makes Ta(t) = llo-T(t)OP = 0. Letting = (aT(t))-1we may rewrite (8.3) as

11(0 T(0)-1 /II 11,711; vqe Rd, 0 < t < T,

a.s., which is equivalent to II(a T(0)-1 II 5 (1/fE). Because II(a(t))' II = Ho-T(0)-111,we have the equivalence of (8.3) and (8.5). We have already proved (8.4), and (8.6)is established similarly.

8.17. Because / is strictly decreasing on [0, U'(0)] and identically zero on [U'(0), oo),we see that X is nondecreasing and for 0 < yl < y, < co:

Y t) = P[ min (Zief°(flo)-*))Yi

as) (0)1

() 7'

in which case X(yi ) = 9C(y2) = 0. The equality 5t (y) = 0 follows. The equality5d'(0) = limylo X(y) = co is a consequence of the monotone convergence theorem.Continuity of X on (0, y], and hence its surjectivity, follows from (8.43) and thedominated convergence theorem.

8.19. Let IN , ) = H(t, et), so Jr is defined and of class CI.2 on [0, T] x R. A bit ofcomputation shows that if solves the Cauchy problem

1 1-. + fl - (/3 - r --21101I2 lee --21101121fee = g(t, el); 0 t T, CcUB

Jrcr,e) o; E

Condition (8.52) on H implies that Ye satisfies (7.20) for everyµ > 0 and M > 0depending on tt. It follows from Problem 7.7 that

T

iot,e) = E[f e-P('-`)g(s,e4)ds L, =

where di, = (I3 -r - I10112 /2) ds - OT dWs. This is the stochastic differentialequation satisfied by log Ys"), and thus

T

H(t, y) = "Jog y) = E[f e-13('-`)g(s, Ys"))dsl.

5.10. Notes

Section 5.1: The term stochastic differential equation was actually introducedby S. Berngtein (1934, 1938) in the limiting study of a sequence of Markovchains arising in a stochastic difference scheme. He was only interested in thedistribution of the limiting process and showed that the latter had a densitysatisfying the Kolmogorov equations. However, according to Gihman &Skorohod (1972), it would be an exaggeration to consider Berngtein thefounder of this theory.

Independently of Ito's work, I. I. Gihman (1947, 1950) developed a theo-ry of stochastic differential equations, complete with results on existence,

5.10. Notes 395

uniqueness, smooth dependence on the initial conditions, and Kolmogorov'sequations for the transition density.

Since the early work of Ito and Gihman, the interest in the methodologyand the mathematical theory of stochastic differential equations has enjoyedremarkable successes. The constructive and intuitive nature of the concept, aswell as its strong physical appeal, have been responsible for its popularityamong applied scientists; for instance, the "dW" term on the right-hand sideof (1.9) has an important interpretation as white noise in statistical com-munication theory. Apart from its original and continued relevance to physics(see the notes to Section 6, as well as Nelson (1967), Freidlin (1985)), the newmethodology became gradually indispensable in fields such as stochasticsystems (Arnold & Kliemann (1983)) and stability theory (Friedman (1976),Khas'minskii (1980)), stochastic control and game theory (Fleming & Rishel(1975), Friedman (1976), Krylov (1980)), filtering (Liptser & Shiryaev (1977),Kallianpur (1980)), communication and dynamical systems (Wong & Hajek(1985)), mathematical economics (cf Section 5.8), mathematical biology andpopulation genetics (cf. examples in Chapter XV of Karlin & Taylor (1981)).Stochastic partial differential equations arise in filtering (Pardoux (1979)) andneurophysiology (Walsh (1984, 1986)).

On the other hand, the analytical approach to diffusions and, more gener-ally, Markov processes, has been further developed in conjunction with theHille-Yosida theory of semigroups; see, for instance, the lecture notes of Ito(1961a) and Chapter I of Ethier & Kurtz (1986).

Section 5.2: Theorems 2.5, 2.9 are standard; they are due to ItO (1942a, 1946,1951) and can be found in several monographs such as those by Skorohod(1965), Mc Kean (1969), Gihman & Skorohod (1972, 1979), Arnold (1973),Friedman (1975), Liptser & Shiryaev (1977), Stroock & Varadhan (1979),Kallianpur (1980), Ikeda & Watanabe (1981). These sources, as well as Fried-man (1976), should be consulted for further reading on the subject matter ofthis chapter and some of its applications.

The study of comparison results for stochastic differential equations startedwith Skorohod (1965). The article by Ikeda & Watanabe (1977) containsimportant refinements of Proposition 2.18 and Exercise 2.19, with applica-tions to stochastic control and to tests for explosions; see also Chapter VI inIkeda & Watanabe (1981), Yamada & Ogura (1981), Hajek (1985).

Doss (1977) and Sussmann (1978) were the first authors who studied thepossibility of pathwise solutions to stochastic differential equations, via anappropriate reduction to an ordinary differential equation in the spirit ofProposition 2.21. Extension of the latter to several dimensions has Lie-algebraic ramifications; see Ikeda & Watanabe (1981), pp. 107-110. Themodeling issue of Subsection 5.2.D was first raised by Wong & Zakai (1965a),who obtained Proposition 2.24 under more restrictive conditions. The one-dimensional equation (2.34) was studied in detail by Chitasvili & Toronjadze(1981).



Stochastic differential equations driven by general semimartingales, insteadof just Brownian motion, have been studied by Do leans-Dade (1976) andProtter (1977, 1984) among others; see also Elliott (1982), Chapter XIV.Stochastic integral equations of the Volterra type have also been considered;see, for instance, Kleptsyna & Veretennikov (1984) and Protter (1985).

Section 5.3: The methodology that leads to Proposition 3.20 and Corollary3.23 was developed by Yamada & Watanabe (1971); we follow Ikeda &Watanabe (1981) in our exposition.

Further instances of one-dimensional equations dX, = a()C,)dW, admittingno strong solution, in the spirit of Example 3.5 but with continuous dispersioncoefficients o( ), have been discovered by Barlow (1982).

Section 5.4: The book by Stroock & Varadhan (1979) should be consultedfor further reading on martingale problems and multidimensional diffusions.For the martingale approach to more specialized questions on diffusions, suchas the support theorem, boundary behavior, degeneracy, and the constructionof diffusion processes on manifolds, the reader is referred to the articles byStroock & Varadhan (1970, 1971, 1972) and the lecture notes of Priouret(1974), respectively.

The "martingale problem" approach of this section can also be employedto characterize more general Markov processes in terms of their correspond-ing infinitesimal operators; a good part of Chapter IV in Ethier & Kurtz (1986)is devoted to this subject.

Section 5.5: Material for the first two subsections was drawn primarily fromthe papers of Engelbert & Schmidt (1981, 1984, 1985). The use of local timeto prove pathwise uniqueness in Theorem 5.9 is due to Perkins (1982c),although Perkins's result has been sharpened by the use of the Engelbert andSchmidt zero-one law. Proposition 5.17 is a time-homogeneous version of amore general result due to Zvonkin (1974); according to the latter, the equa-tion (2.1) with d = 1 has a unique strong solution, provided that

(i) the drift b(t, x) is bounded and Borel-measurable,(ii) the dispersion o-(t, x) is bounded (both above and away from the

origin), is continuous in (t, x), and satisfies a Holder condition of the type(2.24) with h(u) = KO; a > (1/2).

Zvonkin's results were extended to the multidimensional case by Vereten-nikov (1979, 1981, 1982), who showed in particular that the equation (3.2) withd > 1 has a unique strong solution for any bounded, Borel-measurable driftb(t, x).

In another important development, Nakao (1972) showed that pathwiseuniqueness holds for the equation (5.1), provided that the coefficients b, aare bounded and Borel-measurable, and a is bounded below by a positiveconstant and is of bounded variation on any compact interval. For furtherextensions of this result (to time-dependent coefficients), see Veretennikov(1979), Nakao (1983), and Le Gall (1983).

The material of Subsection C is fairly standard; we relied on sources such


5.10. Notes 397

as McKean (1969), Kallianpur (1980), and Ikeda & Watanabe (1981), partic-ularly the latter. A generalization of the Feller test to the multi-dimensionalcase is due to Khas'minskii (1960) and can be found in Chapter X of Stroock& Varadhan (1979), together with more information about explosions.

A complete characterization of strong Markov processes with continuoussample paths, including the classification of their boundary behavior, ispossible in one dimension; it was carried out by Feller (1952-1957) andappears in Ito & Mc Kean (1974) and Dynkin (1965), Chapters XV-XVII. Seealso Meleard (1986) for an approach based on stochastic calculus. The recur-rence and ergodic properties of such processes were investigated by Maruyama& Tanaka (1957); see also §18 in Gihman & Skorohod (1972), as well asKhas'minskii (1960) and Bhattacharya (1978) for the multi-dimensional case.

Section 5.6: Langevin (1908) pioneered an approach to the Brownianmovement that centered around the "dynamical" equation (6.22), instead ofrelying on the parabolic (Fokker-Planck-Kolmogorov) equation for the tran-sition probability density. In (6.22), X, represents the velocity of a free particlewith mass m in a field consisting of a frictional and a fluctuating force, a is thecoefficient of friction, and 62 = 2ockT/m, where T denotes (absolute) tem-perature and k the Boltzmann constant. Langevin's ideas culminated in theOrnstein-Uhlenbeck theory for Brownian motion; long considered a purelyheuristic tool, unsuitable for rigorous work, this theory was placed on firmmathematical ground by Doob (1942). Chapters IX and X of Nelson (1967)contain a nice exposition of these matters, including the Smoluchowski equa-tion for Brownian movement in a force field.

Section 5.7: The monograph by Freidlin (1985) offers excellent follow-upreading on the subject matter of this section, as well as on degenerate andquasi-linear partial differential equations and their probabilistic treatment.

In the setting of Theorem 7.6 with k 0, g E: 0 it is possible to verify directly,under appropriate conditions, that the function

(10.1) u(t, x) = Ef(Xx))

on the right-hand side of (7.15) possesses the requisite smoothness and solvesthe Cauchy problem (7.12), (7.13). We followed such an approach in Chapter4 for the one-dimensional heat equation. Here, the key is to establish"smoothness" of the solution V) to (7.1) in the initial conditions (t, x) so asto allow taking first and second partial derivatives in (10.1) under the expec-tation sign; see Friedman (1975), p. 124, for details.

Questions of dependence on the initial conditions have been investigatedextensively. The most celebrated of such results is the diffeomorphism theorem(Kunita (1981), Stroock (1982)), which we now outline in the context of thestochastic integral equation (4.20). Under Lipschitz and linear growth condi-tions as in Theorem 2.9, this equation has, for every initial position x e Rd,a unique strong solution IX,(x); 0 "i: t < col. Consider now the (d + 1)-dimensional random field d = {X t(x , w); (t, x) e [0, co) x Rd, co e ill. It can beshown, using the Kolmogorov-entsov theorem (Problem 2.2.9) in conjunc-


tion with Problems 3.3.29 and 3.15, that there exists a modification of Isuch that:

(i) (t, x) i- gt(x, w) is continuous, for a.e. w e CI;(ii) For fixed t > 0, xi-, IC,(x, w) is a homeomorphism of D into itself for a.e.

W e

Furthermore, if the coefficients b, a have bounded and continuous deriva-tives of all orders up to k > 1, then for every t > 0,

(iii) x F- fet(x, w) is a C" -diffeomorphism for a.e. w e e.

For an application of these ideas to the modeling issue of Subsection 5.2.D,see Kunita (1986).

Malliavin (1976, 1978) pioneered a probabilistic approach to the questionsof existence and smoothness for the probability densities of Brownian func-tionals, such as strong solutions of stochastic differential equations. Theresulting "functional" stochastic calculus has become known as the stochasticcalculus of variations, or the Malliavin calculus; it has found several exegesesand applications beyond its original conception. See, for instance, Watanabe(1984), Chapter V in Ikeda & Watanabe (1981), and the review articles of Ikeda& Watanabe (1983), Zakai (1985) and Nualart & Zakai (1986). For applica-tions of the Malliavin calculus to partial differential equations, see Stroock(1981, 1983) and Kusuoka & Stroock (1983, 1985).

Section 5.8: The methodology of Subsection A is new, as is the resultingtreatment of the option pricing and consumption/investment problems inSubsections B and C, respectively. Similar results have been obtained indepen-dently by Cox & Huang in a series of papers (e.g., (1986, 1987)). For SubsectionB, the inspiration comes in part from Harrison & Pliska (1981) and Bensoussan(1984); this latter paper, as well as Karatzas (1988), should be consulted forthe pricing of American options. Material for Subsection C was drawn frommore general results in Karatzas, Lehoczky & Shreve (1987). The problem ofSubsection D was introduced by Samuelson (1969) and Merton (1971); it hasbeen discussed in Karatzas et al. (1986) on an infinite horizon and with verygeneral utility functions. An application of these ideas to equilibrium analysisis presented in Lehoczky & Shreve (1986). See also Duffie (1986) and Huang(1987).


CHAPTER 6

P. Levy's Theory of BrownianLocal Time

6.1. IntroductionThis chapter is an in-depth study of the Brownian local time first encounteredin Section 3.6. Our approach to this subject is motivated by the desire toperform computations. This is manifested by the inclusion of the conditionalLaplace transform formulas of D. Williams (Subsections 6.3.B, 6.4.C), thederivation of the joint density of Brownian motion, its local time at the originand its occupation time of the positive half-line (Subsection 6.3.C), and thecomputation of the transition density for Brownian motion with two-valueddrift (Section 6.5). This last computation arises in the problem of controllingthe drift of a Brownian motion, within prescribed bounds, so as to keep thecontrolled process near the origin.

Underlying these computations is a beautiful theory whose origins can betraced back to Paul Levy. Levy's idea was to use Theorem 3.6.17 to replacethe study of Brownian local time by the study of the running maximum (2.8.1)of a Brownian motion, whose inverse coincides with the process of first passagetimes (Proposition 2.8.5). This latter process is strictly increasing, but increasesby jumps only, and these jumps have a Poisson distribution. A precise state-ment of this result requires the introduction of the concept of Poisson randommeasure, a notion which has wide application in the study of jump processes.Here we use it to provide characterizations of Brownian local time in termsof excursions and downcrossings (Theorems 2.21, 2.23).

In Section 6.3 we take up the study of the independent, reflected Brownianmotions obtained by looking separately at the positive (negative) parts of astandard Brownian motion. These independent Brownian motions are tiedtogether by their local times at the origin, a fact which does not violate theirindependence. Exactly this situation was encountered in the Discussion of


400 6. P. Levy's Theory of Brownian Local Time

F. Knight's Theorem 3.4.13, where we observed that intricately connectedprocesses could become independent if we time-change them separately andthen forget the time changes. The first formula of D. Williams (Theorem 3.6)is a precise statement of what can be inferred about the time change fromobserving one of these reflected Brownian motions.

Section 6.4 is highly computational, first developing Feynman-Kac for-mulas involving Brownian local time at several points, and then using theseformulas to perform computations. In particular, the distribution of local timeat several spatial points, when the temporal parameter is equal to a passagetime, is computed and found to agree with the finite-dimensional distributionof one-half the square of a two-dimensional Bessel process. This is the Ray-Knight description of local time; it allows us finally to prove the Dvoretzky-Erdos-Kak utani Theorem 2.9.13.

6.2. Alternate Representations ofBrownian Local Time

In Section 3.6 we developed the concept of Brownian local time as the densityof occupation time. This is but one of several equivalent representations ofBrownian local time, and in this section we present two others. We begin witha Brownian motion W = {147,, ,F,; 0 < t < co} where P[Wo = 0] = 1 and {Ft}satisfies the usual conditions, and we recall from Theorem 3.6.17 (see, inparticular, (3.6.34), (3.6.35)) that

(2.1) = M, - B,, 2L,(0) = Mt; V 0 t < co] = 1,

where B, = sgn(Ws) AK is itself a Brownian motion,

(2.2) M,= max 135; 0 < t < co,o<s<i

and L,(0) is the local time of W at the origin:

(3.6.2) L,(0) = lim 1 meas {0 < s < t; I147,1 < e }.

Thus, a study of the loca;ti0 4time of W at the origin can be reduced to a studyof the more easily conceived process M. The idea of this reduction and muchof what follows originated with P. Levy (1939, 1948).

A. The Process of Passage Times

The process M of (2.2) is continuous, nondecreasing, and "flat" (constant)during excursions of the reflected Brownian motion I WI away from the origin.Because the Lebesgue measure of the set {0 < t < co; I W11 = 0} is zero, P-a.s.

6.2. Alternate Representations of Brownian Local Time 401

(Theorem 2.9.6), the process M has time derivative almost everywhere equalto zero; it is very much like the Cantor function.

We may regard M as a new clock, which stops when I WI is away from theorigin and runs at an accelerated rate when I WI is at the origin. In order topass from this new clock to the original one, we introduce the right-continuousinverse of M, defined as in Problem 3.4.5:

(2.3) Sb inf t 0; M, > b1 = inf {t 0; B, > b}; 0 b < oo.

Each Sb is an optional (and hence also a stopping) time of the right-continuousfiltration f<F,1. The left-continuous inverse of the process M is simply thefamily of passage times

(2.4) Tb inf tt 0; M, = b1 = inf t 0; B, = 0 b < co.

Regarded as processes parametrized by the spatial variable b, T = {Tb;0 5 b < col and S = {Sb; 0 < b < co} are modifications of one another(Problem 2.7.19). The existence of two inverses for M reflects the fact that Midentifies the starting and ending points of the excursions of I WI away fromthe origin. These excursions correspond to jumps of the process T, as we nowelaborate.

Recall from Remark 2.8.16 that for each t > 0, the time in se [0, for whichB, = M, is almost surely unique. Thus, for each w in an event SI* of probabilityone, this assertion holds for every rational t. Fix w e Se, a positive number t(not necessarily rational), and define

(2.5) y,(w) sup {s e [0, t]; Ws(co) = 0} = sup {s e [0, a Bs(co) = M,(w)1,

(2.6) 13,(w) inf {s e [t, co); W,(w) = 0} = inf {s e [t, co); Bs(co) = M,(01.

If 14/,(co) = 0, then /3,(w) = t = y,(w). We are interested in the case W,(w) # 0,which implies

(2.7) < t < A(0).

In this case, the maximum of Bs(w) over 0 < s < t is attained uniquely ats = y,(w), hence 7,,,,,(w)(w) = y,(w). Similarly Tww),(w) = Sw,)(w) = f3,(w), forotherwise there would be a rational q > 13,((o) such that Bfit(w)(w) = Bm,)(w) =WO, a contradiction to the choice of w e Sr. We see then that for w eand t chosen so that W(w) 0 0, the size of the jump in Tb(w) at b = M,(w) isthe length of the excursion interval (y,(w), Ma))) straddling t:

(2.8) TAftoo+ (0) - Tut( (0(0 = - Y*0)

It is clear from (2.4) that Tb is strictly increasing in b, and To = 0. It is lessclear that T grows only by jumps. To see this, consider the zero set of W(w),namely

22'1, A {0 < t < co; Wt(w) = 0},

which is almost surely closed, unbounded, and of Lebesgue measure zero(Theorem 2.9.6). As with any open set, £ n (0, co) can be written as a countable

union of disjoint open intervals

27. n (0, co ) = U 4(0,ae A

and each of these intervals contains a number ta(w). In the notation of (2.5),(2.6), we have

la = (ye., A.); a G A.

Because meas(Aeb,) = 0 for P-a.e. w e S/, we have

= E yi.) = E (Tm. - E (Tx, - Tx);me A aeA ° ° x<M,

almost surely. Now set t = Ta to obtain

E (Tx, - Tx), a < oo;x <a

0 < t < co,

letting a T b and using the left-continuity of T, we see finally that, except on aP-null set,

Tb E (Tx+ - Tx), 0 b < oo.x<b

The reverse inequality is obvious.We summarize our observations thus far.

2.1 Theorem. The processes S = {Sb; 0 < b < co} of (2.3) and T = {Tb;0 < b < col of (2.4) are modifications of one another. Being the right-(respectively, left-) continuous inverses of the process M in (2.2), they arestrictly increasing. Moreover, they increase by jumps only:

(2.9) Sb = > (sx-sx_), T, = E (Tx+ - Tx); 0 b < co,x 0,

(2.11) Ee-asb e-bf2;; 0 < a, b < co.

PROOF. The only new assertions are those contained in the last sentence; thesefollow from Proposition 2.8.5.

It is apparent from Theorem 2.1 that T must have infinitely many jumpson any interval [0, 6], b > 0, but T can have only finitely many jumps whosesize exceeds a given number ( > 0. We want to know the distribution of the(random) number of such jumps. To develop a conjecture about the answer

to this question, we ask another. How large must b be in order for T to have ajump on [0, b] whose size exceeds e? We designate by T, the minimal suchb, and think of it as a "waiting time" (although b is a spatial, rather thantemporal, parameter for the underlying Brownian motion W). Suppose wehave "waited" up to "time" c and not seen a jump exceeding size e; i.e., T, > c.Conditioned on this, what is the distribution of T,? After waiting up to "time"c, we are now waiting on the reflected Brownian motion I = W.+T,WTI to undergo an excursion of length exceeding t; thus, the conditional dis-tribution of the remaining wait is the same as the unconditional distributionof T,:

(2.12) P[T, > c + bit, > c] = P[T, > b].

This "memoryless" property identifies the distribution of T1 as exponential(Problem 2.2).

2.2 Problem. Let / = [0, co) or / = R. Show that if G: 1 -* (0, co) is monotoneand

(2.13) G(b + c) = G(b)G(c); b, c e I,

then G is of the form G(b) = e-'i6; be I, for some constant .1 e R. In particular,(2.12) implies that for some A > 0,

P[-i- > b] = e-'i6; 0 < b < co.

After T,, we may begin the wait for the next jump of T whose size exceedse, i.e., the wait for I W+7-,1 = +7. WT I to have another excursion ofduration exceeding e. It is not difficult to show, using the strong Markovproperty, that the additional wait t2 is independent of Tl. Indeed, the "inter-arrival times" T1, t2, r3, ... are independent random variables with the same(exponential) distribution. Recalling the construction in Problem 1.3.2 of thePoisson process, we now see that for fixed b > 0, the number of jumps of theprocess T on [0, b], whose size exceeds e, is a Poisson random variable. Toformalize this argument and obtain the exact distributions of the randomvariables involved, we introduce the concept of a Poisson random measure.

B. Poisson Random Measures

A Poisson random variable takes values in No A {0, 1, 2, ... and can bethought of as the number of occurrences of a particular incident of interest.Such a concept is inadequate, however, if we are interested in recording theoccurrences of several different types of incidents. It is meaningless, for example,to keep track of the number of jumps in (0, b] for the process S of (2.3), becausethere are infinitely many of those. It is meaningful, though, to record thenumber of jumps whose size exceeds a positive threshold e, but we would like

to do this for all positive simultaneously, and this requires that we not onlycount the jumps but somehow also classify them. We can do this by lettingv((0, b] x A) be the number of jumps of S in (0, b] whose size is in A e.4((0, co)),and then extending this counting measure from sets of the form (0, b] x A tothe collection .4((0, co)2). The resulting measure v on ((0, co)2, .4((0, co)2)) willof course be random, because the number of jumps of {S.; 0 < a < b} withsizes in A is a random variable. This random measure v will be shown to bea special case of the following definition.

2.3 Definition. Let (S1,F, P) be a probability space, (H, Yr) a measurablespace, and v a mapping from SI to the set of nonnegative counting measureson (H, Jr), i.e., v,,,(C)e No u {co} for every co ES./ and C e Jr. We assumethat the mapping coi-+ v,,,(C) is .F/M(No u {co})-measurable; i.e., v(C) is anNo u {co}-valued random variable, for each fixed Ce Ye. We say that v is aPoisson random measure if:

(i) For every C E A', either P[v(C) = co] = 1, or else

A(C) -A Ev(C) < co

and v(C) is a Poisson random variable:

P[v(C) = e--A(c)(A(C))"; neNo.

n!

(ii) For any pairwise disjoint sets C1, ..., Cm in Jr, the random variablesv(C, ), v(Cm) are independent.

The measure A(C) = Ev(C); Cele, is called the intensity measure of v.

2.4 Theorem (Kingman (1967)). Given a o--finite measure A on (H, Jr), thereexists a Poisson random measure v whose intensity measure is A.

PROOF. The case A(H) < co deserves to be singled out for its simplicity. Whenit prevails, we can construct a sequence of independent random variables

. with common distribution PN1 E = A(C)/01(11); C E dr, as well as anindependent Poisson random variable N with P[N = n] = e-411)(A(H))7n!;n e No. We can then define the counting measure

v(C) A hi 1c(4 C Ei=i

It remains to show that v is a Poisson random measure with intensity A. Givena collection C1, ..., Cm of pairwise disjoint sets in Y9, set Co = H Uir=i Cso E,7=0 v(C,,) = N. Let no, n1, nm be nonnegative integers with n = no +ni + + n,. We have

P[v(C0) = no, v(C1) = n1, ..., v(Cm) = nm]

= P[N = n] P[v(C0) = no, v(Ci) ni, v(Cm) = n,,,IN

= e(.1,(H))" n! (A(C0) yo 0.(Ci ) r 0.(Cm)r"

?Tn! no! ni!... ?T.! (1.(H) ) A(H) ) A(H)

nn e_A(co(A(Ck)r"

k =0 nk!

and the claim follows upon summation over no E No.

2.5 Problem. Modify the preceding argument in order to handle the case ofo- finite A.

C. Subordinators

2.6 Definition. A real-valued process N = {Ng; 0 < t < co} on a probabilityspace (1/, <F,P) is called a subordinator if it has stationary, independent incre-ments, and if almost every path of N is nondecreasing, is right-continuous,and satisfies No = 0.

A prime example of a subordinator is a Poisson process or a positive linearcombination of Poisson processes. A more complex example is the processS = {S; 0 < b < co } described in Theorem 2.1. P. Levy (1937) discovered thatS, and indeed any subordinator, can be thought of as a superposition ofPoisson processes.

2.7 Theorem (Levy (1937), Hinein (1937), Ito (1942b)). The moment gen-erating function of a subordinator N = {N1; 0 ._ t < co } on some (SI, F, P) isgiven by

(2.14) Ee'N, = exp[ -t {mot + J (1 - e'e)ii(de)}]; t > 0, a > 0,(o,c0)

for a constant m 0 and a cr-finite measure /..i on (0, co) for which the integralin (2.14) is finite. Furthermore, if i7 is a Poisson random measure on (0, c0)2 (ona possibly different space (0, g", P)) with intensity measure

(2.15)

then the process

A(dt x de) = dt it(de),

(2.16) N, A mt + J ev((0,t] x de); 0 < t < co(0, co)

is a subordinator with the same finite-dimensional distributions as N.

2.8 Remark. The measure it in Theorem 2.7 is called the Levy measure of thesubordinator N. It tells us the kind and frequency of the jumps of N. Thesimplest case of a Poisson process with intensity A > 0 corresponds to m = 0

andµ which assigns mass A. to the singleton { 1 }. If y does not have supporton a singleton but is finite with A. = 14(0, co)), then N is a compound Poissonprocess; the jump times are distributed just as they would be for the usualPoisson process with intensity ;4., but the jump sizes constitute a sequence ofindependent, identically distributed random variables (with common distribu-tion it(de')/),), independent of the sequence of jump times. The importanceof Theorem 2.7, however, lies in the fact that it allows /.4 to be a-finite; this isexactly the device we need to handle the subordinator S = {Sb; 0 < b <which has infinitely many jumps in any finite interval (0, b].

PROOF OF THEOREM 2.7. We first establish the representation (2.14). Thestationarity and independence of the increments of N imply that for a > 0,the nonincreasing function p °(t) A Ee-xl'4; 0 < t < co, satisfies the functionalequation pc,(t + s) = pc,(t)p(s). It follows from Problem 2.2 that

(2.17) Ee'N' = e-"I1(2); t > 0, a > 0,

holds for some continuous, nondecreasing function 0: [0, co) -* [0, co) with0(0) = 0. Because the function g(x) A xe-";x > 0, is bounded for every a > 0,we may differentiate in (2.17) with respect to a under the expectation sign toobtain the existence of ti/ and the formula

(a)e-"1/(2) =1E[N,C°N1

1- 8e-21 P[N,Ed1]; a > 0, t > 0.t t

Consequently, we can write

(2.18) Oa) = lim kck J (1 +k-'c c [0,00)

where

ck E[1 +1N 1, p,(&) A cl-I-k 1

(( P[N,/, e de].

1/,

If N 0 a.s., the theorem is trivially true, so we may assume the contrary andchoose a > 0 so that c E(IVI 1{bri <a}) is positive. For ck we have the bound

1(2.19) ck > E[

1 +1NA

m1{N )] >

1 + aE[N11,1{N<all

i

=1 k

k(1 + a) E E[(N/k -

k(1 + a)

We can establish now the tightness of the sequence of probability measures{ pk } if= 1. Indeed,

P[N, < e] > P[Nfik - Nu_of < e; j = 1, k]

= 1P[Nlik

and thus, using (2.19), we may write

(2.20) MK cc))1 + a

kP[N,,,, >k(1 + a)

{1 (P[N, < tykl.

Because limk k(1 - = -log 0 < < 1, we can make the right-handside of (2.20) as small as we like (uniformly in k), by taking ( large. Prohorov'sTheorem 2.4.7 implies that there is a subsequence {p,5}1°=, which convergesweakly to a probability measure p on ([0, co), d([0, co))). In particular, be-cause the function e + e)e-e is bounded for every positive a, we musthave

lim (1 + e)e-2? pk.,(d()= (1 + e)e-cie p(do.10.c.) [0.c.)

Combined with (2.18), this equality shows that kick, converges to a constantc > 0, so that

(2.21) 1//(a) = c J (1 + e)e-at p(cl()

= cp( {O}) c J (1 + e)e'ep(de); 0 < a < co.op..)

Note, in particular, that i//' is continuous and decreasing on (0, co). From thefundamental theorem of calculus, the Fubini theorem, (2.21), and /i(0) = 0, weobtain now

(2.22) Oa) = acp({0}) + c f (0..)1 + e (1 e-2e)p(de); 0 < a < co.

The representation (2.14) follows by taking m = cp({0}) and

c(1 +p(de) =(2.23) p(de); > 0,

the latter being a a-finite measure on (0, co). In particular, we have from (2.22):

(2.24) tp(c) = ma + (1 - C'e')12(de) < co; 0 < a < cc.(0.00)

We can now use Theorem 2.4 with H = (0, co)2 and lt = ge(H), to constructon a probability space (n, P) a random measure 17 with intensity given by(2.15); a nondecreasing process 13 can then be defined on (n, P) via (2.16).It is clear that Ro = 0, and because of Definition 2.3 (ii), 13 has independentincrements (provided that (2.25), which follows, holds, so the increments are

defined). We show that IV is a subordinator with the same finite-dimensionaldistributions as N. Concerning the stationarity of increments, note that for anonnegative simple function cp(f)=E7=iail,(l) on (0, x), where A1, ...,are pairwise disjoint Borel sets, the distribution of

tp(t) t + h] x (10= + h] x Ai)(o..) i =1

is a linear combination of the independent, Poisson (or else almost surelyinfinite) random variables Iii((t, t + h] x Ai)}7=1 with respective expectations{h1t(Ai)}7=1. Thus, for any nonnegative, measurable co, the distribution of.1.0,00(p(I)v((t,t + h] x dl) is independent of t. Taking (Xi) = t, we have thestationarity of the increment A +h - A.

In order to show that IV is a subordinator, it remains to prove that

(2.25) A < co; 0 t < oo

and that lS is right-continuous, almost surely. Right-continuity will followfrom (2.25) and the dominated convergence theorem applied in (2.16), and(2.25) will follow from the relation

(2.26) Re'" = e-0(x); a > 0, t > 0,

where (I, is as in (2.24). With n > 1, and (1.) A j2-", = (6("1, /1")]; 1 < j < 4",we have from the monotone and bounded convergence theorems:

(2.27)

4^E exp - f ( 17( 0, t] x de)} = lim exp - E 4!), t] x /.;"))} .

(0.oD) n-.co j= 2

But the random variables ii((0, t] x II")); 2 < j < 4" are independent, Poisson,with expectations ty(11")); 2 < j < 4", and these quantities are finite becausethe integral in (2.14) is finite. The expectation on the right-hand side of (2.27)becomes

4" 4"Eexp{ -aer, i7((0, x /l"))} =exp -t (1 - )µ(/1")) ,

! =2 .1 =2

which converges to exp{ -t f(o)(1 - e'f)ki(d1)} as n co. Relation (2.26)follows and shows that for each fixed t > 0, A has the same distribution asN. The equality of finite-dimensional distributions is a consequence of theindependence and stationarity of the increments of both processes.

Theorem 2.7 raises two important questions:

(I) Are the constant m > 0 and the Levy measure it unique?(II) Does the original subordinator N admit a representation of the form

(2.16)?

One is eager to believe that the answer to both questions is affirmative; for

the proofs of these assertions we have to introduce the space of RCLLfunctions, where the paths of N belong.

2.9 Definition. The Skorohod space D[0, co) is the set of all RCLL functionsfrom [0, oo) into R. We denote by AD [0, co)) the smallest a-field containingall finite-dimensional cylinder sets of the form (2.2.1).

2.10 Remark. The space D[0, co) is metrizable by the Skorohod metric insuch a way that M(D [0, co)) is the smallest a-field containing all open sets(Parthasarathy (1967), Chapter VII, Theorem 7.1). This fact will not be neededhere.

2.11 Problem. Suppose that P and P are probability measures on (D[0, co),.4(D [0, co))) which agree on all finite-dimensional cylinder sets of the form

1Y GDP, GO; Ati)Gri, Y(WernI,

where n > 1, 0 < t, < t2 < < < co, and Fie M(R); i = 1, n. Then Pand P agree on .4(D [0, co)). (Hint: Use the Dynkin System Theorem 2.1.3.)

2.12 Problem. Given a set C g_ (0, co)2, let n(; C): D[O, co) -> fro u {co} bedefined by

(2.28) n(y; A # {(t,e)E C; IY(t) - -)I = el,where # denotes cardinality. In particular, n(y; (t, t + h] x ((, co)) is the num-ber of jumps of y during (t,t + h] whose sizes exceed e. Show that n(; C) is.11(D[0, co))-measurable, for every Ce.4((0, co)2). (Hint: First show thatn(; (0, t] x (i' , co)) is finite and measurable, for every t > 0, ( > 0.)

Returning to the context of Theorem 2.7, we observe that the subordinatorN on (SI,,F,P) may be regarded as a measurable mapping from (12,") to(D [0, co), .4(D [0, co))). The fact that Ar defined on (5 , g, P) by (2.16) has thesame finite-dimensional distributions as N implies (Problem 2.11) that N and

induce the same measure on D[0, co):

PEN e A] = P[KI e A]; V A e .4(D [0, oo)).

We say that N under P and under P have the same law. Consequently, forC1, C2, C, in gf((0, co)2), the distribution under P of the random vector(n(N; . , n(N; Cm)) coincides with the distribution under 15 of (n(N; C1),

, n(N; Cm)). But

(2.29) n(g ; C) = ("(C); C e *(0, 00) x 00))

is the Poisson random measure (under P) of Theorem 2.7, so

(2.30) v(C) A n(N; C) = # {(t, eC; N, - Nt_ = e}; C e .4((0, al')

is a Poisson random measure (under P) with intensity given by (2.15).

We observe further that for t > 0, the mapping qv D[0, cc) -* [0, cc] definedby

(My) A f in(y; (0, t] x do)(o,o0)

is Af(D [0, oo))/.4([0, co])-measurable, and

co,(N) = f (v((0, t] x do, (MR) = f mo,t] x de).co,c0) co,c0)

It follows that the differences {N, - pi(N); 0 < t < col and {A - cp,(R);0 < t < col have the same law. But A - co,(N). mt is deterministic, and thusN, - q),(N) = mt as well. We are led to the representation

(2.31) N, = mt + i 0v((0, t] x dO); 0 < t < 00.

We summarize these remarks as two corollaries to Theorem 2.7.

2.13 Corollary. Let N = {Nt; 0 < t < co} be a subordinator with moment gen-erating function (2.14). Then N admits the representation (2.31), where v givenby (2.30) is a Poisson random measure on (0, cc)2 with intensity (2.15).

2.14 Corollary. Let N = INt; 0 < t < col be a subordinator. Then the constantm .. 0 and the a-finite Levy measure it which appear in (2.14) are uniquelydetermined.

PROOF. According to Corollary 2.13, a(A) = Ev((0, 1] x A); A E MO, 00), wherev is given by (2.30); this shows that it is uniquely determined. We may solve(2.31) for m to see that this constant is also unique.

2.15 Definition. A subordinator N = {Nt; 0 < t < co} is called a one-sidedstable process if it is not almost surely identically zero and, to each a > 0,there corresponds a constant 11(a) 0 such that {aNt; 0 < t < co } and {Now;0 < t < col have the same law.

2.16 Problem. Show that the function Ma) of the preceding definition is con-tinuous for 0 < a < cc and satisfies

(2.32)

as well as

(2.33)

Nay) = Q(a)NY); cc > 0, Y > 0

i I t (a) = r i 3 (a); a > 0,

where r = it/(1) is positive and 4./ is given by (2.17), or equivalently, (2.24). Theunique solution to equation (2.32) is 13(a) = a', and from (2.33) we see that fora one-sided stable process N,

(2.34) Ee'N. = exp{ - till(a)} = exp{ - traE}; 0 < a < oo.

The constants r, c are called the rate and the exponent, respectively, of theprocess. Because b is increasing and concave (cf. (2.21)), we have necessarily0 < e < 1. The choice E = 1 leads to m = r, t = 0 in (2.14). For 0 < E < 1, wehave

tit(2.35) m = 0, m(dt) =

r(1rE E) e' +g; e

D. The Process of Passage Times Revisited

The subordinator S of Theorem 2.1 is one-sided stable with exponent E = (1/2).Indeed, for fixed a > 0, (1/ot)Sb,A; is the first time the Brownian motion(Lemma 2.9.4 (i)) W* = {14/,* (1/.,/a) Wcu; 0 < t < oo } reaches level b; i.e.,

1-az

Sb,A,= inf It 0; 147,* > bl.

Consequently, {otSb; 0 < b < co} has the same law as laSt, = Sb,";; 0 < b < col,from which we conclude that Noz) appearing in Definition 2.15 is NAi Com-parison of (2.11) and (2.34) shows that the rate of S is r = .\/2, and (2.35) givesus the Levy measure

deit(de) = /2n83;2e3

; e > 0.

Corollary 2.13 asserts then that

Sb eq(0,13] x de); 0 < b < oo,to. co)

where, in the notation of (2.1)-(2.4), (2.30)

(2.36) v(C) = # {(b,e) e C; Sb Sb- = el

= # {(b,e) e C; WI has an excursion of duration e starting attime Tb }; CeI((0, co )2 ),

is a Poisson random measure with intensity measure (dt de'1,/27re'3). In par-ticular, for any I eM((0, co)) and 0 < 5 < E < oo, we have

(2.37) Ev(I x [5,0) = meas(/)27c,13

= meas(/)2 (dA

For 0 <6<e<co and b > 0, let us consider the random variables

(2.38) Ne, v((0,13] x [6,0)

= Number of jumps of size e e [6,E) suffered by 0 < a < 131,

1[

(2.39) Lb'' A tv ((0,6] x di')a, e)

= Total length of jumps of size 1 eR v) suffered by{Sa; 0 a 5 b}.

We also agree to write

Nt, A N,- = v40,13] x R cc)), 14 A L'.2,'' = 1 ev((0,13] x de).(o,e)

The process Np = {Np; 0 < b < op} is Poisson with intensity02170411.A - (1/,/i)); the process La- = {L:- ; 0 < b < op} is a sub-ordinator which grows only by jumps (see the last paragraph of the proof ofTheorem 2.7). Furthermore,

rr(2.40) Ee-'11` = exp -b (1 - e-'1)

de 1,L e) ,./21re3 _I

and these assertions concerning LV hold even if 6 = 0.The behavior of N: as 6 1 0 merits some attention. Of course, limayo N: = co

a.s., and any meaningful statement will require some normalization.

2.17 Proposition. For almost every we SI,

tr6(2.41) lim -NI°, (o)) =- b; V 0 5 b < 00.

PROOF. The process

Q, 4 v (0, 13] x[12 , co)); 0 5 t < op,

has nondecreasing, right-continuous paths and independent increments. For0 s < t, the increment Qt -Qs= v((0,13] x [t-2, s-2)) is Poisson withexpectation

al.()

S

-2de

E(Qt - Qs) = b ft-2 2tre3

We conclude that Q is a Poisson process, for which the strong law of largenumbers of Remark 1.3.10 gives lim (Q,It)= b \/(2 /n) a.s. By definition wehave NI, = r ,Q 1/si a so (2.41) holds a.s. for each fixed b. Except for w in a nullset A g. Q, this relation must hold for all rational, nonnegative b. The mono-tonicity in b of sides of (2.41) then gives us its validity for w et-1\A, 0 < b < o o.

2= -(t - s).

it

0

As c 10, the dominated convergence theorem shows that L 0 a.s. In otherwords, since Sb is finite, the total length of its jumps of size less thane must

approach zero with E. We thus normalize Lb in a manner "opposite" to thenormalization of (2.41).

2.18 Proposition. For almost every we SI,

(2.42) lim .\/1 Lb(w) = b; V 0 0 and 0 1:

(2.43 2k)E - v((0,b] x L ep2k, cp2(k -1)p ))

k=1

co< E cp2(k-1)V(0, 6] x [8p2k,cp2(k--1))).k=1

The left-hand side of (2.43) may be written as

EI pk.s/ep2k pk+1N/ep2(k-1)Nbep2(k-1))

k=1

which converges as el 0 (see (2.41)) to

"b E (pk _pk+i)= bk=1

2 pn+1)

It follows that limg10(1//04 > b.0211r)(p - pn+1), a.s., and letting firstn co and then p T 1, we obtain

lim1

> a.s.Lb2

Now let 1(g) suPo<s <eiNF5Nt - b021701, so that limeio 1/(g) = 0 a.s.(Proposition 2.17). The right-hand side of (2.43) is subject to the bound

E .ip2(k-1)1#0, b]X

[gp2k, gp2(k-1)))k=1

Ipk-2,/,p2kNo2k_ pk-l.s/spk=1

k-=-1

\fitpk-2 (b".,(8)) pk-1 (b

- ti(E)/ }

b_ 2 17(0(1

1

2

-PIt follows that lime4.0 (1/N/i)Li, < (b/P),./(2/n). Letting p T 1, we obtain

1im -1/-Lb b a.s.

e.Lo .s/E

This proves that (2.42) holds a.s. for each fixed b. We conclude as in Propo-sition 2.17.

2.19 Exercise. Show that for fixed, positive 6, the process X,(6) A N,6, -b,/2/7r6; 0 b = biir6; 0 b < co.

Furthermore, we have the law of large numbers

1

N't), = .\/- 2, a.s.

6* co TEO

2.20 Exercise. Show that for fixed, positive c, the process Vb(e) A 4, -13\/(2E/n); 0 b = -3

Establish the representation

(2.45) = + NI, d(; 0 < b < co, a.s.,f(o,e)

as well as the law of large numbers

(2.46) lim1-14 =b

a.s.

(Hint: Obtain the moment generating function

121 1 )(2.47) Ee'b(e) = exp ab - -b (1 - e-'1)J

NI 21E1'3 _I

E. The Excursion and Downcrossing Representationsof Local Time

Let us now discuss the significance of Propositions 2.17, 2.18, for Brownianlocal time. Returning to the context of (2.1)-(2.4) and with v the Poissonrandom measure given by (2.36), we may recast these propositions as

(2.48) P in: IMO, Mt] x LE, CO)) = Mt; V 0 5_ t < cd= 1,e 2

(2.49) P[lim fli f (v((0,4] x do) = M,; d0 t < cod = 1.40 LE (0,0

But, by (2.36),

v((0, M,] x [c, co)) = # {jumps of size > e suffered by IS,,; 0 < b < AI}

= # excursion intervals away from the origin,of duration ze, made by I Wu; 0 < u < SAO '

As observed earlier, if we have W 0 0 so that (2.7) holds, then Sm, = A is thetime of conclusion of the excursion of W straddling t, and

{excursionintervals away from

(2.50a) v((0, Mt] x [e, co)) = # the origin, of duration >c, .

initiated by W before t

On the other hand, if W = 0, then

{excursionintervals away from the

/14(2.50b) v((0, ,] x [c, oo)) = # origin, of duration .c, initiatedby W at or before time t

Instead of the expressions on the right-hand side of (2.50a, b), it is perhapseasier to visualize the "number of excursion intervals away from the origin,of duration >c, completed by W at or before time t." This expression differsfrom v((0, Mt] x [e, co)) by at most one excursion, and such a discrepancy incounting would be of no consequence in formulas (2.48), (2.49): in the former,it would be eliminated by the factor ,,/i as c 10; in the latter, the effect on theintegral would be at most e, and even after being divided by 4, the effectwould be eliminated as E 10. Recalling the identifications (2.1), we obtain thefollowing theorem.

2.21 Theorem (P. Levy (1948)). The local time at the origin of the Brownianmotion W satisfies

L,(0) = lim \i-c- #,l.o 8 duration -_c, completed by {Ws; 0 s t}

excursion intervals away from the origin, of

Total duration of all excursion intervals away= from the origin of individual duration <c, ,

40 completed by {Ws; 0 _<_ s < t}

V 0 < t < co, a.s.

2.22 Remark. The notion of local time 2L,(0) as occupation density suggeststhat this quantity is determined by the behavior of the Brownian path W near,rather than at, the origin. Theorem 2.21 shows that local time is actuallydetermined by the way in which Brownian motion spends time at the origin,

for both representations in that theorem can be computed from knowledgeof the zero set J1, = 10 < t < W, = 01 alone.

A third representation of local time in the spirit of Theorem 2.21, thedowncrossings theorem, was conjectured by Levy (1959) and proved by Ito &McKean (1974). We offer a proof taken from Stroock (1982).

Recalling the notation introduced immediately before Theorem 1.3.8, let

Di(E) = E; I WI)

be the number of downcrossings of the interval [0, a] by the Brownian motionWsl; 0 s tl.

2.23 Theorem (P. Levy's Downcrossings Representation of Local Time). Thelocal time at the origin of the Brownian motion W satisfies

(2.51) 24(0) = lim ENE); 0 < t < co, a.s.0

PROOF. For fixed a > 0, let us define recursively the stopping times To 0 and

an 4 inf {t > tn-i; I II = e }, to itif{t > an; III = 0}

for n > 1. With nn = an - rn -1 = rn an, we have from the strong Markovproperty as expressed by Theorem 2.6.16 that the pairs (n 1> areindependent and identically distributed. Moreover, Problem 2.8.14 assertsthat

(2.52) E1 =E2'

and we also have

(2.53) { j < NO} = {-5 < t} is independent of {(rh, ,)},T=J+,.

Suppose that t E [an, t,,) for some n > 1; thenn-i

IlWtAt,} -1WtAn,11 = -1wQ,11+ 147,IJ =1 j=1

= -EDi(E) 1Wil e

If t U,T=1 [an, r), then ET=1{1141tAr,} -1WtAn,1} = aDt(e). In either case,

(2.54) W, T I - iWt,n,11 = -0,(a) + WI - a) l[ni,)(t)J =1 J=1

On the other hand, the local time L.(0) is flat on U;,'°_1 [a, c) (cf. Problem3.6.13 (ii)), and thus from the Tanaka formula (3.6.13) we obtain, a.s.:

"(2.55) {1WtA - I WtAt7,11 = 1Wt1 2L,(0) - E sgn(147,)dWs.iJ =1 J=1 tAr.,

From (2.54), (2.55) we conclude that

(2.56) el),(e) - 2L,(0) = -I WI - 11,,r5)(t)J=1 J=1,,

+ E sgn(14/s)dWs, a.s..i=1 A 5_,

2.24 Problem. Conclude from (2.56) that, for some positive constant C(t)depending only on t, we have

(2.57) EleD,(E) - 2L,(0)12 < C(t)s.

Ceby§ev's inequality and (2.57) give

P[In-2A(n-2) - 2L,(0)I > n-114] < C(t)n- 312 ,

and this, coupled with the Borel-Cantelli lemma, implies

lim n-2D,(n-2) = 2L,(0), a.s.n-oco

But for every 0 < E < 1, one can find an integer n = n(e) 1 such that(n + 1)-2 < E < n-2, and obviously

(n + 1)-2D,(n-2) < eD,(c) < n-2 D,((n + 1)-2)

holds. Thus, (2.51) holds for every fixed t E [0, o0); the general statement followsfrom the monotoncity in t of both sides of (2.51) and the continuity in t of

2.25 Remark. From (2.51) and (2.1), (2.2) we obtain the identity

(2.58) lim ED[o ti(0; E; M(w) B(w)) = 114,(w); 0 < t < ooE

for P-a.e. w E a The gist of (2.58) is the "miraculous fact," as Williams (1979)puts it, that the maximum-to-date process M of (2.2) can be reconstructedfrom the paths of the reflected Brownian motion M - B, in a nonanticipativeway. As Williams goes on to note, "this reconstruction will not be possible forany picture you may draw, because it depends on the violent oscillation of theBrownian path." You should also observe that (2.58) offers just one way ofcarrying out this reconstruction; other possibilities exist as well. For instance,we have from Theorem 2.21 that

limo 2

- #

lim40

excursion intervals away from theorigin, of duration > c, completedby {M, - 135; 0 s t}

total duration of all excursionintervals, away from the origin,

M,; V 0 t < cc,

of individual duration <c, = M;completed by { M, - B5; 0 < s < t}

hold almost surely.

V 0 < t < co,

6.3 Two Independent Reflected Brownian Motions

Our intent in this section is to show how one can create two independent,reflected Brownian motions by piecing together the positive and negativeexcursions of a standard, one-dimensional Brownian motion. This result hasimportant consequences, and we develop some of them, most notably the firstformula of D. Williams (Theorem 3.6). Our basic tools will be the F. KnightTheorem 3.4.13 and the theory of Brownian local time as developed in Section3.6; we shall retain the setting, assumptions, and notation of that section.

A. The Positive and Negative Parts of a Brownian Motion

We start by examining the Tanaka formulas (3.6.11-12) a bit more closely.With a = z = 0, L(t) A L1(0), and

(3.1) /+(t) f 100)(WO dW,, LW A ft 1(-00,0")d117,,.3 0

these formulas read

(3.2) W ± = -/ +(t) + L(t); 0 _5 t < oo

a.s. P°. The processes in (3.1) are continuous, square-integrable martingales,with quadratic variations

(3.3) <1 + >(t) = T +(t) meas {0 s 5 t; Ws> 0}

(3.4) <1_>(t) = meas {0 s < t; W, 5 0}

and cross-variation equal to zero:

(3.5) <4, /_ > (t) = 1( )(Ws) 1(,03(14/)ds = 0.

We also have

(3.6) lim t +(t) = co, a.s.t-'cO

from Problem 3.6.30.On the other hand, WI are nonnegative processes which satisfy

(3.7) 1(0,00)(W±(s))dL(s) = 0J

a.s. P° (Problem 3.6.13 (ii ). It becomes evident then from (3.2), (3.7) that thepairs (L, W±-) are solutions to the Skorohod equation of Lemma 3.6.14 for thefunctions -1 +, respectively. From the explicit form (3.6.32) of the solution tothis equation, we deduce

(3.8) L(t) = max / +(s), 147,± = max / +(s) - /±(t); 0 t < co, a.s. P°.osst oss<,

6.3. Two Independent Reflected Brownian Motions 419

Now let us introduce the right-continuous inverses of the occupation timesT+ of (3.3), (3.4), namely

(3.9) (T) = inf {t 0; I" + (t) > r}; 0 < t < GC.

Theorem 3.4.13 asserts, in conjunction with (3.3)-(3.6), that the processes

(3.10) ±(t) A +(I-V(0); 0 5 r<are independent, standard, one-dimensional Brownian motions under P°, andfrom Theorem 3.4.6 we also have the representations

(3.11) I+(t) = B ±(I ±(t)); 0 < t < co.

Here then is the fundamental result of this section.

3.1 Theorem. The processes

(3.12) W+ (T) A + W-_0(,); 0 < t < cc

are independent, reflected Brownian motions under P°.

PROOF. We start by introducing

(3.13) L +(t) -A- L(TV(r))

and observing that because of (3.8), (3.11), and Problem 3.4.5 (ii), (iii) we have,a.s. P°:

(3.14) L +(t) = max /+(s) = max B+(1" +(s)) = max B +(u)05s5r.V(T) 0<r+(s)<r (.114<.r

for all 0 < t < CC; in particular, L are independent, continuous nondecreas-ing processes.

Now for each 0, I"±(T-+1 (T)) = t < 1-41"-±1(r) + (5) for all (5 > 0, andconsequently 14/-L(.0 > 0, 14/1-,1(.0 0 hold a.s. P°. It follows then that

(3.15) W +(t) = Wrt,i(r) = L +(t) -B +(t) = max B ±(u) -B ±(t)O<u<r

also hold a.s. P°, first for a fixed t > 0, and then by continuity, for all t > 0simultaneously. From Theorem 3.6.17, each of the processes W+ is a reflectedBrownian motion starting at the origin; W+ are independent because B are.

3.2 Remark. Theorem 3.6.17 also yields that the pairs {(WAT), L +(T));0 < t < CO} have the same law as {(1W,I, 24(0)); 0 < t < col, under P°.

In particular, we obtain from (3.6.36) and (3.14), (3.15):

1

(3.16) L + (T) = lim meas {0 5_ o- < t; W+ (0) < a.s. P°40 LE

for every T e [0, co). The processes L± are thus adapted to the augmentationsof the filtrations Wzi- 9 a(W+(u); 0 < u < r).


3.3 Remark. As suggested by the accompanying figure, the construction ofW +() = W,-;,(.) amounts to discarding the negative excursions of the Brownianpath and shifting the positive ones down on the time-scale in order to closeup the gaps thus created. A similar procedure, with a change of sign, gives theconstruction of W_() = W-74.). Theorem 3.1 asserts then, roughly speaking,that the motions of Won the two half-lines (0, co) and ( -co, 0) are independent.

wtW +(T) = w

3.4 Problem. Derive the bivariate densities

(3.17) P°[14/, e da; 24(0)e db] = b + 1611 exp (b + la1)2}da db; a eR2trt3 2t

(3.18) P°[ da; 24(0)e db] =2(a + b)exp

(a + 6)2}da db; > 0

.,./27rt3 2t

for b > 0.

3.5 Exercise. Show that the right-hand side of (5.3.13), Exercise 5.3.12, is givenby Sr q,(a) da, where

(3.19) qt(a) = (Ial + t)2} (ll

2tdv .Thd I 'D exp.{

,/21rtexp

2te-

lal


6.3. Two Independent Relected Brownian Motions 421

B. The First Formula of D. Williams

The processes L, of (3.13) are continuous and nondecreasing (cf. (3.14)), withright-continuous inverses

(3.20) L-,1(b) inf {z > 0; L +(r) > 6} = inf 0; max B +(u) > b} .0<u<r

For every fixed b e [0, co), L7,1 (b) is P°-a.s. equal to the passage time

(3.21) Tb-k A TbB± = inf {t > 0; B +(t)_ b}

of the Brownian motion B+ to the level b; cf. Problem 2.7.19 (i).

3.6 Theorem (D. Williams (1969)). For every fixed a > 0, t > 0 we have

(3.22) E°[e'+'(')I W,(u); 0< u< co] = .12aL,(v); a.s. P°.

PROOF. The argument hinges on the important identity

(3.23) f+-1 (r) = t + L:1 (L,(T)) = z + TL-Ao; a.s. P°

which expresses the inverse occupation time TV (r) as t, plus the passage timeof the Brownian motion B_ to the level L,(r). But L +(t) is a random variablemeasurable with respect to the completion of a(W +(u); 0 < u < co), and henceindependent of the Brownian motion B_ . It follows from Problem 2.7.19 (ii) that

(3.24) Lii (L+(r)) = a.s. P°,

and this takes care of the second identity in (3.23). The first follows from thestring of identities (see Problem 3.7)

(3.25) L 11 (L,(r)) = inf t 0; L_(t) > L,(T)}

= inf It > 0; L(Fil(t)) > L(r VW)}

= inf It 0; 1".=1(t), FV(r)}

=1--(1"-Ti(T))=F;1(t)- r+(rV(T))

ET1(7)- t, a.s. P°.

Now the independence of {B_(u); 0 < u < a)} and { W,(u); 0 < u < cc}, alongwith the formula (2.8.6) for the moment generating function for Brownianpassage times, express the left-hand side of (3.22) as

e-cts-.1.2iLAr)E° CiTb b= +(t) a.s. P°. 111

3.7 Problem. Establish the third and fourth identities in (3.25).

Following McKean (1975), we shall refer to (3.22), or alternatively (3.23), asthe first formula of D. Williams. This formula can be cast in the equivalent

forms

(3.26) PITZ1(r) < t I W+(u); 0 < < ao] = [Tb < t - lb-L+(t)

= 2 [1 -(I) ( Li+ (t) )1

rr

t

e -Li (r)/2(0-0= L+(t) dO

2140 - r)3

a.s. P°, which follow easily from (3.23) and (2.8.4), (2.8.5). We use the notation(120(z) = e'2/2 du.

We offer the following interpretation of Williams's first formula. The re-flected Brownian motion { W+(u); 0 < u < co} has been observed, and then atime r has been chosen. We wish to compute the distribution of TV (r) basedon our observations. Now 147_, consists of the positive part of the originalBrownian motion W, but W+ is run under a new clock which stops wheneverW becomes negative. When r units of time have accumulated on this clockcorresponding to W+, r +' (r) units of time have accumulated on the originalclock. Obviously, F+-1(r) is the sum of T and the occupation time 1-1(1-+-1(r)).

Because W_ is independent of the observed process W+, one might surmisethat nothing can be inferred about r_ (t) from W. However, the independencebetween W+ and W_ holds only when they are run according to their respec-tive clocks. When run in the original clock, these processes are intimately con-nected. In particular, they accumulate local time at the origin at the samerate, a fact which is perhaps most clearly seen from the appearance of the sameprocess L in both the plus and minus versions of (3.2). After the time changes(3.12) which transform W ± into W+, this equal rate of local time accumulationfinds expression in (3.13). (From (3.16) we see that L+ is the local time of W+.)In particular, when we have observed W+ and computed its local time L+(r),and wish to know the amount of time W has spent on the negative half-linebefore it accumulated r units of time on the positive half-line, we have arelevant piece of information: the time spent on the negative half-line wasenough to accumulate L+(t) units of local time.

Suppose L+(c) = b. How long should it take the reflected Brownian motionW- to accumulate b units of local time? Recalling from Theorem 3.6.17 thatthe local time process for a reflected Brownian motion has the same distribu-tion as the maximum-to-date process of a standard Brownian motion, we seethat our question is equivalent to: How long should it take a standardBrownian motion starting at the origin to reach the level b? The time requiredis the passage time Tb appearing in (3.23), (3.26). Once L +(T) = b is known,nothing else about W. is relevant to the computation of the distribution of1--(r+-1(-0).

3.8 Exercise. Provide a new derivation of P. Levy's arc-sine law for F+ (t)(Proposition 4.4.11), using Theorem 3.6.

6.3. Two Independent Reflected Brownian Motions 423

C. The Joint Density of (W(t), L(t), r+(t))

Here is a more interesting application of the first formula of D. Williams. WithT > 0 fixed, we obtain from (3.26):

be-b2/2(`-')(3.27) P°[1-_-,-i CO E dt1W+(t) = a; L+(t) = b] - , dt; t < t < co,

- \ / 2n(t - -03

for almost every pair (a, b) of positive numbers. Remark 3.2 and the bivariatedensity (3.18) give

P°[W,(T)e da; L+(t) e db] =2(a, + b)

e_(a+b)2121 da db; a > 0, b > 0,,./2trr3

and in conjunction with (3.27):

(3.28) P° [ W, (r) e da; L_F(T)e db;IT1(t)e dt] = f(a, b; t, 'Oda db dt;

a > 0, b > 0, t > r

where

(3.29) f(a, b; t, t)ba + b) b2 (a + 6)21

_°nr3/2

(

(t - -0312 exP 1 2(t - r) 2t j .

We shall employ (3.28) in order to derive, at a given time t e (0, co), thetrivariate density for the location W, of the Brownian motion; its local timeL(t) = L1(0) at the origin; and its occupation time r+(t) of (0, co) as in (3.3),up to t.

3.9 Proposition. For every finite t > 0, we have

(3.30) P°[ W, e da; L(t) e db; r+(t)e dt]

f(a, b; t, -r)da db dr; a > 0, b > 0, 0 0, 0 0 need be established; the one fora < 0 follows from the former and from the observation that the triples(W L(t), r,(0) and ( -W L(t), t - F+(t)) are equivalent in law.

Now in order to establish (3.30) for a > 0, one could write formally

1

dtr[147+(t)eda; L+(r)e db; rvcoe dt]

1=

dt '-PqW e da; L(t)e db;1-.,(t)e dt]; a > 0, b > 0, 0 < r < t,

and then appeal to (3.28). On the left-hand side of this identity, r is fixed andwe have a density in (a, b, t); on the right-hand side, t is fixed and we have adensity in (a, b, r). Because the two sides are uniquely determined only up tosets of Lebesgue measure zero in their respective domains, it is not clear howthis identity should be interpreted.

We offer now a rigorous argument along these lines; we shall need to recallthe random variable fl, of (2.6), as well as the following auxiliary result.

3.11 Problem. For a e gl, t > 0, c > 0 we have

(3.31) P° [ max Ws > a + c; min Ws < a - ci = o(h)t<s<t-i-h t<s<t+h

and for a > 0, r > 0:

(3.32) P°[W+(r) > a; t TV (r) < t + h; fl, < t + h] = o(h)

(3.33) P°[14/, > a; fl, < t + h] = o(h)

as h 1 O.

PROOF OF PROPOSITION 3.9. For arbitrary but fixed a > 0, b > 0, t > 0, r e (0, t)we define the function

(3.34) F(a, b; t, r) Aco

6 a

which admits, by virtue of (3.28), (3.32), the interpretation

f(a, 13; t, r) da d13

(3.35) F(a, b; t, r)

1= lim -P°[W+(r) > a; L+(t) > b; t _<_ F471(r) < t + h]

hl.o h

1= lim -P°[W+(r) > a; L+(t) > b; t ._<_ 1"471(r) < t + h; 13, t + h].

1,13 h

For every h > 0 we have

1"+(s) = 1"+(t) + s - t, L(s) = L(t); V s e [t, t + h]

on the event 114/, > 0, fl, > t + h }. Therefore, with 0 < c < a and

(3.36) A A {L(t) > b; T - h <1"+(t) r; 13, > t + hl,

we obtain

(3.37)

P°[W.,(r)> a+ e; L +(r) >b; t 51T1(r)<t + h; 13,t +h]- P°[Wt> a; A]

< P° [ max Ws > a + c; 241-P°[ min Ws > a; Al = o(h),t<s<t+h t<st+h

6.4. Elastic Brownian Motion 425

by virtue of (3.31). Dividing by h in (3.37) and then letting h 10, we obtain inconjunction with (3.35), (3.32):

(3.38) F(a + e, b; t, t) < lim -1

PqW, > a; L(t) > b; t -h < r+(t) < aT-1.7i h

Similarly,

(3.38)

P°[Wt> a; A] - PG1W+(z)> a- e; L+(z)>b; tl-Z1(T)<t+h; A > t+h]

< P° [ max Ws > a; /1]- P°[ min Ws > a -E; Al = o(h),t<s<t+h r<s<14-h

and we obtain from (3.33):

1

(3.39) F(a - E, b; t, T) _>_ lim -P°[1,11, > a; L(t) > b; T - h < F+(t) t].h.Lo h

Letting E ,I, 0 in both (3.38), (3.39) we conclude that

1

F(a, b; t, T) = lim -P°[W, > a; L(t) > b; i - h < F,(1) _.<_ r],hlo h

from which (3.30) for a > 0 follows in a straightforward manner. 0

3.12 Remark. From (3.30) one can derive easily the bivariate density

bte-'21"`-')(3.40) P° [24(0) e db; r+ (0 e dr] =

4irr 3/2 (t - -r)3/2db di; b > 0, 0 < r < t

as well as the arc-sine law of Proposition 4.4.11.

The reader should not fail to notice that for a < 0, the trivariate densityof (3.30) is the same as that for (W M 0,) in Proposition 2.8.15, for M, =maxo.(s , Ws, et = sup {s .5 t; Ws = AI. This "coincidence" can be explainedby an appropriate decomposition of the Brownian path {Ws; 0 < s < t }; cf.Karatzas & Shreve (1987).

6.4. Elastic Brownian Motion

This section develops the concept of elastic Brownian motion as a tool forcomputing distributions involving Brownian local time at one or severalpoints. This device allows us to study local time parametrized by the spatialvariable, and it is shown that with this parametrization, local time is relatedto a Bessel process (Theorem 4.7). We use this fact to prove the Dvoretzky-Erdi5s-Kakutani Theorem 2.9.13.

We employ throughout this section the notation of Section 3.6. In par-ticular, W = {W, F; 0 5_ t < co}, (C/, F), {Px}x RI will be a one-dimensionalBrownian family with local time

(4.1) L,(a) = lim -1

meas {0 s t; 1W- al 5 a}; 05t <co,aeIR.40 4a

On a separate probability space (ST, , P'), let R1, R be independent,exponential random variables with (positive) parameters yi , , y,,, respec-tively, i.e.,

P' (R1 e dri, R e dr) = dr,.i=i

We consider n distinct points al, an on the real line and define a newprocess 1P, which is the old Brownian motion W "killed" when local time atany of these points ai exceeds the corresponding level R,. More precisely, with

Tr(at) = inf { t > 0; L,(a,) > r}; r > 0, and

(4.2) inf { t 0; L,(a,) > R, for some 1 < i 5. = min T1<i<n

we define the new process

if A {Wt; 0 t < C,

A; t C,

and call it elastic Brownian motion with lifetime C. Here, A is a "cemetery" stateisolated from R. We may regard l' as a process on C/ g S2 x = F 0 .Y7',fix = Px x P'.

A. The Feynman-Kac Formulas for Elastic Brownian Motion

Our intent is to study the counterpart

(4.3) u(x) = Ex J f(1P-t) exp { -at -J t k(Ifs)ds} dt

= f f(W,) exp --1 - ft k(IP) ds} dt dP' dPx.n n' o

of the function z in (4.4.14). The constant a is positive, and the functions f andk are piecewise continuous on R (Definition 4.4.8), mapping l u {A} into Rand [0, co), respectively, and with

(4.4) f(A) = k(A) = 0.

Hereafter, we will specify properties of functions restricted to R; condition

(4.4) will always be an unstated assumption. In order to obtain an analyticalcharacterization of the function u, we put it into the more convenient form

min r, (a,)

U(X) = r .fw k(K)ds dt dri ...drdPxn 0 0 0 1=1

co rc.f(W)e- - k(K)ds fl dri ...drdt dPx,

f0 1,,(a 1) JL,(a,,) 1=1

whence

(4.5) u(x) = EX f°D f(W)exp {- - ft k(Ws)ds - E yiL,(a,)}dt.1=1

4.1 Theorem. Let f: 01 and k: fly [0, co) be piecewise continuous func-tions, and let D = Df u Dk be the union of their discontinuity sets. Assume that,for some a > 0,

(4.6) EX e' I f(W,)I dt < co; V X E kR,

and that there exists a function Ii: R R which is continuous on R, C1 onR\{ai,...,a,,}, C2 on DIVD u {a1,... , an}), and satisfies

1(4.7) (a + = -2a" + f on OlVD u {a ,a}),

(4.8) (a,+) - -) = ylii(a,); I u.

PROOF. An application of the generalized ItO rule (3.6.53) to the process

X, A ti(W,)exp {-at -jt k(Ws)ds - y,L,(a,)}; 0 t < co,1 =1

yields

nAs,,EX f(W,) exp {- - f k(Ws)ds - y,L,(a,)} dt = a(x) - Ex X. A s,,,

o i=1

where

(4.9) S,, = inf ft 0; IV

If f and u are bounded, we may let n op and use the bounded convergencetheorem to obtain u = u. If f and a are nonnegative, then Ex X A sn > 0 andwe obtain a > u from the monotone convergence theorem.

4.2 Remark. Theorem 4.1 is weaker than its counterpart Theorem 4.4.9 be-cause the former assumes the existence of a solution to (4.7), (4.8) rather thanasserting that u is a solution. The stronger version of Theorem 4.1, whichasserts that u defined by (4.5) satisfies all the conditions attributed to u, isalso true, but the proof of this result is fairly lengthy. In our applications,the function u with the required regularity will be explicitly exhibited. Thesecomments also apply to Theorem 4.3.

A variation of Theorem 4.1 can be obtained by stopping the Brownianmotion W when it exits from the interval [b, c], where -co < b < c < oo arefixed constants not in {al, , With the convention Tx, = co, define

(4.10)

Tb

v(x) = EX [1{Tb <T,} eXP {- Tb - ds y,LTb(ai)}1; b x c.0 i=1

4.3 Theorem. Let k: [b, c] [0, co) be piecewise continuous, a > 0, and assumethat there exists a function 17: [b, c] which is continuous on [b, c], Cl on(b, c)\{ai, , a}, C2 on (b, c)\(LI, u {al, , a}), and satisfies

1

(4.11) (a + k)i5 = -26 on (b, c)\(D, u la , , a}),

(4.12) +) - nai-) = yib(ai); 1 5 i 5 n,

(4.13) 17(b) = 1,

(4.14) i5(c) = 0,

(except that (4.13) should be omitted if c = oo). If 15 is bounded, then 15 is thefunction v of (4.10); if i3 is nonnegative, then 15 > v.

PROOF. With

Y, i5( exp -at - k(14/s)ds - > yiLt(ai)}; 0 t < co,o i=i

the generalized Ito rule (3.6.53) yields i5(x) = Ex X.Tn AT, A /I A Sn; b < x < c, whereS is given by (4.9). If 1i-is bounded, we may let n co to conclude that i5(x) =

1 {Tb <T, }YTb = v(x). If 15 is nonnegative, Fatou's lemma gives

i5(x) > Ex 1{ Tb< YTb = v(x).

The following exercises illustrate the usefulness of Theorem 4.3 in computa-tions of distributions.

4.4 Problem. For any positive numbers a, /3, y, b, justify the formula

6.4. Elastic Brownian Motion

(4.15) Ex exp( -ca'b - 13f +(Tb) - YLTA)

Y + .i2c4 sinh(x.,/2(a + /3)) +.N./2(a + 13)

Y + N/2c4 sinh(b,/2(a + II)) +.,./2(a + /3)

ex 2a

cosh(x.,./2(a + /3))

cosh(b.,/2(a + 11)

Y + ,.xsinh(b,./2(a + /3)) + cosh(b2(a + /9))

.,./2(a + /3)

With x = 0, we obtain in the limit as al 0, iq i 0:

429

; 0 < x < b,

; x < 0.

(4.16) E° exp(-yL,b(0)) = ; y > 0.

In other words, LTb(0) under P° is an exponential random variable with expecta-tion E° LTb(0) = b. On the other hand, as a 10, y 10, (4.15) becomes

(4.17) Exe-or.(To =cosh(x.i2/3);

0 < x < b,cosh(b.,/249)

and we recover (4.4.23) by setting x = 0. Can you also derive (4.4.23) from(2.8.29) and Theorem 3.1 without any computation at all?

4.5 Exercise. Let R, be the first time that the Brownian path W falls b > 0units below its maximum-to-date M, A- max,,,<, Ws; i.e.,

R, = inf {t _- 0; Mt -W, = b}.

Show that

(4.18) E° exp( - aRb -yMRb)=---../2a cosh(bN/2a) + y sinh(bN/2a)

holds for every a > 0, y > 0. Deduce the formula

1(4.19) E° exp(-yMRb) = yb; y > 0,

which shows that MRb is an exponential random variable under P° with expecta-tion E° M Rb = b. (Hint: Recall Theorem 3.6.17.)

.1S

The following exercise provides a derivation based on Theorem 4.1 of thejoint density of Brownian motion, its local time at the origin, and its occupa-tion time of [0, co). This density was already obtained from D. Williams's firstformula in Proposition 3.9.

4.6 Exercise.

(i) Use Theorem 4.1 to justify the Laplace transform formula

(4.20) E° lia'm)(W)e -2' -flf.(t)-yL,(0)dto

= 2e-'12('+9)

.,./2(oc + /3) [y + ,./2a + .12(a + /3)]

for positive numbers a, /3, y, and a.(ii) Use the uniqueness of Laplace transforms, the Laplace transform formula

in Remark 4.4.10, the formula

f.0

b2e-At

bexp 1 1 dt = exp{ - b,./2.1.}; b > 0, A > 0,

o .,/2/rt3

and (4.20) to show that for a > 0, b > 0, 0 < T < t:

(4.21) P°[W, > a; WO) e db; F+(t)e di

7t,t112(t

bW/2 exP { 2(tb-2 T)

(a +2-rb)2}db d- r.

(iii) Use (4.21) to prove (3.30).

B. The Ray-Knight Description of Local Time

We now formulate the Ray-Knight description of local time, evaluated at timet = Tb, and viewed as a process in the spatial parameter.

4.7 Theorem (Ray (1963), Knight (1963)). Let IR .A; 0 < t < ool, (S, °.),{Pr}r>o be a Bessel family of dimension 2, and let b > 0 be a given number.Then 111?; 0 < t < b} under P° has the same law as {L,.(b - t); 0 < t < b}under P°.

In order to prove Theorem 4.7, it is necessary to characterize the finite-dimensional distributions of {R,2; 0 < t 1,

(4.23) f(ti, Yi; tz; Yz; ; 4,, yn) = fn-i(tz - t1, Yz; t3 - tl, Y3; ; to - tl, Yn)n

+ t 1 E Ykfn-k(tk+1 - tk, Yk+1; tk+2 - tk, Yk+2; ; to - tk, yn).k=1

Although we will not need this fact, the reader may wish to verify that

fn(ti, Yi; t2, Yz; ; tn, yn) = 1 + E Ytti E E vivito), - t1)1 <<sn 1 <i<j<n

+ E E E yiyiykti(t; - - +1 < i<J<k <n

+ Y1 Y2 Ynt 1 (t2 t 1 ) (tn 4-1).

We will, however, need the relation

(4.24) fn+i(t yi; t :t: n, , vn, n1, , vn+1)

( Yn+1= [1 + Yn+1(tn+1 - tn)]fn t1, Yl; . ; tn-1, 'A-1; tn, Yn +

1 + Yfl-F1 (4+1 - tn)

valid for n > 1, which is easily proved from (4.22), (4.23) by induction.

4.8 Lemma. For 0 < t1 < t2 < < tn < co and positive numbers yl, , yn,

the Laplace transform of the finite-dimensional distributions of the two-dimensional Bessel process is given by

1 n 1

(4.25) [expL f=E

yiR4}] =1 fn(t1, Yl; ; tn, Yn).

PROOF. The straightforward computation

1 yx 2Ex e-vivt212 = exp

\/ 1 + yt 2(1 +x

gives

(4.26) fir e- YR1/2 = 1 exp Yr2 y > 0, t > 0, r > 0,1 + yt 2(1 + yt)

which proves (4.25) for n = 1. Assume that (4.25) holds for some value of n,and choose 0 t1 < < tn < tn+i < co and positive numbers v,i, Yn, Yn+lFrom the strong Markov property, (4.26), and (4.24), we obtain

1 n+1 }1E° [exp

2 1Eyi/q

=1 '

exp { -- E yin C [exp(--2

yn+/ 12,2.+,) R,. = rni= i n 2} 0

f[0,co),, 2 1=1

X P° [R,1 e dri,... , Rt. e dr.]

exp -! i yiri2 Ern exp --= 21 Y.-Filq.1--t.f[0,00). 21-1

x [R, edri,..., Rt e dr.]

1exp 21

"

LEI(

y,r,2Yn+i r,i/2

1 + Yn+1tn+1 tn)y+i(t+1 -x [R, E dr , , Rt. E dr n]

= yi; ; t., Y.; t.+1, Y.+1).

4.9 Lemma. With 0 = S 1 < <have the Laplace transform formula

(4.27)

< b and positive numbers 61, ..., (5, we

1[exp - OiLT(0}] = - (5.; b - (5.-1; ..; b - 61).

PROOF. Theorem 4.3 implies that the function

V(X) -4 Ex [expf- E SiLTA)}1; < x bi=i

can be found by seeking a bounded, continuous function i3 on (-co, b) whichis linear in each of the intervals (-cc, 0), (0, n b) and whichsatisfies

- = 1 < i < n,

01,) = 1.

Thus, /3 must be of the form

ti(x) = Ci Lix on i4-1], 0 < i < n,

where 4 = -co, = b, and we must have

(4.28) Lo = 0,

(4.29) Ci-i + = Ci + 1/.1; 1 < i < n,

(4.30) Li - = (51(C1_1 + 1 < i < n,

(4.31) C + bL = 1.

These 2n + 2 equations can be solved as follows. Let 131_1 = (C1_1 + 1L1_1)/Co; 1 <i < n + 1, so Bo = 1. We have

Li(4.32) Bi = B1_1 + - 0- 1 i n,

Co

from (4.29), as well as

(4.33) L, = L1_1 + oiCoBi_i; 1 < i < n,

from (4.30). This last identity, along with (4.28), gives

Li = co E afpfri; 1 < i < n,J=1

which, substituted in (4.32), yields

(4.34) B, = 131_1 + - E 6/3;_i; 1 n.J=1

The recursion (4.34) determines B1, B2, ... , B, and (4.31) together with thedefinition of B gives Co = (1/B). The constants L1, L2, ... L are nowdetermined by (4.33), and C1, can be found from (4.29).

Having thus solved the equations (4.28)-(4.31), we may conclude fromTheorem 4.3 that

1Co = = v(0) = v(0) = E° [exp - oiLTigi)}1.

Bn i=i

But comparison of the recursion (4.34) with (4.23) shows that

13; = JilSitl - bi, Si, Sitl - 6i-1; - ; Sitl - S1, 61); 1 < 1 < n. El

PROOF OF THEOREM 4.7. We simply make the identifications t, = b -Yi = in Lemmas 4.8 and 4.9.

Armed with the description of local time in Theorem 4.7, one can providea very simple proof for the Dvoretzky- Erdos- Kakutani Theorem 2.9.13 con-cerning the absence of points of increase on the Brownian path.

PROOF OF THEOREM 2.9.13. We first show that

(4.35) P° [co e W (w) has a point of strict increase] = 0.

The event in (4.35) is equal to Ur,pe(2 Ar where(1r<p

{co ea; 3 0 e[r,p), such that W.(co) < Wo(co) < Wv(w),

V ue[r, 0), V ve (0, p]},

and thus it suffices to prove, for given rationals 0 < r < p, that P°(A,,p) = 0.Considering, if necessary, the Brownian motion { Wr-ht - Wr, t-Fri-t; 0 S t < oo },we may assume that r = 0.

Because of the continuity of Brownian local time, Problem 3.6.13(H),Theorem 4.7, and Proposition 3.3.22, we may choose an event n. S./ withP°(S2 *) = 1 such that for every w e

(4.36) the mapping (t, a)1-* L,(a, co) is continuous,

(4.37) for every b e Q, f 151\ {b}(Ws(co)) dLs(b, co) = 0, and

(4.38) for every beQn (0, co), LT,,,,,)(a, co) > 0; V 0 < a < b.

Suppose that w e SP n A0,,, for some p > 0, let 0 be as in the definition of A 0and choose a (positive) rational number b e (14/e(w), Wy(w)). Let {6,,} ;,°=1 be anondecreasing sequence of nonnegative rational numbers with lim, 6,, =Wo(w). From (4.38) we have

(4.39) 0 < LTb(c,)(We(w), w) = [LT00(14/0(0), w) - LTb(c,)(b, w)]

+ [LTb(c,)(b, w) - Le(b, co)]

+ [Le(b, w) - LTb(c4(b, w)]; V n > 1,

thanks to (4.37). From the nature of 0, the second difference on the right-handside of (4.39) vanishes for all n > 1, and we have lim, Tb.(w) = 0.

We can now let n co in (4.39), and use the joint continuity property (4.36),to arrive at the contradiction 0 < LTb(c,)(Wo(w), w) = 0. This shows thatP°(A0,) = 0, and leads ultimately to (4.35).

Consider now the process 11(w) Wt(w) + t; 0 < t < co, w e O. FromCorollary 3.5.2, there exists a probability measure P on (S2, w) with

1

P (A) = E°[ lA exp { -WT. - Ti]; VA e .FT

holding for every finite T > 0, and such that iP is standard Brownian motionunder P. For every w e 0, the points of increase of W(w) become points ofstrict increase of 1,r(w), and (4.35) shows that

P[w e 0; 14 (w) has a point of increase] = 0.

But the two measures P° and P are equivalent on .FT, and consequently

P° [w e 0; W(w) has a point of increase on [0, T]] = 0

for every fixed T e (0, co). The assertion of the theorem follows easily.

4.10 Exercise. Verify the Cameron & Martin (1945) formulab

1

E° exp(

-13 j. W2 dt} = ; 13 > 0, b > 0o N/cosh(b N/213)

for the Laplace transform of the integral Jo 14/,2 dt. (Hint: Use (4.17).)

C. The Second Formula of D. Williams

n

The first formula of D. Williams (relation (3.22)) tells us how to compute thedistribution of the total time elapsed F+-1(r), given that W+ is being observedand that r units of time have elapsed on the clock corresponding to W. Therelevant information to be gleaned from observing 14/, is its local time 1.+(r).The second formula of D. Williams (relation (4.43)) assumes the same observa-

tions, but then asks for the distribution of the local time 4;4,0) of W at apoint b < 0. Of course, if b = 0, the local time in question is just L +(r), whichis known from the observations. If b < 0, then L,-, i(T)(b) is not known, but asin Williams's first formula, its distribution depends on the observation of W,only through the local time L+(r) in the manner given by (4.43).

4.11 Problem. Under P°, the process {LTb(0); 0 s b < a)} has stationary,independent increments, and

(4.40) E° exp{ -otLTb(0)} =1

aim; a > 0, be R.

4.12 Lemma . Consider the right-continuous inverse local time at the origin

(4.41) p, A inf ft >_ 0; Lt(0) > 4; 0 s s < oo.

For fixed bP°, and

0 0, the process N, A Lp.(b); 0 ._ s < co is a subordinator under

(4.42) Eoe-aNs = expas

a, s > 0.1 + alb'

PROOF. Let Lt = L,(0), and recall from Problem 3.6.18 that lim, L, = a), P°a.s. The process N is obviously nondecreasing and right-continuous withN, = 0, a.s. P°. (Recall from Problem 3.6.13 (iii) that po = 0 a.s. P°.) We havethe composition property p, = p3 + p, o O; 0 < s < t, which, coupled withthe additive functional property of local time, gives P ° -a.s.:

isi, - Als = Lpt_.08..(b) 0 Op.; 0 < s < t.

Note that W, = 0 a.s. According to the strong Markov property as expressedin Theorem 2.6.16, the random variable Lp,_.00..(b) o Op. is independent of Fi,.

and has the same distribution as L ts (b) = N,,. This completes the proof thatN is a subordinator.

As for (4.42), we have from Problem 3.4.5 (iv) for a > 0, /3 > 0:

q --'A E° ro° exp{ -As - ocLp.(b)} ds = E° f exp{ - 134 - aL,(b)} dL,o

T1,E0 e-pc., dLt

fo

+ E° [e-PLTb .i'm exp { -13L, 0 OTb - aL,(b) o 0Tb} d(L, 0 OT,).o

The first expression is equal to [1 - E° e-PLTb]/ 11, and by the strong Markovproperty, the second is equal to ee-flurb times

Eb J exp { 134 - aL,(b)} dL,

03

= Eb f exp{ /IL, - aL,(b)} dL,To

= Eb[e-21-70(b) f exp{ - fL, o °To aLt(b) OTo} d(Lt o 0To)03

oo

= Eqe-"-To(b)]- E° f exp { -13L, - aL,(b)} dL,

= q Eb[e'LT0(b)] = q- E°[e'LTb].

Therefore,

q =1-[1 - E°e-PLTb] + q E° [e-I3LTb] E°[e-2LTb],

and (4.40) allows us to solve for

CO

qe-fisE°(e-"i's)ds = (13 +

i

1 + 001

Inversion of this transform leads to (4.42).

4.13 Remark. Recall the two independent, reflected Brownian motions WI_of Theorem 3.1, along with the notation of that section. If b < 0 in Lemma4.12, then the subordinator N is a function of W_, and hence is independentof W.+. To see this, recall the local time at 0 for W_: L_(t) A Li(t)(0), and let

Lb_(T) L,w(b)

be the local time of W_ at b. Both these processes can be constructed from W_(see Remark 3.2 for L_), and so both are independent of The same is truefor

FAO = inf {T > 0: LAO >

and hence also for

Lb-(F-(P.0) = Lrit(rApo)(b).

But Wp. = 0 a.s., and so fi(ps + a) > 11(0 for every e > 0 (Problem 2.7.18applied to the Brownian motion W o Op.). It follows from Problem 3.4.5 (iii)that TT' (1"_(ps)) = p and so Lb_.(f_(ps))=-- N,.

A comparison of (4.42) with (2.14) shows that for the subordinator N, wehave m = 0 and Levy measure u(d() =

4.14 Proposition (D. Williams (1969)). In the notation of Section 3, and forfixed numbers a > 0, t > 0, b < 0, we have a.s. P°:

6.5. Transition Probabilities of Brownian Motion with Two-Valued Drift 437

(4.43) E° [exp { - al,,,I(t)(b)} I W+(u); 0 '(i); Ls(0) = L i (0(0)}

= inf {s _.- IT'(x); Ws = 0}, a.s. P°.

Because W. > 0 for r_ +' (0 _< u _< p,r.v.(0) = pw,), the local time 4(6) mustbe constant on this interval. Therefore, with N the subordinator of Lemma4.12, Nwo = L +(,)(b) = 1-;,(0(b), a.s. P°. Remark 4.13, together with relation(4.42), gives

E° [exp { -akso} I W+ (u); 0< u< co] = E° [exp{ - aNt}

4.15 Exercise.

]It=1.-,.(r)

= exp {+(t) }

a.s. P°.1

(xL

+ albl '

(i) Show that for # > 0, a > 0, b < 0, we have

fiL T.(0) 1

1 + flibi Y1E0 [exp { - fil.. T(b)} I W+(u); 0 u < co }]

(ii) Use (i) to prove that for a > 0, /3 > 0, a > 0, b < 0,

Eb[expl -13LT.(b) - aLT.(0)}]

where f2 is given by (4.22), (4.23). (Thisprovide an alternate proof of Lemma 4.9;

1

a.s. P°.

f2(a, a; a + Ibl, 11)'

argument can be extended tosee McKean (1975)).

6.5. An Application: Transition Probabilities ofBrownian Motion with Two-Valued Drift

Let us consider two real constants Coo < Os, and denote by V the collection ofBorel-measurable functions b(t, x): [0, co) x 01 [On, Or]. For every b E W andx ell, we know from Corollary 5.3.11 and Remark 5.3.7 that the stochasticintegral equation

(5.1) X, = x + 1 t b(s, Xs) ds + 14(t; 0 s t < coo

has a weak solution (X, W), (CI, ,Px), {,i} which is unique in the sense of

probability law, with finite-dimensional distributions given by (5.3.11) for0 ti < t2 < < t = t < co, I- e ROI"):

(5.2) Px[(X,,,. . . , xtn) e F]

= Ex [1{(}v,,

t1

}f,1

)E ,-) exp If b(s, Ws)dW, -0

b2(s, Ws) ds}1.o

Here, {147 .97,}, (0,39, {P},,E R is a one-dimensional Brownian family.We shall take the point of view that the drift b(t, x) is an element of control,

available to the decision maker for influencing the path of the Brownianparticle by "pushing" it. The reader may wish to bear in mind the special case00 < 0 < 0,, in which case this "pushing" can be in either the positive or thenegative direction (up to the prescribed limit rates 0, and 0,, respectively). Thegoal is to keep the particle as close to the origin as possible, and the decisionmaker's efficacy in doing so is measured by the expected discounted quadraticdeviation from the origin

J(x; b) = Ex J e- xi X, dto

for the resulting diffusion process X. Here, a is a positive constant. The controlproblem is to choose the drift b. e 0/i for which J(x; b) is minimized over allb eqi:

(5.3) J(x; b.) = min J(x; b), V x e R.bell

This simple stochastic control problem was studied by Bend, Shepp &Witsenhausen (1980), who showed that the optimal drift is given by b.(t,x) =u(x); 0 < t < co, x e R and

(5.4) u(x) 4- 101;x < 6}

'-4-

1 1

Bo; x > 6 '6

N/ q + 2a + 0, q + 2aL - BoThis is a sensible rule, which says that one should "push as hard as possibleto the right, whenever the process Z, solution of the stochastic integralequation

(5.5) Z, = x + f t u(Zs) ds + W,; 0 < t < oo,o

finds itself to the left of the critical point 6, and vice versa." Because there isno explicit cost on the controlling effort, it is reasonable to push with fullforce up to the allowed limits. If 0, = -0, = 0, the situation is symmetric and6 = 0.

Our intent in the present section is to use the trivariate density (3.30) inorder to compute, as explicitly as possible, the transition probabilities

(5.6) fit(x, z) dz = Ex [Z, E dz]

of the process in (5.5), which is a Brownian motion with two-valued, state-dependent drift. In this computation, the switching point 6 need not be relatedto 00 and 0,. We shall only deal with the value 6 = 0; the transition prob-abilities for other values of 6 can then be obtained easily by translation.

The starting point is provided by (5.2), which puts the expression (5.6) inthe form

(5.7) z)dz = Ex[l {w, caz} exP f u(Ws) dllis - ,1 J u2(Ws) ds}1.- IL o

Further progress requires the elimination of the stochastic integral in (5.7).But if we set

0 z; z < 0f(z) u(y)dy =

100z; z > 0,1

we obtain a piecewise linear function, for which the generalized Ito rule ofTheorem 3.6.22 gives

f(W) = J(Wo) + J u(His)dW, + (00 - 0,)L(t)0

where L(t) is the local time of W at the origin. On the other hand, with thenotation (3.3) we have

0

(Ws) ds = tfi t + (01, - EDF.,(t),

and (5.7) becomes

(5.8) /3,(x; z)dz = exp[f(z) - f(x) - Od

itiO /10 exp{NO, - 00) + (eq -

I:" [W, e dz; L(t)e db; 1-+(t)e dr].

It develops then that we have to compute the joint density of (W L(t), 1-+(t))under F.', for every x e R, and not only for x = 0 as in (3.30). This is accom-plished with the help of the strong Markov property and Problem 3.5.8; inthe notation of the latter, we recast (3.30) for b > 0, 0 < t < t as

P°[W,eda; L(t) e db;1-+(t)e dr]

J2h(t; b, 0)h(t - r; b - a, 0) da db d-r; a < 012h(t - b, 0)h(r; b + a, 0) da db d-r; a > 0,

and then write, for x > 0 and a < 0:

(5.9) Px[W,e da; L(t) e db; 1"÷(t)e d-r]

= Px[W, e da; L(t) e db;F+(t)e di; T0

= jo Px[Wie da; L(t)e db; 1-+(t)e d-r1T0 = s] 1' [T0 e ds]

= J i P° [W,_, e da; L(t - s) e db; F +(t - s)e dt - s]- h(s; x, 0) ds0

= 2h(t; b + x, 0)h(t - t; b - a, 0) da db ch.

For x > 0, a > 0 a similar computation gives

(5.10)

Px[W,e da; L(t)e db; FAO e dr] = 2h(t - t; b, 0)h(t; b + a + x, 0)dadb dr,

and in this case we have also the singular part

(5.11) Px[Wieda; L(t) = 0, FAO = t] = Px[W,e da; T0 > t] = p_(t; x,a)da

[p(t; x, a) - p(t; x, - a)] da

(cf. (2.8.9)). The equations (5.9)-(5.11) characterize the distribution of the tripleL(t),1",(t)) under Px. Back in (5.8), they yield after some algebra:

(5.12) f),(x, z) =

2 c° ft e2be, h(t - r; b - z, -0,)h(t; x + b, -00)d-c db; x > 0, z < 0J0

2 J.te200,+zoo)h(t_ t; b, -01)h(t; x + b + z, -00)dr db1:0 0

(X -Z 0002}2004 ];1 [exp exp (x z2t"2

2nt 2t

x > 0, z > 0.

Now the dependence on 00, 0, has to be invoked explicitly, by writingz; 00,0,) instead of 15,(x, z). The symmetry of Brownian motion gives

(5.13) /5,(x, z; 00, 0, ) = fit( -x, -z; -0,, -00),

and so for x < 0 the transition density is obtained from (5.12) and (5.13). Weconclude with a summary of these results.

5.1 Proposition. Let u: R [00, 0,] be given by (5.4) for arbitrary real 5, andlet Z be the solution of the stochastic integral equation (5.5). In the notation of(5.6), + 5, z + 5) is given for every z e R, 0 < t < oo by

(i) the right-hand side of (5.12) if x Z 0, and

(ii) the right-hand side of (5.12) with (x, z, 00, 01) replaced by (- x, - z, -01,- 00), if x 0.

5.2 Remark. In the special case 01 = -00 = 0 > 0 = (5, the integral term inthe second part of (5.12) becomes

2 f e20(b-z)h(t - T; b, - 0)h(t; x + b + z, 0)dt dbso t

0 0

= 2 ft e_20zh(t - T; b, 0)h(t; x + b + z, 0) dt d00 0

=2 e'h(t; x + 2b + z, 0)db

(x + z + 0021[exp 20x}\/2nt2t

c°96,-26.1 exp

(v 0021dvi,

x+.

where we have used Prob em 3.5.8 again. A similar computation simplifiesalso the first integral in (5.12); the result is

(5.14) "Mx, z) =

1 [exp(x -z - 002}

exp(v - 0021 dvi

9e-20z,/2nt 2t x+ z 2t

x z > a

1[exp 120x

(x -z 002/exp (V -2t9t)2}dvi;

.12nt 2t x-z

x > 0, z < O.

When 0 = 1 and x = 0, we recover the expression (3.19).

5.3 Exercise. When 01 = -00 = 1, S = 0, show that the function v(t, x)Ex(Z,2); t > 0, x ER is given by

1 t(5.15) v(t, x) = +

.N/- t - 1)exp

(I xi 2-t t)2

{03(1 - t .1}(121(lxi t)

ezixi2_)[i (D(Ixl

with 41)(z) = (1/.\/2n)Sz e-u212 du, and satisfies the equation

(5.16)

as well as the conditions

v, = iv - (sgn x) v,

1

lim v(t, x) = x2, lim v(t, x) =(4.0 t-oco 2

5.4 Exercise (Shreve (1981)). With 01 = -0o = 1, 6 = 0, show that the func-tion v of (5.15) satisfies the Hamilton-Jacobi-Bellman equation

(5.17) v, =1-2

vx + min (uvx) on (0, co) x R.1u15.1

As a consequence, if X is a solution to (5.1) for an arbitrary, Borel-measurableb: [0, co) x [ - 1, 1] and Z solves (5.5) (under Ex in both cases), then

(5.18) Ex Z,2 < Ex X i2; 0 < t < CO.

In particular, Z is the optimally controlled process for the control problem(5.3).


2.5. If A is a-finite but not finite, then there exists a partition {1-11}711 g_ if of H with0 < 2(H1) < co for every i > 1. On a probability space (SI, P), we set up inde-pendent sequences 10); j e IN,}711 of random variables, such that for every i 1:

(1) a), are independent,(ii) N, A a) is Poisson with EN; = 2(H1), and(iii) P[c ji)E C] = 2(C n H;)12(H;); = 1, 2, ....

As before, MC) E7±1 1,(0)); CE if, is a Poisson random measure for everyi > 1, and v1, v2, ... are independent. We show that v v1 is a Poissonrandom measure with intensity A. It is clear that Ev(C) = E Er_1 v;(C n =Er=, 2(C n H;) = A(C) for all C E if, and whenever 2(C) < o o , v(C) has the properdistribution (the sum of independent Poisson random variables being Poisson).Suppose 2(C) = co. We set 2 = 2(C n H.), so v(C) is the sum of the independent,Poisson random variables Ivi(C)IT_I, where Evi(C) = A,. There is a number a > 0such that 1 -e > 2/2; 0 < 2 < a, and so with b A 2(1 - ca):

CO CO

E P[1,;(C) > 1] = E (11=1 1=1

because Er=, 2 = 2(C) = co. By the(Chung (1974), p. 76), P[v(C) = co]This completes the verification that vverification of (ii) is straightforward.

1 co- - E (A, Ab) =oo2 i=1

second half of the Borel-Cantelli lemma= P[vi(C) 1 for infinitely many i] = 1.satisfies condition (i) of Definition 2.3; the

2.12. If for some ye D[0, co), t > 0, and e > 0 we have n(y; (0, t] x ((, co)) = co, thenwe can find a sequence of distinct points {tk }T=1 g (0, t] such that I y(th ) -y(tk - )1 > (; k > 1. By selecting a subsequence if necessary, we may assumewithout loss of generality that {tk}ic_, is either strictly increasing or else strictlydecreasing to a limit Be [0, t]. But then y(t k)} ,T=, and { y(t, -) }k 1 converge tothe same limit (which is y(0 -) in the former case, and y(0) in the latter). In eithercase we obtain a contradiction.

For any interval (t, t + h], the set

Ar.,+h(e) { y E D [0, co); 3 s E (I, h] such that I y(s) - y(s -)I > e}

.0 .0

=un U D[O, oo); y(q) - y(r)I > ek=1 m=1 i<r<q<ti-h

q-r<(11m).reQqQorq=i+h

is in .4(D[0, co)), as is

1

{y D[O, oo); n(y; (0, t] x (/, co)) > ml = Uo=q0<q1<<qm, i=1

Let us now fix 0 < t < oo and / > 0 and let be be the Dynkin system of allsets CE MOO, I] x (e, COO for which n( ; C) is measurable. The a-field MOO, x((, co)) is generated by the sets of the form (0, -r] x (A, co); 0 < T < t, e < CO,

the collection of such sets is closed under finite intersections, and each such setbelongs to gt.e. It follows from Theorem 2.1.3 that n( ; C) is measurable forevery CE .4(03, ti x (C, CO)). For CE g((0, co)2), we write C as the disjoint union

c = u U ic n x

/1- 111}.k =1 .i =1

2.16. From (2.17) and Definition 2.15 we have EC" = e 0124' 1 , which gives (2.33)and the continuity of 13( ). Because N is not identically zero, kfr(1) is strictlypositive. Furthermore, for every y > 0,

e-`114")411) = Ee-",a,,, = EC"'N, = EC"A,) = Cr9(Y)4")

and (2.32) follows from (2.33). To see that #(a) = a', set G(a) = fl(ea) and applyProblem 2.2. For 0 < E < 1, comparison of (2.24) and (2.33) yields

f(

ras = ma + (1 - e-'1)p(de).o..)

Corollary 2.14 asserts that the constant m > 0 and a-finite measure p satisfyingthe equation are unique. If E .-- 1, they are given by m = r, p = 0. If 0 < e < 1,we set m = 0, p(d/)= red( /f'(1 - t)/I+` and integrate by parts to reduce theintegral to a standard Laplace transform:

(1 -C )p(cli) = fo.,e) e-2e Cr' de = ea'.(o..)

2.24. We have from (2.56)

2

EIED,(e) - 24(0)12 < 2e2 + 2E sgn( Ws) d Ws) .

1=1 ItA

This last expectation is computed asco 1+.)4(c)

E E {(t A (3)) - A 21_1 )1 E 2, coi- E E 1{J,Dt(."+1i=1 1 =1 j=0

= E2(ENE) + 1)

by virtue of (2.52), (2.53). But now the downcrossing inequality (Theorem 1.3.8

(iii)), applied to the submartingale I WI, gives ED,(E) < -1

EIW,I =1

3.7. The following are obviously valid, modulo P°-negligible events:

{t > 0; L_(t) > L +(t)} = {t 0; (t)) > L(ITI (T))}

g {t 0; Ff1(t) (T))

{t 0; L(F1I(t)) L(F;1(r)))

= {t 0; L_(t) > L +(t) }.

It

Therefore, we have a.s. P°:

= inf It >_ 0; L_(t) > L +(t)} < inf It 0; 1-11(t) (r)}

inf (t 0; L_(t) > L+(t)} = L:1(L+(t)).

The third identity in (3.25) follows now from (3.24). For the fourth, it suffices toobserve F., (s) = inf {t 0; 1---,.(t)> s }; 0 < s < co, a.s. P°, which is a conse-quence of Lemma 3.4.5 (v).

3.11. Using the reflection principle in the form (2.6.2) and the upper bound in (2.9.20),we obtain

P°[ max Ws a + e; min Ws < a - E]

t<s<t+h t<s<t+h

a

< Px[ max Ws > a + e[W, Edx]-ao 0 <s<lo

J Px[ min Ws a - e] e[W, E dx]0<s<h

<Pa[max Ws > a + Pa[ min Ws <a -EJ0<s<h 0<s <h

OD

= 213° [ max Ws > E0 <s<h

= 4P° [Wh > e] <h

e-'2/2h = o(h).

For (3.32), we notice the inequality

P°[W+(t) > a; t I (r) < t + h; Mt) < t + h]

< P° [ max Ws > a; min W < 0],t<s<t+h t<s<t+h

where this term is o(h) as h 10, thanks to (3.31); a similar argument proves (3.33).

6.7. Notes 445

4.11. Let 4 = L,(0) and assume without loss of generality that b > 0. From theadditive functional property of local time, its invariance under translation, andthe composition property Tb = Ta + Tb 0 OT.; 0 < a < b, we see that P ° -a.s.:

LT. - LT. = LT.08,..0 OT. = L Tb on, 0 Ot; 0 < a < b,

where 07(a))(s) 'A- oi(t + s) - (0(t); s > 0. The stationarity and independence ofincrements follow from the strong Markov property as expressed by Theorem2.6.16. Formula (4.40) is just a restatement of (4.16).

6.7. Notes

Section 6.2: Most of the material here is due to P. Levy (1937, 1939, 1948).The representation (2.14) is a special case of a general decomposition resultfor processes with stationary, independent increments (Levy processes) intoBrownian and Poisson components, obtained by P. Levy (1937); see alsoLoeve (1978). In our exposition of Theorem 2.7, we follow Ito & McKean(1974) and Williams (1979). Both these books, as well as McKean (1975) andChapter VII of Knight (1981), may be consulted for further reading onBrownian local time. Chung & Durrett (1976) and Williams (1977) also dealwith the subject of Theorem 2.23. Taking up a theme of Levy, Ikeda &Watanabe (1981), pp. 123-136, show how to use local time to constructBrownian motion from a Poisson random measure on the space of excursionsaway from the origin. This should be read together with Chung's (1976)excellent treatise on excursions. Ito (1961b) applied Poisson random measuresto the study of Markov processes. The characterizations of Theorems 2.21 and2.23 have been generalized to Markov processes by Fristedt &,Taylor (1983);see also Kingman (1973). Invariance principles are discussed by Perkins(1982a) and Csaki & Revesz (1983); see also Borodin (1981). Perkins (1982b)showed that Brownian local time is a semimartingale in the spatial parameter;McGill (1982) studied its Markov properties.

Section 6.3: Theorem 3.1 appears in Ito & McKean (1974), section 2.11,and in Ikeda & Watanabe (1981), pp. 122-124. Proposition 3.9 is taken fromKaratzas & Shreve (1984a); the bivariate density (3.40) was obtained byPerkins (1982b), using different methodology. The use of local time in decom-position of Brownian paths is illustrated by the work of Williams (1974) andHarrison & Shepp (1981).

Section 6.4: The strong version of Theorem 4.1 discussed in Remark 4.2,but with n = 1, appears in Ito & McKean (1974), pp. 45-48. For more generalresults along the lines of Theorems 4.1 and 4.3, the reader should consultKnight (1981), Theorem 7.4.3. Exercise 4.5 comes from Taylor (1975), whereapplications to finance and process control are discussed; see also Williams(1976), Lehoczky (1977), and Azema & Yor (1979). We follow Ito & McKean(1974) in our approach to the Ray-Knight theorem 4.7 and in Lemma 4.12,and Knight (1981) for the proof of the Dvoretzky-Erthis-Kakutani Theorem2.9.13.


Section 6.5: This material is taken from Karatzas & Shreve (1984a). Thecontrol problem of this section was introduced and solved by Bend, Shepp,& Witsenhausen (1980) and has also been solved in the symmetric case ofExercises 5.3, 5.4 by martingale methods (Davis & Clark (1979)), stochasticcomparison methods (Ikeda & Watanabe (1977)), and the stochastic maxi-mum principle (Haussmann (1981)). See also Balakrishnan (1980). Stochasticcontrol problems in which the optimal control process is a local time havebeen studied by a number of authors, including Bather & Chernoff (1967);Benek Shepp & Witsenhausen (1980); Chernoff (1968); Chow, Menaldi &Robin (1985); Harrison (1985); Karatzas (1983); Karatzas & Shreve (1984b,1985); and Taksar (1985).


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France 52, 569-578.WIENER, N. (1924b) The Dirichlet Problem. J. Math. Phys. 3, 127-146.WILLIAMS, D. (1969) Markov properties of Brownian local time. Bull. Amer. Math.

Soc. 75, 1035-1036.WILLIAMS, D. (1974) Path decomposition and continuity of local time for one-

dimensional diffusions. Proc. London Math. Soc. 28, 738-768.WILLIAMS, D. (1976) On a stopped Brownian motion formula of H. M. Taylor. Lecture

Notes in Mathematics 511, 235-239. Springer-Verlag, Berlin.WILLIAMS, D. (1977) On Levy's downcrossings theorem. Z. Wahrscheinlichkeitstheorie

verw. Gebiete 40, 157-158.WILLIAMS, D. (1979) Diffusions, Markov Processes and Martingales, Vol. 1: Foundations.

J. Wiley & Sons, Chichester, England.WONG, E. & HAJEK, B. (1985) Stochastic Processes in Engineering Systems. Springer-

Verlag, New York.WONG, E. & ZAKAI, M. (1965a) On the relation between ordinary and stochastic

differential equations, Intern. J. Engr. Sci., 3, 213-229.WONG, E. & ZAKAI, M. (1965b) On the convergence of ordinary integrals to stochastic

integrals. Ann. Math. Stat. 36, 1560-1564.YAMADA, T. & OGURA, Y. (1981) On the strong comparison theorems for solutions of

stochastic differential equations. Z. Wahrscheinlichkeitstheorie verw. Gebiete 56,3-19.

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YOR, M. (1977). Sur quelques approximations d' integrates stochastiques. LectureNotes in Mathematics 581, 518-528. Springer-Verlag, Berlin.

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Index

A

Abel transform 266Adapted process 4Additive functional 202-203Adelman, 0. 109Aldous, D. 46American option 376, 398Andre, D. 79Arc-sine law

for last exit of Brownian motion fromzero 102

for occupation time by Brownian mo-tion of (0,co) 273, 422

for the time Brownian motion obtainsits maximum 102

Arnold, L. 236, 395Arzela-Ascoli theorem 62Ash, R.B. 1 1 , 8 5

Augmented filtration (see Filtration)Azerna, J. 238, 445

BBachelier, L. 126Backward Ito integral 148

Backward Kolmogorov equation (see

Kolmogorov equations)Balakrishnan, A.V. 446Barlow, M.T. 396

Barrier at a boundary point 247-248Bather, J.A. 446Bayes' rule 193

Bellman equation (see Hamilton-Jacobi-Bellman equation)

Bena, V.E. 200, 438, 446Bensoussan, A. 398BernKtein, S. 394Besse! process 158ff

in the Ray-Knight description of localtime 430

minimum of 162

scale and speed measure for 346stochastic integral equation for 159

strong Markov property for 159

Bhattacharya, R.N. 397Billingsley, P. 11, 126Black & Scholes option pricing

formula 378-379Blumenthal, R.M. 127, 238Blumenthal zero-one law 94Borel set 1

Borel cr-field I

Borodin, A.N. 445Brosamler, G.A. 237Brown, R. 47Brownian bridge 358-360

d-dimensional 361maximum of 265


460 Index

Brownian family 73universal filtration for 93with drift 78

Brownian functional 185

stochastic integral representationof 185-186, 188-189

Brownian local time 126, 399ffas a semimartingale in the spatial

parameter 445as a Bessel process in the spatial

parameter 430defined 203downcrossing representation of 416excursion interval representation

of 415existence of 207joint density with Brownian motion

and occupation time of(0,00) 423

Markov properties of 445Ray-Knight description of 430, 445Tanaka formula for 205, 215

Brownian martingale 182

Ito integral representation of 182, 184Brownian motion

absorbed 97-100, 265augmented filtration for 91

boundary crossing probabilitiesfor 262ff

complex-valued 253construction of 48

as a weak limit of randomwalks 59ff

from finite-dimensionaldistributions 49ff

Levy-Ciesielski method 56ffd-dimensional 72

recurrence properties of 161-162rotational invariance of 158

elastic 425ffequivalence transformations 103-104excursion intervals of 401

as a Poisson random measure 411used to represent Brownian local

time 415used to represent maximum-to-

date 417existence of 48filtration for 48

geometric 349last exit time of 100-103Markov property for 75martingale characterization of 156-

157, 178, 314maximum (running maximum, maxi-

mum-to-date) of 95-96, 102-103, 417, 429

negative part of 418-420occupation time of (0,00) 273, 274,

422joint density with Brownian motion

and local time 423one-dimensional 47passage time of (see Brownian passage

time)positive part of 418-420quadratic variation of 73, 106reflected 97, 210, 418-420scale and speed measure for 344scaling of 104

semimartingale local time for 225symmetry of 104

time-inversion of 104

time-reversal of 104

transition density for (see Gaussiankernel)

with drift 196ffminimum of 197

passage times of 196

scale and speed measure for 346with two-valued drift 437ff

Brownian oscillator 362Brownian passage time 79, 95, 126

density of 80, 88, 96for drifted Brownian motion 196

Laplace transform of 96process of passage times 96, 400ff

as a one-sided stable process 411Levy measure for 411

Brownian sample pathlaw of the iterated logarithm for 112,

127

level sets of 105-106, 201-202, 209Levy modulus of continuity

for 114ff, 127local Holder continuity of 56, I I l ,

113-114, 126-127local maxima of 107


Index 461

nondifferentiability of 110, 203non-monotonicity of 106

no point of increase for (see alsoDvoretzky-Erdos-Kakutani theo-rem)

quadratic variation of 106

slow points of 127

unbounded first variation of 106

zero set of 104-105Burkholder, D.L. 44, 166, 237Burkholder-Davis-Gundy

inequalities 166

CCad lag process (see RCLL process)Cameron, R.N. 191, 237Cameron-Martin formula 434Cameron-Martin-Girsanov formula (see

Girsanov's theorem)Canonical probability space 71

Capital asset pricing (see Optimal con-sumption and investment problem)

Cauchy problem (see Partial differentialequations)

Causality 286, 311Centsov, N.N. 53, 126Chaleyat-Maurel, M. 237Change of measure 129 (see also Girsa-

nov's theorem)Change-of-variable formula 128

for functions of continuoussemimartingales 149, 153

for convex functions of Brownianmotion 214

for convex functions of continuoussemimartingales 218-219

Chernoff, H. 446Chitashvili, R.J. 395Chow, P.L. 446Chow, Y.S. 46Chung, K.L. 4, 6, 11, 45, 46, 125,

127, 236, 445Ciesielski, Z. 48, 126Cirel'son, B.S. 304Cinlar, E. 237Clark, J.M.C. 446Class D 24Class DL 24

Cocozza, C. 237Communication 395Comparison of solutions to stochastic dif-

ferential equations (see Stochasticdifferential equations)

Consistent family of finite-dimensionaldistributions 50, 126

Consumption process 373Contingent claim 376

fair price for 377valuation process of 378

Continuous local martingale 36cross-variation of 36integration with respect to I45ffmoment inequalities for 166

quadratic variation of 36represented as an ItO integral 170ffrepresented as time-changed Brownian

motion I73ffContinuous semimartingale 149

change-of-variable formula 149, 153convex function of 218local time for 218Tanaka-Meyer formulas for 220

Controllability 355-356Controllability matrix 356Convergence in distribution 61

Convex function 212-213Ito formula for 214of a continuous semimartingale 218second derivative measure of 213

Coordinate mapping process 52, 71Copson, E.T. 277Countably determined o-field 306Countably generated cr-field 307Courrege, P. 236Covariance matrix 103,355Cox, J. 398Cramer-Wold device 61

Cross-variation processas a bilinear form 31-32for continuous local martingales 36for continuous square-integrable

martingales 35

for d-dimensional Brownianmotion 73, 157

for square-integrable martingales 31

Csaki, E. 445Cylinder set 49


462 Index

DDambis, K.E. 174

Daniell, P.J. 50. 126Davis, B. 127, 166, 253Davis, M.H.A. 446Dellacherie, C. 10, 45, 46, 236Determining class of functions 325

Diffeomorphism theorem 397-398Diffusion matrix 281, 284Diffusion process 281-282

boundary behavior 396degenerate 327, 396killed 369on a manifold 396support of 396

Dini derivates 109

Dirichlet problem 240, 243ff, 364-366Dispersion matrix 284Doleans-Dade, C. 236, 396Donsker, M.D. 126, 280Donsker's theorem 70Doob, J.L. 4, 45, 46, 170, 236, 237,

239, 278, 397Doob-Meyer decomposition of a

submartingale 24ffDoob's maximal inequality for

submartingales 14

Doss, H. 295, 395Downcrossings 13-14

representation of Brownian localtime 416

Drift vector 281, 284Dubins, L.E. 174

Dudley, R.M. 188

Duffle, D. 398Durrett, R. 239, 445Dvoretzky, A. 108, 127Dvoretzky-Erdos-Kakutani theorem 108,

433, 445Dynamical systems 395Dynkin, E.B. 98, 127, 242. 269, 397Dynkin system 49

EEconomics equilibrium 398Einstein, A. 126Elastic Brownian motion 426El Karoui, N. 237

Elliott, R.J. 10, 46, 236, 237. 396Elliptic operator 364Elliptic partial differential equations (see

Partial differential equations)Emery. M. 189

Engelbert, H.J. 329, 332, 335. 396Engelbert-Schmidt zero-one law 216,

396Equilibrium (see Economics equilibrium)Erdos. P. 108, 433, 445Ergodic property (see Recurrence: Posi-

tive recurrence)Ethier. S.N. 127, 395, 396European option 376Excursion intervals of Brownian motion

(see Brownian motion)Exit distribution

of Brownian motion from asphere 252

Exit timeof Brownian motion from a

sphere 253Expectation vector (see Mean vector)Exponential supermartingale 147, 153.

I98ffnot a martingale 201

Extension of a probability space 169

Extension of measure (see Kolmogorovextension theorem)

Exterior sphere property 365

FFeller. W. 282, 397Feller property 127

Feller's test for explosions 348-350,396

Feynman, R.P. 279Feynman-Kac formula (see also Kac for-

mula) 366for Brownian motion 267fffor elastic Brownian motion 426-428

Filtering 395Filtration 3

augmentation of 89, 285completion of 89enlargement of 127

generated by a process 3

left-continuous 4, 90


Index 463

right-continuous 4, 90"universal" 93usual conditions for 10

Finite-dimensional distributions 2, 50,64

Fisk, D.L. 46, 236Fisk-Stratonovich integral 148, 156,

295, 299Fleming, W.H. 381, 395Fokker-Planck equation (see Kolmogorov

equations)Forward Kolmogorov equation (see Kol-

mogorov equations)Fourier, J.B. 279Freedman, D. 125, 127Freidlin, M. 395,397Friedman, A. 236, 269, 327, 365, 366,

368, 395, 397Fristedt, B. 445Fubini theorem for stochastic

integrals 209, 225Fundamental solution of a partial differ-

ential equation 255, 368as transition probability density 369

GGame theory 395Gaussian kernel 52Gaussian process 103, 355, 357Gauss-Markov process 355ffGeman, D. 237Generalized Ito rule for convex

functions 214Getoor, R.K. 127, 238Gihman, 1.1. 236, 394, 395, 397Gilbarg, D. 366Girsanov, I.V. 237, 292Girsanov's theorem 190ff, 302 (see also

Change of measure)generalized 352

Green's function 343Gronwall inequality 287-288Gundy, R. 166

HHaar functions 57Hajek, B. 395Hale, J. 354

Hall, P. 46Hamilton-Jacobi-Bellman (HJB)

equation 384-385, 442Harmonic function 240Harrison, J.M. 398, 445, 446Hartman, P. 279Haussmann, U.G. 446Heat equation 254ff, 278-279

backward 257, 268fundamental solution for 255nonnegative solutions of 257Tychonoff uniqueness theorem 255Widder's uniqueness theorem 261

Heat kernel (see Gaussian kernel)Hedging strategy against a contingent

claim 376Hermite polynomials 167

Heyde, C.C. 46Hincin, A. Ya. 112, 405Hitting time 7, 46Hoover, D.N. 46Horowitz, J. 237Huang, C. 398Hunt, G.A. 127, 278

Ikeda, N. 46, 236, 237, 238, 299, 306,395, 396, 398, 445, 446

Imhof, J.P. 127

Increasing process 23

integrable 23

natural 23Increasing random sequence 21

integrable 21

natural 22Independent increments 48Indistinguishable processes 2

InequalitiesBurkholder-Davis-Gundy 166

Doob's maximal 14

downcrossings 14

Gronwall 287-288Jensen 13

Kunita-Watanabe 142

Lenglart 30martingale moment 163ffsubmartingale 13-14uperossings 14


464 Index

Infimum of the empty set 7

Infinitesimal generator 281

Integration by parts 155

Intensity measure (see Poisson randommeasure)

Invariance principle 70. 311. 445Invariant measure 353, 362Irregular boundary point 245Ito. K. 80. 127. 130, 149. 236. 278,

279, 283, 287, 293. 394, 395.397. 405, 416, 445

Ito integral (see also Stochastic inte-gral) 148

backward 148

multiple 167

Ito's rule 128

for C functions of continuoussemimartingales 149. 153

for convex functions of Brownianmotion 214

for convex functions of continuoussemimartingales 218-219

JJacod. J. 237Jensen inequality 13

Jeulin, T. 127, 238

KKac. M. 279Kac formula (see also Feynman-Kac for-

mula) 267, 271-272Kahane. J.P. 127

Kakutani, S. 108, 278Kallianpur, G. 395, 397Karatzas, I. 279, 398, 425. 445. 446Karlin. S. 395Kazamaki, N. 237Kellogg. O.D. 278Kelvin transformation 252Khas'minskii, R.Z. 395. 397Kingman, J.F.C. 404. 445Kleptsyna, M.L. 396Kliemann, W. 395Knight. F.B. 126, 127. 179. 430. 445Kolmogorov, A.N. 126. 236. 239. 282

Kolmogorovtentsov continuity theoremfor a random field 55for a stochastic process 53

Kolmogorov equationsbackward 282, 369, 397forward 282. 369. 397

Kolmogorov extension theorem 50Kopp, P.E. 236Kramers-Smoluchowski

approximation 299Krylov, N.V. 327. 395Kunita, H. 130. 149. 156, 236, 397,

398Kunita-Watanabe inequality 142

Kurtz, T.G. 127, 395, 396Kussmaul, A.U. 236Kusuoka. S. 398

LLamperti. J. 294Langevin, P. 358. 397Laplace, P.S. 279Large deviations 280Last exit time 100-103Law of a continuous process 60Law of an RCLL process 409Law of large numbers

for Brownian motion 104

for Poisson processes 15

Law of the iterated logarithm 112, 127LCRL process 4Lebesgue. H. 278Lebesgue's thorn 249Le Gall. J.F. 225. 353. 396Lehoczky. J.P. 398. 445Lenglart. E. 44Lenglart inequality 30Levy P. 46, 48. 97. 126. 127, 202.

211. 236. 237. 253. 400, 405,415. 416. 445

Levy martingale characterization ofBrownian motion 156-157, 178

Levy measure 405, 410Levy modulus of continuity 114

Levy process 445Levy zero-one law 46Lifetime 369


Index 465

Liptser, R.S. 23. 46. 200. 236. 395Localization 36

Local martingale 36

not a martingale 168, 200-201Local martingale problem (see also Mar-

tingale problem) 314

existence of a solution of 317-318uniqueness of the solution of 317

Local maximum (see Point of local maxi-mum)

Local timefor Brownian motion (see Brownian lo-

cal time)for continuous semimartingales 218.

396

for reflected Brownian motion 219

in stochastic control 446Loeve, M. 126, 445

MMalliavin's stochastic calculus of

variations 398

Marchal, B. 237

Markov, A.A. 127

Markov family 74

Markov process 74

Markov property 71. 127equivalent formulations of 75-78for Brownian motion 74-75for Poisson processes 79

Markov time (see Stopping time)Martin. W.T. 191, 237Martingale (see also Continuous local

martingale: Local martingale;Square-integrable martingale; Sub-martingale) II, 46

convergence of 17-19convex function of 13

exponential (see Supermartingale, ex-ponential)

last element of 18. 19moment inequalities for I63ff. 166transform 21. 132uniform integrability of 18. 19

Martingale problem (see also Local mar-tingale problem) 311, 318,396

existence of a solution of 317-318.323. 327

strong Markov property for the solutionof 322

time-homogeneous 320

uniqueness of the solution of 327

well-posedness 320, 327Maruyama. G. 237. 397Mathematical biology 395

Mathematical economics 395

Maximum principle for partial differentialequations 242, 268

Maximum principle for stochasticcontrol 446

McGill, P. 445

McKean. H. 80, 125, 127, 236, 237,278, 279. 396. 397. 416. 437.445

Mean-value property 241

Mean vector 103. 355Measurable process 3

Meleard, S. 397

Menaldi. J.L. 446

Merton. R.C. 381, 398Mesure du voisinage (see Brownian local

time)Metivier, M. 236

Meyer, P.A. 5, 12. 45, 46, 212, 218,236, 237

Millar, P.W. 236

Minimum of the empty set 13

Modification of a stochastic process 2

Modulus of continuity 62. 114Mollifier 206

Moment inequalities formartingales I63ff

Multiple It6 integral 167

N

Nakao, S. 396

natural scale 345

Nelson. E. 47. 282. 299. 395. 397Neurophysiology 395

Neveu. J. 46Novikov, A.A. 279

Novikov condition I98ffNualart, D. 398


466

0Occupation time 203, 273, 274Ogura, Y. 395One-sided stable process 410Optimal consumption and investment

problem 379ff, 398Optimal control (see Stochastic optimal

control)Option pricing 376ff, 398Optional sampling 19, 20Optional time 6Orey, S. 237Ornstein, L.S. 358Ornstein-Uhlenbeck process 358, 397

PPaley, R.E.A.C. 110, 236Papanicolaou, G. 238Parabolic partial differential equation (see

Partial differential equation)Pardoux, E. 395Parthasarathy, K.R. 85, 306, 322, 409Partial differential equations

Cauchy problem 366ff, 397fundamental solution for 368-369

degenerate 397Dirichlet problem 240, 243ff, 364- 366elliptic 239, 363parabolic 239, 267, 363quasi-linear 397

Passage time for Brownian motion (see

Brownian passage time)Pathwise uniqueness (see Stochastic dif-

ferential equation)Pellaumail, J. 236Perkins, E. 396, 445Picard-LindelOf iterations 287Piecewise C' 271

Piecewise continuous 271

Piecewise-linear interpolation of theBrownian path 57, 299

Pitman, J. 46, 237Pliska, S.R. 398Poincare, H. 278Point of increase 107

for a Brownian path 108

Point of local maximum 107for a Brownian path 107

Index

Poisson equation 253Poisson family 79Poisson integral formulas

for a half-space 251

for a sphere 252Poisson process 12

compensated 12, 156compound 405filtration for 91

intensity of 12

strong law of large numbers for 15

weak law of large numbers for 15

Poisson random measure 404intensity measure for 404on the space of Brownian

excursions 445used to represent a subordinator 410

Pollack, M. 352Population genetics 395Port, S. 253, 278Portfolio process 373Positive recurrence 353, 371Potential 18

Predictableprocess 131

random sequence 21-22stopping time 46

Principal eigenvalue 279Priouret, P. 396Process (see Stochastic process)Progressive measurability 4

of a process stopped at a stoppingtime 9

Progressively measurablefunctional 199-200

Prohorov, Yu.V. 126Prohorov's theorem 62Projection mapping 65Protter, P. 237, 396p-th variation of a process 32Pyke, R. 126

QQuadratic variation

for a continuous local martingale 36

for a process 32for a square-integrable martingale

31


Index 467

R Shiryaev, A.N. 23, 46, 200, 236, 395

Random field 55, 126 Shiryaev-Roberts equation 168

Random process (see Stochastic process) Shreve, S.E. 398, 425, 442, 445, 446Random shift 83 Siegmund, D. 43, 197, 261, 279, 352

Random time 5 a-fieldRandom walk 70, 126 Borel 1

Rao, K.M. 46Ray, D. 127, 430RCLL process 4Realization 1

Recurrence 370, 397of d-dimensional Brownian

motion 161-162of a solution to a stochastic differential

equation 345, 353Reflected Brownian motion (see Browni-

an motion, reflected)Reflection principle 79-80Regular boundary point 245, 248-250Regular conditional probability 84-85,

306-309Regular submartingale 28

Resolvent operator 272

Revesz, P. 445

Revuz, D. 44Riesz decomposition of a

supermartingale 18

Rishel, R.W. 381, 395Robbins, H. 43, 197, 261, 279Robin, M. 446Rosenblatt, M. 279

S

Sample path (see also Brownian samplepath) I

Sample space 1

Samuelson, P.A. 398

Scale function 339, 343Schauder functions 57

Schmidt, W. 329, 332, 335, 396Schwarz, G. 174

Sekiguchi, T. 237Semigroup 395

Semimartingale (see Continuous semi-martingale)

Sharpe, M.J. 237Shepp, L.A. 438, 445, 446Shift operators 77, 83

generated by a process at a randomtime 5

of events immediately after an optionaltime 10

of events immediately after t,.?-121 4

of events prior to a stopping time 8

of events strictly prior to t>0 4

universal 73

Simon, B. 280Simple process 132

Skorohod, A.V. 236, 323, 395, 397Skorohod equation 210Skorohod metric 409Skorohod space of RCLL functions 409Smoluchowski equation 361, 397Sojourn time (see Occupation time)Speed measure 343, 352

as an invariant measure 353, 362Square-integrable martingale

cross-variation of 31, 35ItO integral representation of

Brownian 182

metric on the space of 37

orthogonality of 31

quadratic variation of 31

Stability theory 395

State space 1

Stationary increments 48

Stationary process 103

Statistical communication theory 395

Statistical tests of power one 279

Stochastic calculus 128, 148, 150Stochastic calculus of variations 398

Stochastic control (see Stochastic optimalcontrol)

Stochastic differential 145, 150, 154Stochastic differential equation 281ff,

394

approximation of 295ff, 395comparison of solutions of 293, 395,

446explosion time of 329functional 305


468 Index

Stochastic differential equation (cont.)Gaussian process as a solution

of 355, 357linear 354ffone-dimensional 329ff

Feller's test for explosions 348-350, 396

invariant distribution for 353nonexplosion when drift is

zero 332pathwise uniqueness 337, 338,

341, 353, 396strong existence 338, 341, 396strong uniqueness 338, 341, 396uniqueness in law 335, 341weak existence 334, 341weak solution in an interval 343

pathwise uniqueness 301, 302, 309-310

strong existence 289, 310-311, 396strong solution 283, 285strong uniqueness 286, 287, 291, 396uniqueness in law 301, 302, 304-

305, 309, 317weak existence 303, 310, 323, 332weak solution 129, 300weak solution related to the martingale

problem 317-318well-posedness 319-320with respect to a semimartingale 396

Stochastic integral 129ffcharacterization of 141-142definition of 139

with respect to a martingale having ab-solutely continuous quadraticvariation 141

Stochastic integral equation (see Stochas-tic differential equation)

of Volterra type 396Stochastic maximum principle 446Stochastic optimal control 284, 379,

395, 438, 446Stochastic partial differential

equations 395Stochastic process 1

adapted to a filtration 4finite-dimensional distributions of 2Gaussian 103

Gauss-Markov 355ffLCRL 4measurable 3

modification of 2

of class D 24of class DL 24progressively measurable 4

RCLL 4sample path of I

simple 132

state space of 1

stationary 103

stopped at a stopping time 9zero-mean 103

Stochastic systems 395Stone, C. 253, 278Stopping time 6

accessible 46predictable 46totally inaccessible 46

Stratonovich integral (see Fisk-Stratonov-ich integral)

Stricker, C. 168, 189Strong existence (see Stochastic differen-

tial equation)Strong Markov family 81-82

universal filtration for 93

Strong Markov process 81-82augmented filtration for 90-92classification of boundary behavior

of 397Strong Markov property 79, 127

equivalent formulations of 81-84extended 87for Brownian motion 86, 127for Poisson processes 89for solutions of the martingale

problem 322Strong solution (see Stochastic differen-

tial equation)Strong uniqueness (see Stochastic differ-

ential equation)Stroock, D.W. 127, 236, 238, 283,

311, 322,397, 398,

323,416

327, 395, 396,

Submartingale (see also Martingale) 11

backward 15

convergence of 17-18


Index 469

Doob-Meyer decomposition ofinequalities for 13-14last element of 12-13, 18maximal inequality for 14

optional sampling of 19-20path continuity of 14, 16regular 28uniform integrability of 18

Subordinator 405Supermartingale (see also Martin-

gale) 11

exponential 147, 153, 198ffnot a martingale 201

last element of 13

Sussmann, H. 395Synonimity of processes 46

24ff Trotter, H.F. 207, 237Trudinger, N.S. 366Tychonoff uniqueness theorem 255

TTaksar, M. 446Tanaka, H. 237, 301, 397Tanaka formulas for Brownian

motion 205, 215Tanaka-Meyer formulas for continuous

semimartingales 220Taylor, H. 395, 445Taylor, S.J. 445Teicher, H. 46Tied-down Brownian motion (see-

Brownian bridge)Tightness of a family of probability

measures 62Time-change

for martingales 174

for one-dimensional stochastic differen-tial equations 330ff

for stochastic integrals 176-178Time-homogeneous martingale problem

(see Martingale problem)Toronjadze, T.A. 395Trajectory 1

Transformation of drift (see Girsanov'stheorem)

Transition densityfor absorbed Brownian motion 98for Brownian motion 52for diffusion process 369for reflected Brownian motion 97

UUhlenbeck, G.E. 358Uniformly elliptic operator 364Uniformly positive definite matrix 327Uniqueness in law (see Stochastic differ-

ential equation)Universal filtration 93

Universal cr-field 73

Universally measurable function 73

Uperossings 13-14Usual conditions (for a filtration) 10

Utility function 379

V

Value function 379Van Schuppen, J. 237Varadhan, S.R.S. 127, 236, 238, 279,

283, 311, 322,397

Variation of a processVentsel, A.D. 237Veretennikov, A. Yu.Ville, J. 46

Wald identities 141,

323,

32

396

168,

327,

197

396,

Walsh, J.B. 395Wang, A.T. 212, 237Wang, M.C. 358Watanabe, S. 46, 130, 149, 156, 232,

236, 237, 238, 291, 292, 293,299, 306, 308, 309, 395, 396,397, 398, 445, 446

Weak convergence of probabilitymeasures 60-61

Weak solution (see Stochastic differentialequation)

Weak uniqueness (see Stochastic differ-ential equation, uniqueness in law)

Wentzell, A.D. 39, 82, 117, 127White noise 395


470 Index

Widder, D.V. 256. 277, 279Widder's uniqueness theorem 261Wiener. N. 48. 110. 126, 236. 278Wiener functional (see Brownian func-

tional)Wiener martingale (see Brownian martin-

gale)Wiener measure

d-dimensional 72one-dimensional 71

Wiener process (see Brownian motion)Williams. D. 127. 445

first formula of 421-422. 423second formula of 436-437

Williams, R. 236Wintner, A. 279Witsenhausen, H.S. 438. 446Wong. E. 169, 237. 395

YYamada. T. 291. 308, 309. 395. 396Yan. J . A . 189Yor, M. 44, 46. 126. 127. 168. 237,

238. 445Yushkevich, A.A. 98. 127, 242

Zakai, M. 169. 395. 398Zaremba, S. 278Zaremba's cone condition 250Zero-one laws

Blumenthal 94Engelbert-Schmidt 216, 396Levy 46

Zvonkin, A.K. 396Zygmund, A. 110. 236


second edition - tsinghua universityfaculty.math.tsinghua.edu.cn/~zliang/paper/ioannis karatzas,...

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