MATHEMATICAL AND ANALYTICAL TECHNIQUES WITH … · Theory of Stochastic Differential Equations with Jumps and Applications: Mathematical and Analytical Techniques with Applications

THEORY OF STOCHASTIC DIFFERENTIAL EQUATIONS WITH JUMPS AND APPLICATIONS

MATHEMATICAL AND ANALYTICAL TECHNIQUES WITH APPLICATIONS TO ENGINEERING


Alan Jeffrey, Consulting Editor

Published:

Inverse Problems A. G. Ramm

Singular Perturbation Theory R. S. Johnson

Methods for Constructing Exact Solutions of Partial Differential Equations with Applications S. V . Meleshko

Stochastic Differential Equations with Applications R. Situ

Forthcoming:

The Fast Solution of Bounda~y Integral Equations S. Rjasanow and 0. Steinbach

THEORY OF STOCHASTIC DIFFERENTIAL EQUATIONS WITH JUMPS AND APPLICATIONS


RONG SITU

a - Springer

Library of Congress Cataloging-in-Publication Data

Theory of Stochastic Differential Equations with Jumps and Applications: Mathematical and Analytical Techniques with Applications to Engineering By Rong Situ

ISBN-10: 0-387-25083-2 e-ISBN 10: 0-387-25 175-8 Printed on acid-free paper. ISBN-1 3: 978-0387-25083-0 e-ISBN-13: 978-0387-251 75-2

O 2005 Springer Science+Business Media, Inc. All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, Inc., 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. U se in connection w ith any form o f in formation storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now know or hereatler developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if the are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights.

Printed in the United States of America. (BSIDH)

9 8 7 6 5 4 3 2 1 SPIN 1 1399278

Contents

Preface xi

Acknowledgement xvii

Abbreviations and Some Explanations xix

I Stochastic Differential Equations with Jumps inRd 1

1 Martingale Theory and the Stochastic Integral for PointProcesses 31.1 Concept of a Martingale 31.2 Stopping Times. Predictable Process 51.3 Martingales with Discrete Time 81.4 Uniform Integrability and Martingales 121.5 Martingales with Continuous Time 171.6 Doob-Meyer Decomposition Theorem 191.7 Poisson Random Measure and Its Existence 261.8 Poisson Point Process and Its Existence 281.9 Stochastic Integral for Point Process. Square Integrable Mar-

tingales 32

2 Brownian Motion, Stochastic Integral and Ito's Formula 39

Contents

2.1 Brownian Motion and Its Nowhere Differentiability 392.2 Spaces C° and £2 432.3 Ito's Integrals on C2 442.4 Ito's Integrals on £2-'oc 472.5 Stochastic Integrals with respect to Martingales 492.6 Ito's Formula for Continuous Semi-Martingales 542.7 Ito's Formula for Semi-Martingales with Jumps 582.8 Ito's Formula for d-dimensional Semi-Martingales. Integra-

tion by Parts 622.9 Independence of BM and Poisson Point Processes 642.10 Some Examples 652.11 Strong Markov Property of BM and Poisson Point Processes 672.12 Martingale Representation Theorem 68

Stochastic Differential Equations 753.1 Strong Solutions to SDE with Jumps 75

3.1.1 Notation 753.1.2 A Priori Estimate and Uniqueness of Solutions . . . 763.1.3 Existence of Solutions for the Lipschitzian Case . . . 79

3.2 Exponential Solutions to Linear SDE with Jumps 843.3 Girsanov Transformation and

Weak Solutions of SDE with Jumps 863.4 Examples of Weak Solutions 99

Some Useful Tools in Stochastic Differential Equations 1034.1 Yamada-Watanabe Type Theorem 1034.2 Tanaka Type Formula and Some Applications 109

4.2.1 Localization Technique 1094.2.2 Tanaka Type Formula in d—Dimensional Space . . . 1104.2.3 Applications to Pathwise Uniqueness and Conver-

gence of Solutions 1124.2.4 Tanaka Type Formual in 1-Dimensional Space . . . 1164.2.5 Tanaka Type Formula in The Component Form . . . 1214.2.6 Pathwise Uniqueness of solutions 122

4.3 Local Time and Occupation Density Formula 1244.4 Krylov Estimation 129

4.4.1 The case for 1—dimensional space 1294.4.2 The Case for d—dimensional space 1304.4.3 Applications to Convergence of Solutions to SDE with

Jumps 133

Stochastic Differential Equations with Non-Lipschitzian Co-efficients 1395.1 Strong Solutions. Continuous Coefficients with p— Conditionsl405.2 The Skorohod Weak Convergence Technique 145

Contents vii

5.3 Weak Solutions. Continuous Coefficients 1475.4 Existence of Strong Solutions and Applications to ODE . . 1535.5 Weak Solutions. Measurable Coefficient Case 153

II Applications 161

6 How to Use the Stochastic Calculus to Solve SDE 1636.1 The Foundation of Applications: Ito's Formula and Girsanov's

Theorem 1636.2 More Useful Examples 167

7 Linear and Non-linear Filtering 1697.1 Solutions of SDE with Functional Coefficients and Girsanov

Theorems 1697.2 Martingale Representation Theorems (Functional Coefficient

Case) 1777.3 Non-linear Filtering Equation 1807.4 Optimal Linear Filtering 1917.5 Continuous Linear Filtering. Kalman-Bucy Equation . . . . 1947.6 Kalman-Bucy Equation in Multi-Dimensional Case 1967.7 More General Continuous Linear Filtering 1977.8 Zakai Equation 2017.9 Examples on Linear Filtering 203

8 Option Pricing in a Financial Market and BSDE 2058.1 Introduction 2058.2 A More Detailed Derivation of the BSDE for Option Pricing 2088.3 Existence of Solutions with Bounded Stopping Times . . . . 209

8.3.1 The General Model and its Explanation 2098.3.2 A Priori Estimate and Uniqueness of a Solution . . . 2138.3.3 Existence of Solutions for the Lipschitzian Case . . . 215

8.4 Explanation of the Solution of BSDE to Option Pricing . . 2198.4.1 Continuous Case 2198.4.2 Discontinuous Case 220

8.5 Black-Scholes Formula for Option Pricing. Two Approaches 2238.6 Black-Scholes Formula for Markets with Jumps 2298.7 More General Wealth Processes and BSDEs 2348.8 Existence of Solutions for Non-Lipschitzian Case 2368.9 Convergence of Solutions 2398.10 Explanation of Solutions of BSDEs to Financial Markets . . 2418.11 Comparison Theorem for BSDE with Jumps 2438.12 Explanation of Comparison Theorem. Arbitrage-Free Market 2508.13 Solutions for Unbounded (Terminal) Stopping Times . . . . 2548.14 Minimal Solution for BSDE with Discontinuous Drift . . . . 258

viii Contents

8.15 Existence of Non-Lipschitzian Optimal Control. BSDE Case 2628.16 Existence of Discontinuous Optimal Control. BSDEs in R1 . 2678.17 Application to PDE. Feynman-Kac Formula 271

9 Optimal Consumption by H-J-B Equation and LagrangeMethod 2779.1 Optimal Consumption 2779.2 Optimization for a Financial Market with Jumps by the La-

grange Method 2799.2.1 Introduction 2809.2.2 Models 2809.2.3 Main Theorem and Proof 2829.2.4 Applications 2869.2.5 Concluding Remarks 290

10 Comparison Theorem and Stochastic Pathwise Control 29110.1 Comparison for Solutions of Stochastic Differential Equations 292

10.1.1 1—Dimensional Space Case 29210.1.2 Component Comparison in d—Dimensional Space . . 29310.1.3 Applications to Existence of Strong Solutions. Weaker

Conditions 29410.2 Weak and Pathwise Uniqueness for 1-Dimensional SDE with

Jumps 29810.3 Strong Solutions for 1-Dimensional SDE with Jumps . . . . 300

10.3.1 Non-Degenerate Case 30010.3.2 Degenerate and Partially-Degenerate Case 303

10.4 Stochastic Pathwise Bang-Bang Control for a Non-linearSystem 31210.4.1 Non-Degenerate Case 31210.4.2 Partially-Degenerate Case 316

10.5 Bang-Bang Control for d—Dimensional Non-linear Systems 31910.5.1 Non-Degenerate Case 31910.5.2 Partially-Degenerate Case 322

11 Stochastic Population Control and Reflecting SDE 32911.1 Introduction 33011.2 Notation 33211.3 Skorohod's Problem and its Solutions 33511.4 Moment Estimates and Uniqueness of Solutions to RSDE . 34211.5 Solutions for RSDE with Jumps and with Continuous Coef-

ficients 34511.6 Solutions for RSDE with Jumps and with Discontinuous Co-

efficients 34911.7 Solutions to Population SDE and Their Properties 35211.8 Comparison of Solutions and Stochastic Population Control 363

Contents ix

11.9 Caculation of Solutions to Population RSDE 372

12 Maximum Principle for Stochastic Systems with Jumps 37712.1 Introduction 37712.2 Basic Assumption and Notation 37812.3 Maximum Principle and Adjoint Equation as BSDE with

Jumps 37912.4 A Simple Example 38012.5 Intuitive thinking on the Maximum Principle 38112.6 Some Lemmas . 38312.7 Proof of Theorem 354 386

A A Short Review on Basic Probability Theory 389A.I Probability Space, Random Variable and Mathematical Ex-

pectation 389A.2 Gaussian Vectors and Poisson Random Variables 392A.3 Conditional Mathematical Expectation and its Properties . 395A.4 Random Processes and the Kolmogorov Theorem 397

B Space D and Skorohod's Metric 401

C Monotone Class Theorems. Convergence of Random Processes407C.I Monotone Class Theorems 407C.2 Convergence of Random Variables . . . . . . . . . . . . . . 409C.3 Convergence of Random Processes and Stochastic Integrals 411

References 415

Index 431

Preface

Stochastic differential equations (SDEs) were first initiated and developed by K. Ito (1942). Today they have become a very powerful tool applied to Mathematics, Physics, Chemistry, Biology, Medical science, and almost all sciences. Let us explain why we need SDEs, and how the contents in this book have been arranged.

In nature, physics, society, engineering and so on we always meet two kinds of functions with respect to time: one is determinstic, and another is random. For example, in a financial market we deposite money rt in a bank. This can be seen as our having bought some units r)! of a bond, where the bond's price Pf satisfies the following ordinary differential equation

dPf = Pfrtdt, P: = 1, t E [O,T], where rt is the rate of the bond, and the money that we deposite in the

t bank is r t = r)!Pf = r)! exp[Jl r,ds]. Obviously, usually, Pf = exp[& r,ds] is non-random, since the rate rt is usually deterministic. However, if we want to buy some stocks from the market, each stock's price is random. For simplicity let us assume that in the financial market there is only one stock, and its price is P i . Obviously, it will satisfy a differential equation as follows:

dPi = Pi(btdt + d(a stochastic perturbation)), Po' = Po', t E [O,T], where all of the above processes are 1-dimensional. Here the stochastic perturbation is very important, because it influences the price of the stock, which will cause us to earn or lose money if we buy the stock. One important' problem arises naturally. How can we model this stochastic perturbation? Can we make calculations to get the solution of the stock's price Pi, as we do in the case of the bond's price Pf? The answer is positive, usually a

continuous stochastic perturbation will be modeled by a stochastic integral/ 0 asdws, where Wt,t > 0 is the so-called Brownian Motion process (BM),or the Wiener process. The 1—dimensional BM wt, t > 0 has the followingnice properties: 1) (Independent increment property). It has an indepen-dent increment property, that is, for any 0 < t\ < • • • < tn the system{wo, w^ — wo, w42 — Wt,, • • • , Wtn — tvtn^i, } is an independent system. Orsay, the increments, which happen in disjoint time intervals, occured inde-pendently. 2) (Normal distribution property). Each increment is Normallydistributed. That is, for any 0 < s < t the increment Wj — ws on this timeinterval is a normal random variable with mean m, and variance a2(t — s).We write this as wt — ws ~ N(m,a2(t — s)). 3) (Stationary distributionproperty). The probability distribution of each increment only depends onthe length of the time interval, and it does not depend on the starting pointof the time interval. That is, the m and a2 appearing in property 3) areconstants. 4) (Continuous trajectory property). Its trajectory is continuous.That is BM wt, t > 0 is continuous in t.

Since the simplest or say, the most basic continuous stochastic perturba-tion, intuitively will have the above four properties, the modeling of the gen-eral continuous stochastic perturbation by a stochastic integral with respectto this basic BM Wt,t > 0 is quite natural. However, the 1—dimensionalBM also has some strange property: Even though it is continuous in t, itis nowhere differentiate in t. So we cannot define the stochastic integral/0 as(u})dws{(jj) for each given w. That is why K. Ito (1942) invented acompletely new way to define this stochastic integral.

Our first task in this book is to introduce the Ito stochastic integral anddiscuss its properties for later applications.

After we have understood the stochastic integral jQ as(uj)dws(uj) we canstudy the following general stochastic differential equation (SDE):

xt = xo + / 0 *Z(s , x s )ds + Jo a(s,xs)dws, t>0,

or equivalently, we write

dxt — b(t,xt)dt + <f(t,xt)dwt,xo = xo,t > 0. (1)

Returning to the stock's price equation, we naturally consider it as thefollowing SDE:

dPl = P}(btdt + ctdwt),Pj = P o \* € [0,T\. (2)

Comparing this to the solution of Pt°, one naturally asks could the solutionof this SDE be P / = PQ exp[J0 bsds + JQ crsdws}'! To check this guess,obviously if we can have a differential rule to perform differentiation onPQ expxt, where Xt = f0 bsds + JQ crsdws, then we can make the check. Ormore generally, if we have an f(x) € C2(R) and dxt = btdt + atdwt, canwe have

df(xt) = f'(xt)dxt = f'(xt) (btdt + crtdwt)f

If as in the deterministic case, this differential rule holds true, then weimmediately see that P/ = PQ exp[/0 bsds + JQ asdws] satisfies (2). Unfor-turnately, such a differential rule is not true. K. Ito (1942) has found thatit should obey another differential rule - the so-called Ito's formula:

df'(xt) = f'(xt)dxt + |/"(x t) Wtf dt.By this rule one easily checks that

P / = PQ1 exp[/0* bsds +jl asdws - \ /„* |<rs|2 ds]

is a solution of (2), and P / = PQ exp[/0 bsds + JQ asdws] actually satisfiesanother SDE:

dPl = Pi [/„* bsds + /„* asdws + I f* K | 2 ds], Pi = PQ1 , W € [0, T].Our second task in this book is to establish the Ito formula and discuss

its applications: solving SDE and solving other problems.However, even if we have a powerful Ito formula (or say, Ito's differential

rule) in our hand, we still need to discuss how to solve the general SDE,because usually, the form of the solution of SDE is not easy to guess.Moreover, for solutions of SDE, we actually meet a more complicated andhence also a more interesting case. Consider some physical quantity xjdetermined by dynamics. If this dynamics is deterministic, that is, it is notdisturbed by any random noises, say such that dxf = b(t, Xt)dt, xo = XQ, t >0; then solving this ODE we immediately get this quantity xt- However,if the dynamics are disturbed by some continuous random noise, say suchthat dxt = b(t,Xt)dt + a(t,xt)dwt,xo = XQ,t > 0, then for the amountXt, or say, the solution of this SDE, two situations can arise. The first oneis, if we think that the random noise - BM ws,s < t, is an input, thenafter disturbing the dynamics we get an output xt. This means that thesolution xt is a functional of the given noise - BM ws,s < t for each t.We will call such a solution a strong solution. Another situation is thatfor a given noise we cannot immediately find the solution. However, wecan find a random process xt, t > 0, and maybe another random noisethat is also a BM wt,t > 0, such that (xt,Wt),t > 0 satisfy the SDEdxt = b(t,xt)dt + a(t,xt)dwt,xo = xo,t > 0. In this case we will call(xt,wt),t > 0 or simply, Xt,t > 0, a weak solution of the original SDE.Obviously, from the engineering point of view the strong solutions is morerealistic and useful. However, since, if the strong solution xt, t > 0 exists,then all finite dimensional probability ditributions of (xt,wt)t>0 are thesame as that of (St, w t ) t > 0 . So, if we do not know the existence of a strongsolution, but we do know the existence of a weak solution (xt,Wt),t > 0,then from the probability point of view the weak solution can still help usin some sense. Therefore, for solutions of SDEs there are two kinds thatneed to be considered: strong solutions and weak solutions.

Our third task in this book is to introduce the concepts of solutions andto discuss their existence and uniquess and the related important theory.(For example, Girsanov's theorem and the matingale representation theo-rem, the first of which can help us find weak solutions, while the second

one is necessary for finding the solutions of backward SDE and the filteringproblem considered later).

Since, actually, in the realistic world we will always meet some jump typestochastic perturbation, in this book we also consider stochastic integralswith respect to a Poisson counting measure (which is generated by a Poissonpoint process), the Ito formula and SDE for this case. (To find the reasonwhy we consider the Poisson point process and its related integral as ajump type stochastic perturbation see the subsection "The General Modeland its Explanation" in Chapter 8 - "Option Pricing in a Financial Marketand BSDE").

The first three Chapters are intended to solve the above three tasks.They are the basic foudation of the SDE theory and its applications.

However, interesting and important things for SDE do not only comefrom the above mentioned three chapters, where they exhibit the followingfacts: the definition of Ito's stochastic integrals and Ito's differential ruleare completely different from the deterministic case, etc. The interestingand important things also come from the following facts: __

1) For an ordinary differential equation (ODE) dxt — b(t,Xt)dt,xo —XQ, t > 0 if b(t, x) is only bounded and jointly continuous, then even thoughthe solution exists, is not necessary unique. However, for the SDE (1) in1—dimensional case if 6(£, a;) and a(t, x) are only bounded and jointly Borel-measurable, and |5(i,a;)|~ is also bounded and a(t, x) is Lipschitz continu-ous in x, then (1) will have a unique strong solution. (Here "strong" meansthat Xt is #2"—measurable). This means that adding a non-degenerate sto-chastic perturbation term into the differential equation, can even improvethe nice property of the solution.

2) The stochastic perturbation term has an importaqnt practical mean-ing in some cases and it cannot be discarded. For example, in the invest-ment problem and the option pricing problem from a Financial Market,the investment portfolio actually appears as the coefficient of the stochas-tic integral in an SDE, where the stochastic integral acts like a stochaticperturbation term.

3) The solutions of SDEs and backward SDEs can help us to explain thesolutions of some deterministic partial differential equations (PDEs) withintegral terms (the Feynman-Kac formula) and even to guess and find thesolution of a PDE, for example, the soluition of the PDE for the price of aoption can be solved by a solution of a BSDE - the Black-Scholes formula.

4) More and more.So we have many reasons to study the SDE theory and its applications

more deeply and carefully. That is why we have a Chapter that discussesuseful tools for SDE, and a Chapter for the solutions of an SDE with non-Lipschitzian coefiicients. These are Chapter 4 and 5.

The above concerns the first part of our book, which represents the theoryand general background of the SDE.

The second part of our book is about the Applications.We first provide a short Chapter to help the reader to take a quick look

at how to use Stochastic Analysis (the theory in the first part), to solve anSDE.

Then we discuss the estimation problem for a signal process : the so-calledfiltering problem, where the general linear and non-linear filtering problemfor continous SDE systems and SDE systems with jumps, the Kalman-Bucy filtering equation for continuous systems, and the Zakai equation fornon-linear filtering, etc. are also considered.

Since, now, research on mathematical finance, and in particular on theoption pricing problem for the financial market has become very popular,we also provide a Chapter that discusses the option pricing problem andbackward SDE, where the famous Black-Scholes formulas for a market withor without jumps are derived using a probability and a PDE approach,respectively; and the arbitrage-free market is also discussed. The interestingthing here is that we deal with the mathematical financial problem by usingthe backward stochastic differential equation (BSDE) technique, which nowbecomes very powerful when treating many problems in the finacial market,in mathematics and in other sciences.

Since deterministic population control has proved to be important andefficient, and the stochastic population control is more realistic, we alsoprovide a Chapter that develops the stochastic population control problemby using the reflecting SDE approach, where the existence, the comparisonand the calculation of the population solution and the optimal stochasticpopulation control are established.

Besides, the stochastic Lagrange method for the stochastic optimal con-trol, the non-linear pathwise stochastic optimal control, and the MaximumPrinciple (that is, the necessary conditions for a stochastic optimal control)are also formulated and developed in specific Chapters, respectively.

For the convenience of the readers three Appendixes are also provided:giving a short review on basic probability theory, space D and Skoro-hod's metric, and monotone class theorems and the convergence of randomprocesses.

We suggest that the reader studies the book as follows:For readers who are mainly interested in Applications, the following ap-

proach may be considered: Appendix A —> Chapter 1 —+ Chapter 2 —>Chapter 3 •—> Chapter 6 —» Any Chapter in The second part "Applications"except Chapter 10, and at any time return to read the related sections inChapters 4 and 5, or Appendixes B and C, when necessary. However, toread Chapter 10, knowledge of Chapters 4 and 5 and Appendixes B and Care necessary.

Acknowledgement

The author would like to express his sincere thanks to Professor Alan Jeffrey for kindly recommending publication of this book, and for his interest in the book from the very beginning to the very end, and for offering many valuable and important suggestions. Without his help this book would not have been possible

Abbreviations and Some Explanat ions

All important statements and results, like, Definitions, Lemmas, Propo- sitions, Theorems, Corollaries, Remarks and Examples are numbered in sequential order throughout the whole book. So, it is easy to find where they are located. For example, Lemma 22 follows Definition 21, and Theo- rem 394 is just after Remark 393, etc. However, the numbers of equations are arranged independently in each Chapter and each Appendix. For example, (3.25) means equation 25 in Chapter 3, and (C.4) means the equation 4 in Appendix C.

The following abbreviations are frequently used in this book. a.e. almost everywhere. a.s. almost sure. BM Brownian Motion. BSDE backward stochastic differential equation. FSDE forward stochastic differential equation. H-J-B equation Hamilton-Jacobi-Bellman equation. IDE integral-differential equation. ODE ordinary differential equation. PDE partial differential equation. RCLL right continuous with left limit. SDE stochastic differential equation. P - a.s almost sure in probability P. a+ max {a, 0) . a- max {-a, 0). a v b max{a,b). a A b min{a,b).

fj. « v measure /i is absolutely continuous with respect to v\ that is, forany measurable set A, v(A) = 0 implies that n(A) = 0.£n —> £, a.s. £n converges to £, almost surely; that is, £n(w) —> £(w) for allUJ except at the points w € A, where P(A) = 0.£n —> ^, in P £n converges to ^ in probability; that is, Ve > 0,

lim^oo P(w : |C(w) - e(w)| > E) = 0.# {•} the numbers of • counted in the set {•} .cr(xs, s < t) the smallest cr—field, which makes all a;s, s < t measurable.S[̂ |?7] E[£\(T(T})]. It means the conditional expectation of ^ given cr(r)).

The following notations can be found on the corresponding pages. Forexample, #, 4,387 means that notation J can be found in page 4 and page387.0,4,387S, 4,387

Thp

C ,

c\

11 (

343434

oc,Rdl

43434343

Rd{

) ( /

34

48)51

Zd®d2

f,P)-

212

212), 21212

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O, 7T5, 7P, 4,387

, 212

Part I

Stochastic Differential Equations with .. Jumps in

Martingale Theory and the Stochastic Integral for Point Processes

A stochastic integral is a kind of integral quite different from the usual deterministic integral. However, its theory has broad and important applications in Science, Mathematics itself, Economic, Finance, and elsewhere. A stochastic integral can be completely charaterized by martingale theory. In this chapter we will discuss the elementary martingale theory, which forms the foundation of stochastic analysis and stochastic integral. As a first step we also introduce the stochastic integral with respect to a Point process.

1.1 Concept of a Martingale.

In some sense the martingale conception can be explained by a fair game. Let us interprete it as follows:

In a game suppose that a person a t the present time s has wealth x, for the game, and at the future time t he will have the wealth xt. The expected money for this person a t the future time t is naturally expressed as E[xt15,], where E[.] means the expectation value of ., 8, means the information up to time s , which is known by the gambler, and E[.lSs] is the conditional expectation value of - under given 5,. Obviously, if the game is fair, then it should be

E[xt15sl = xs,W 2 s. This is exactly the definition of a martingale for a random process xt, t >

0. Let us make it more explicit for later developement.

4 1. Martingale Theory and the Stochastic Integral for Point Processes

Let (R, 5 , P ) be a probability space, {St)t,o be an information family ( in Mathematics, we call it a a-algebra f a d y or a a-field family, see Appendix A ) , which satisfies the so-called "Usual Conditions":

( 2 ) 5, c St, as 0 5 s 5 t; (ii) 5t+ =: nh>05t+h. Here condition (i) means that the information increases with time, and condition (ii) that the information is right continuous, or say, 1 gt, as h 1 0. In this case we call a a-field filtration. -

Definition 1 A real random process { x ~ ) , , ~ is called a martingale (supermartingale, submartingale) with respect 15 {5t}t,o - , or {x t , &}t20 is a martingale (supermartingale, submartingale), i f (2) xt i s integrable for each t 1 0; that is, E lxtl < w , V t 1 0; (ii) xt is gt-adapted; that is, for each t 2 0, xt is &-measurable; (iii) E(xtlTs] = x,, (respectively, <, 2), a.s. VO 5 s 5 t.

For the random process {xt)tE~O,Tl and the random process { x , ) ~ = ~ with discrete t ime similar definitions can be given.

Example 2 If { ~ t ) , , ~ is a random process with independent increments; that is, VO < tl < t2 7 . . . < t,, the family of random variables

{xo, X t l - 2 0 , xtz - X t l , . . . , xt, - is independent, and the increment xt - x,, W > s , is integrable and with non-negative expectation, moreover, xo is also integrable, then {xt)t>o is a submartingale with respect to {$;),Lo, where 5; = a(x, , s 5 t ) , whizh is a a-field generated by {x,, s 5 t ) (that is, the smallest a-field which makes all x,, s 5 t measurable) and makes a completion.

In fact, by independent and non-negative increments, 0 5 E(x t - x S ) = E[(xt - ~ ~ ) 1 5 : ] , V t 2 S.

Hence the conclusion is reached.

Example 3 If is a submartingale, let yt := xt V 0 = max(xt,O), then {yt)t20 is still a submartingale.

In fact, since f ( x ) = x V 0 is a convex function, hence by Jensen's inequality for the conditional expectation

E[x t V OISs] 1 E[xtlSs] V E[Ol5s] 2 2 s V O,Vt 2 S .

So the conclusion is true.

Example 4 If { x ~ } , , ~ is a martingale, then { I ~ t l ) ~ , ~ is a submartingale. - -

In fact, by Jensen's inequality E[Ixtl 15,] > IE[xtlSsll = Ix.4 ,W 1 S.

Thus the conclusion is deduced. Martingales, submartingales and supermartingales have many important

and useful properties, which make them become powerful tools in dealing with many theoretical and practical problems in Science, Finance and elsewhere. Among them the martingale inequalites, the limit theorems, and the

1.2 Stopping Times. Predictable Process 5

Doob-Meyer decomposition theorem for submartingales and supermartingales are most helpful and are frequently encountered in Stochastic Analysis and its Applications, and in this book. So we will discuss them in this chapter. However, to show them clearly we need to introduce the concept called a stopping time, which will be important for us later. We proceed to the next section.

1.2 Stopping Times. Predict able Process

Definition 5 A random variable ~ ( w ) E [O, co] is called a &-stopping time, or simply, a stoping time, if for any (co >)t > 0, { ~ ( w ) 5 t ) E St.

The intuitive interpletation of a stopping time is as follows: If a gambler has a right to stop his gamble at any time ~ ( w ) , he would of course like to choose the best time to stop. Suppose he stops his game before time t , i.e. he likes to make ~ ( w ) I t , then the maximum information he can get about his decision is only the information up to t , i.e { T ( w ) < t ) E zt. The trivial example for a stopping time is ~ ( w ) = t , Vw E 0. That is to say, any constant time t actually is a stopping time.

For a discrete random variable T ( W ) E {0,1,2,. . . , CO) the definition can be reduced to that ~ ( w ) is a stopping time, if for any n E {0,1,2,. - . ) , {T(w) = n ) E zn, since { ~ ( w ) = n ) = { ~ ( w ) < n ) - { ~ ( w ) < n - I ) , and {T(w) 5 n ) = UZzl { ~ ( w ) = k) . The following examples of stopping time are useful later.

Example 6 Let B be a Bore1 set i n R' and {xn)z="=le a sequence of real gt-adapted random variables. Define the first hitting time T B ( W ) to the set B (i.e. the first time that { x , ) ~ = ~ hits B) by

T B ( w ) = inf { n : xn(w) E B ) . Then T B ( w ) is a discrete stopping time.

In fact, {TB(W) = n ) = nr:; {xk E B C ) n { x , E B ) E 8,. For a general random process with continuous time parameter t we have

the following similar example.

Example 7 Let xt be a d- dimensional right continuous zt - adapted process and let A be an open set i n Rd. Denote the first hitting time OA(W) to A by

a A ( w ) = inf { t > O : xt (w) E A ) . Then a A ( w ) k a stopping time.

In fact, by the open set property and the right continuity of xt one has that

{ ~ A ( w ) I t ) = n ? = l { ~ A ( w ) < t + i) - 00 - u T ~ ~ , r < t + ~ / n { ~ ( w ) E A ) E gt+o = zt,


where Q is the set of all rational numbers. The following properties of general stopping times will be useful later.

Lemma 8 T(W) is a stopping time, if and only if { ~ ( w ) < t ) E 5t, W.

Proof. 3: { ~ ( w ) < t) = UFZ1 { ~ ( w ) < t - i) E 5t . +==: {T(w) < t ) = flF=l {T(w) < t + i) E $t+o = %t. H

Lemma 9 Let a, T, a,, n = 1,2, + . . be stopping times. Then (i) a A ~ , a V r , (ii) a = limn,, a,, when a, T or a, 1,

are all stopping times.

Proof. (i): {a A T < t) = {a < t) U {T < t ) E &, { a v ~ < t) ={a t ) n { ~ < t) E St. (ii): If a, f a , then {a < t) = flFZl {a, < t) E 3t. If a, 1 a, then {a < t) = UFZl {a, < t ) E 5t .

By Lemma 8 a is a stopping time. H Now let us introduce a a- field which describes the information obtained

up to stopping time T. Set 5, ={A E 5,: Vt E [O,oo),Afl{~(w) 5 t) E 5t),

where we naturally define that 5, = Vtlo3t, i.e. the smallest a-field including all Zt, t E [0, m). Obviously, 3, is a a- algebra, and if ~ ( w ) = t, then 8, = &.

Proposition 10 Let a, T, a,, n = 1,2, . . - be stopping times. (1) If 4 ~ ) < T(w), Vw, then 5, c &. (2) If an(^) 1 4w) , vw, then nF=15,, = 5,. (3) a E 5,. (We use f E 5, to mean that f is 5,-measurable).

Proof. (1): A n {T I t ) = (A n {a < t)) fl { r < t) E 5t. (2): By (1) 5, c n,"!15,,. Conversely, if A E n~=15,,, then Afl{o, < t ) = ~ ~ ~ ( A n { a , < t - ~ ) ) ~ S ~ , V t 2 0 , V n .

Hence A n {a < t) = u,"!~(A n {a, < t)) E &, and A n { a < t) = fl&(An {a < t + i ) ) E &+o = 5t, i.e. A E 5,. (3): For any constant 0 < c < oo one has that {a < c)n{a < t) E StAc c

&,so { a < c ) E&. . It is natural to ask that if { x ~ ) , , ~ is &-adapted, and a is a stopping

time, is it true that z, E go? Generally speaking, it is not true. However, if {xt)tZo is a progressive measurable process, then it is correct. Let us introduce such a related concept.

Definition 11 An valued process { ~ t ) ~ , ~ is called measurable (respectively, progressive measurable), if the mapping

(t, w ) E (0, oo) x Q -+ xt(w) E Rd

1.2 Stopping Times. Predictable Process 7

(respectively, for each t > 0 , ( s ,w) E [0, t] x R -+ xt(w) E Rd) is %([O, m)) x 5 / % ( R d ) -measurable

(respectively, %([0, t ] ) x gt / % ( R d ) -measurable); that is, { ( t ,w) : xt(w) E B ) E %([0, m)) x S ,VBE % ( R d ) (respectively, { ( s , w ) : s E [O, t ] , x,(w) E B ) E %([0, t ] ) x & , V B E % ( R d ) ) .

Let us introduce two useful a-algebras as follows: Denote by P (respectively, 0) as the smallest a-algebra on [O, oa) x R such that all left- continuous (respectively, right-continuous) $t-adapted processes

vt(w) : [ o , ~ ) x -+ yt(w) E Rd are measurable. P (respectively, 8) is called the predictable (respectively, optional) a-algebra. Thus, the following definition is natural.

Definition 12 A process { x ~ ) , , ~ - is called predictable (optional), i f the mapping

( t , ~ ) E [o, W ) x n -+ x ~ ( w ) E R~ is P / % ( R d ) -measurable (respectively 0 / % ( R d ) -measurable).

Let us use the notation f E P t o mean that f is P - measurable; etc. I t is easily seen that the following relations hold:

f E P + f E 0 + f is progressive measurable + f is measurable and St -adapted.

W e only need t o show the first two implications. T h e last one is obvious. k Assume that {xt)t>o - is left-continuous, let x: = x $ , as t E [ F , 2n ),

k = 0,1, . - . ; n = 1,2, . . . . Then obviously, x: is right-continuous, and by the left-continuity o f x t , x:(w) -+ xt(w) , as n -+ co,Vt,Vw. So {xt)tlo E 8. From this one sees that PC 0. Let us show that {xt)tyo E 8 implies that {xt)t20 is progressive measurable. For this for each given t > 0 we show that {x , ) ,>~ restricted on ( s , w ) E [0, t ] x R is %([0, t ] ) x 5t- measurable. In fact, without loss o f generality we may assume that {xt)t>o is right- continuous. Now for each given t > 0 , let x> x+ as s E [&, w), k =

0,1, - . . ,2" - 1; n = 1,2, . . . . Then obviously, { x ~ ) , ~ [ ~ , ~ ~ is %([0, t ] ) x 3t-measurable, so is { x ~ ) , , ~ ~ , ~ ~ , since by the right continuity o f xt we have that as n -+ m , x y ( w ) -+ x,(w) ,Vs E [O,t],Vw.

Let us show the following

Theorem 13 If { x ~ ) , , ~ is a Rd-valued progressive measurable process, then for each stopping f me a, Z,I,<, is &-measurable.

W e will use the compositon o f measurable maps t o show this theorem. For this we need the following lemma.

Lemma 14 If fi is a measurable mapping from ( R , 5) to (R; , g:), i = 1,2, . . - ; then

f ( w ) = ( f1 (w) , f2 (w) , is a measurable mapping from (a, 5) to (0; x R i x . . . ,zi x 5; x - . - ).


In fact, for any Bi E $,i = 1,2, ... , f-l(B1 x B 2 x . - . ) = ng1 fz:l(~i) E 5. So f-l(k-7; x x ; x ...) c5.

Now let us prove Theorem 13. Proof. Let 52, = {a < co). We need to show that x, is a measurable

mapping from (au , 3,) to (Rd, 123(Rd)). For any given t 2 0 by Proposition 10 a E 5,. So by the definition of 5,, a is a measurable mapping from ({a < t) ,St) to ([O,tl, B([O,tl)). Hence by Lemma 14 gl(w) = (a(w),w) is a measurable mapping from ({a < t) ,St) to ([0, t] x 52, 123([0, t]) x s t ) . Note that by the progressive measurability of { x ~ ) ~ , ~ , g2(s, W) = xs(w) is a measurable mapping from ([O, t] x a, 123([0, t]) x St) to (Rd, 123(Rd)). Hence x,(,)(w)I,<, = g2 o gl(w) is a measurable mapping from ({a < t) , z t ) to ( R ~ , % ( R ~ ) ) This shows that for any B E 123(Rd), {x,I,<, E B) n {a < t) E ?jt.Since t 2 0 is arbitrary by definition {x,I,<, E B) E 3,.

1.3 Martingales with Discrete Time

First we will show the Doob's stopping theorem (or called Doob's optional sampling theorem) for bounded stoping times.

Theorem 15 Let { x ~ ) ~ = ~ , ~ , ~ , . . be a martingle (supermartingle, submartingale), a < T be two bounded stopping times. Then { x ~ ) , = ~ , ~ , ~ , . . . is a strong martingle (respectively, strong supermartingle, strong submartingle), i.e.

E[x71&] = xu (respectivly, 5 , >), a.s.

Proof. We only prove the conclusion for the case of submartingle. By assumption there exists a natural number 0 < no such that T < no. So lxT1 <max{1xnl,n=0,1,2,... ,no) < x z o l x n l . So E lxT1 < o . By the same manner E I x , ~ < co. Note that by the definition of a stopping time and for A E 5, and 0 < n 5 no

A ~ { a = n ) f l { ~ > n } E ~ ~ . Now suppose T -a < 1 in addition. Then by the definition of a submartingale

SA(xu - xT)dP = C 2 o SAn{o=n)n{T>n)(~n - xn+l)dP I 0. In the general case set Tn = T A (a + n), n = 1,2, . - . , no. Then all T, are stopping times, and

a < T l < T 2 < ~ ~ ~ < T n , = ~ , T l - a < 1 , T n + 1 - T n < 1 , n = 1 ,2 , - . . ,no - 1.

Let A E 3, C zT,,,. Then by the above conclusion JA x,dP < JA xT,dP < . . . < JA xTdP.

The proof is complete. H Now we have the following martingale inequality:

Theorem 16 Let { x ~ ) ~ = ~ , ~ , . . . be a submartingale. Then for every X > 0 and natural number N

1.3 Martingales with Discrete Time 9

Proof. Let us use the first hitting time technique and strong submartingale property to show this theorem. Set u = min{n < N : x, 2 A); u = N, i f {.) = 4.

Then a is a bounded stopping time. By Theorem 15 E X N L Exu = E ~ u I r n a x ~ ~ , , < ~ x , ? ~ + E x ~ l X n a X o < n < N z , < ~ 2 AP(m=o<n<N X n 2 A) + Ex~Irnax~<,<~ %,<A. Transferring the last term to the left hand side, we obtain the first in-

equality. Now set T = min{n < N : x, < -A); T = N, if {.) = 4.

Then Ex0 < EXT = E~~Irnin~<,<Nx~<-X + E~NIrnin~<,<~ xn>-X < -A~(mino<~<N xn < -A) + E ( ~ N I r n i n ~ ~ , < ~ %,>-A).

Thus the second inequality is derived.

Corollary 17 1) Assume that {x,),,~,~,... is a real submartingale such that ~ ( ( x : ) ' ) < m, n = 0,1,. . . , for some p > 1. Then for every N, and A > 0,

+ P AP P(ma%<n<N ~2 2 A) < E ( ( x N ) )/ , and if p > 1,

P E(m=O<n<N ( x : ) ~ ) < (5) E((x&)').

2) If {x,),=~,~ ,... is a real martingale such that E(lxnlP) < m, n = 0,1, - . , then the conclusions in 1) hold true for x: and x$ replaced by lxnl and IxN I , respectively.

Proof. 1): By Example 3 {x:),=~,~ ,... is a non-negative submartingale. Using Jensen's inequality again one has that { ( x ~ ) ~ ) ~ = ~ , ~ , . , , is still a non- negative submartingale. Hence the first inequality is obtained from Theo- rem 16. Now if p > 1, set [ = maxo<,<N (x:) . then by Theorem 16 again one has that

A P ( ~ 2 A) I E X : I ~ ~ ~ Hence using Fubini's theorem and Hblder's inequality one derives that

E ( [ ~ ) = E S ; ~ X P - ~ ~ A = E S , O O ~ X P - ~ I ~ < ~ ~ A = pS,OO A ~ - ' P ( ~ A ) ~ A - p-1-+ < P S; p - 2 ~ ( ~ t ~ < > x ) d ~ = z N ] < 6 [ E ( X $ ) P ] ~ / ~ [ E ~ ~ ] ( ~ ~ ) / P .

Now if E (tP) = 0, then the second inequality is trivial. I f E (tp) > 0, dividing both sides by [ E [ ~ ] ( P - ~ ) I P , the second inequality is also obtained.

2): I f {x,),,~,~,... is a real martingale, then by Jensen's inequality we

have that { ~ x ~ ~ ) ~ = ~ , ~ , . . . is a submartingale, and lxn(+ = lx,l. So by 1 ) the conclusions are derived in this case.


In the following we will show the upcrossing inequality for a submartingale, which is the basis for proving the important limit property of a submartingale. First we introduce some notations.

For a real &- adapted process { x ~ ) ~ = ~ , ~ , . . and an interval [a, b], where b > a , let

71 =min{n > 0 : x, < a ) , 7 2 = min{n > 71 : x, > b), ...... 72,+l = min{n > 72, : x, < a), 72n+2 = min{n > 72n+l : xn > b), ...... . I

where we recall that min 4 = +co. Then (7,) is an increasing sequence of stopping times. In fact, Vk 2 0,

{TI = k) = {XO > a, x1 > a , . . . , xk-1 > a, xk < a) E Sk; (72 = k) = U:=: {TI = j, 7 2 = k)

= u;:: (71 = j, x j < a, X j + l < b, , xk-1 < b, xk 2 b) € 8k; {73 = k) = u;;: ( 7 2 = j, 73 = k) = U : Z : { T ~ = ~ , X ~ >bb,xj+l > a , . . - , x ~ - I > U , X I , < U ) E ~ ~ .

Hence r 1 , ~ 2 , and 73 are stopping times. The proofs for the rest are similar. Now set

~,b[x(.), N](w) = max {k 2 1 : T ~ ~ ( W ) < N) , D: [x(.), N] (w) = max {k 2 1 : T ~ ~ - ~ (w) < N) .

N Obviously the first one is the number connected to the upcrossing of {x,),=, for the interval [a, b], and the second one is the number connected to the downcrossing of { x , ) ~ = ~ for the interval [a, b].

Proof. For a submartingale { x , ) ~ = ~ by Example 3 one sees that {Yn)~=o = {(x, - a)+):=o is a non-negative submartingale. Clearly ~;-"[y(.), N](w) = U,b[x(.), N](w). Again define 71 ,72 , . . + as above, but with x, a, and b replaced by y, 0, and b - a respectively. Then if 2k > N

k E(YN-YO) = EC?=~(Y~,,AN-YT~-IAN) = ECn=1(~7znAN-~Tzn-1AN)

k-1 + Cn=o E ( ~ ~ z n + ~ ~ ~ - Y T ~ ~ A N ) 2 (b -a )~@-~[y ( . ) ,N] , where we have used the fact that {yn}F=o is a submartingale, and hence a strong submartingale (Theorem 15), so E (y72n+ lA~ - y T Z n A ~ ) > 0; and y, 2 0, Vn. The first inequality is proved. Now observe that

0 2 E(YTZ,.AN - Y ~ ~ + ~ A N ) - - E [ ( ~ ~ ~ n ~ ~ - ~72n+lh~)(I72n1N<TZn+1 +'TZ~+I<N)I

1.3 Martingales with Discrete Time 11

= E[(b - a - Y N ) ~ T Z , , < N < T Z ~ + ~ f (b - ~)ITz,,+I<N] = E(b - a)&ansN - EYNITZ,,~N<T~,,+~.

Since {~;-"[y(.), N] 1 n ) = {N 2 72,) and

( ~ 2 n I N < 72n f 1) C (72, I N < 72n+2) = {@-"[Y(.), N] = n) . Hence we find that E Y ~ I ~ : [ ~ ( , ) , N I = ~ 2 (b - a)~(~;-"[y( . ) , N] 2 n).

For the downcrossing inequality we have to discuss {~,)r=~ itself di- rectly, since {x, A O)r=o is not a submartingale. Let us set yn = x, - b. Then {vn}~=o is still a submartingale, and

DO_(b-,)I~(.), NI(w) = D%(.), NI(w). Again define r l , r 2 , . . . as above but with x, a, and b replaced by y, - (b -

a), and 0 respectively. We will now use another method to show the last two inequalities. First, for the fourth inequality we have that as n 2 1

0 2 E ( ~ ~ 2 n ~ ~ - YT~,,+IAN) = E[(O - (XN - ~))IQ,,<N<T~.,+~ + (b - ~)IT~,,+~<NI.

Since {D%(.), Nl 1 n + 1) = {D:(,-,)[Y(.), NI 2 n + 1) = {N 2 72n+2) C {N 2 72n+l)

and (72, I N < ~ 2 ~ + 1 ) C (72, 5 N < 72n+2) = {D:[x(.),N] = n ) . Hence it follows that

E(XN - b)+ID:[x(.),N]=n 2 (b - a)P(@[x(.), N] 2 n + 1). The fourth inequality holds. Now taking the summation for n 1 0 it yields

E(XN - b)+ 1 (b - a) C,"=oP(Dt[x(.), N] 2 n + 1) = (b - a) C r = o n ~ ( ~ : [ x ( . ) , N] = n) = (b - ~)ED;[X(.), N].

The third inequality is also established. rn

Proof. Let yn = -x,. Then {y,)r==, is a submartingale. Hence u M . > , Nl = DIba[Y(.), Nl,

and D m . ) , Nl = UZb"[Y(.>, Nl.

Applying Theorem 18 we arrive a t the results. w Theorem 18 and Corollary 19 are the classical crossing theorems on mar-

tingale. We can derive some other useful crossing results which are very useful in the mathematical f i n a n ~ e . [ ~ ~ ] ~ [ ~ ~ ] ~ [ ~ * ~ ] Here we apply some of them to derive the important limit theorem on martingales.


Theorem 20 If {x,)~'~ is a submartingale such that there exists a sub- sequence of {n) , denote it by {nk) , such that

then x, = limn,,xn exists a s . , and x, is integrable. In particular, i f x, < O,Vn, then condition (1.1) is obviously satisfied, and in this case Qn

E [ ~ m l 5 n ] 2 xn, a.8.

Proof. First, clearly condition (1.1) u supnEx: < oo ++ sup,Elxn) < oo. In fact, by the properties of submartingales one has that EX: 5 Ex&, Vlc.

and E lxnl = 2Ex: - EX, < 2Ex: - EX^.

Hence the equivalent relations hold. Now let U,b(x(.)) = lim~,, U,b(x(.), N). Then by Theorem 18

EU,~(X(.)) I & SUPN E(XN - a)+ < m.

Hence U,b(x(.)) < oo, a.s. Let W = Ua,b€Q,a<bwa,b = Ua,b€Q,a<b { h n x n < a < b < h n x n ) .

Then P(W) I Ca,bEQ,a<b P(Wa,b) I C a , b E Q , a < b P(U,b(x(')) = W) = O'

Now we can let x,(w) = lim,,,x,(w), as w $! W; and x,(w) = 0, as w E W. By Fatou's lemma

E 1xm1 I SUP, E Ian1 < oo. Hence x, is integrable. In the case x, < O,Vn, by the definition of a submartingale

0 > E[x,l5,] 2 xn, a s . Vm. Again by Fatou's lemma letting m -4 oo one reaches the final conclusion.

1.4 Uniform Integrability and Martingales

It is well known in the theory of real analysis that if a sequence of measurable functions is dominated by an integrable function, then one can take the limit under the integral sign for the function sequence. That is the famous Lebesgue's dominated convergence theorem. However, sometimes it is difficult to find such a dominated function. In this case the uniform integrability of that function sequence can be a great help. Actually, in many cases it is a powerful tool . Definition 21 A family of functions A c L1(R,5, P ) is called uniformly integrable, i f limx,, s u p f , ~ E(f Il = 0, where L1(R, 5, P) is the to- tality of random variables <, (that is, all J are 5-measurable) such that E IJI < m.

1.4 Uniform Integrability and Martingales 13

Lemma 22 Suppose that {x,)r==, c L1(R, 5, P ) is uniformly integrable, and as n + oo,

x, -+ x, i n probability, i.e. VE > 0, limn,, P(lx, - X I > E) = 0, then

limn,, E 12, - X I = 0. (i.e. x, 4 x, i n ~ l ( R , 5 , P)). In particular, limn,, Ex, = Ex

Proof. In fact, VE > 0, E Ixn - 21 < E(1xn - X I ~ l x n - x l > ~ ) +E(lxn - X ( I lx , -x l<~) = I:" +I;". Hence one can take a X large enough such that I:" < e/2, since clearly

{x, - x)rZl is uniformly integrable. Then for this fixed X by using Lebesgue's dominated convergence theorem one can have a sufficiently large N such that as n 2 N, I:" < ~ / 2 .

For the sufficient conditions of uniform integrability of a family A we have

Lemma 23 Suppose that A c L1(S1,5, P) . Any one of the following conditions makes A uniformly integrable: 1) There exists an integrable g E L1 (R, 5, P ) such that

1x1 1 9 , Vx E A. 2) There mists a p > 1 such that SUPxE~ E lx(w)IP < oo.

Proof. 1): Since as X + oo SUP,EAP(~X(W)I > A) 5 f SUPXEAE~X(W)I 5 ;EI~(w)I So by the integrability of g one has that as X -, oo E lx(w)l Ilx(w)l>x 5 E ~ ( W ) I ~ , ( ~ ) ~ > ~ -+ 0, uniformly w.r.t. x E A. 2): Since SUPxE~ P(~X(W)~ > A) 5 SUPXE~ E Ix(w)I --, 0, as x 4 oo. SO

a s X 4 o o

SUPXEA E ~x(w)I IIX(W)I>X 1 S U P , ~ ~ ( E Ix(w)I~)~IP SUP,~~[P(~X(W)I > ~)](p-')/p 0. w Now we know that the uniform integrability condition is weaker than

the domination condition. Actually, it is also the neccessary condition for the L1 -convergence of the sequence of integrable random variables or, say, integrable functions.

Theorem 24 Suppose that {x,):=~ c L1(S1,5, P). Then the following two statements are equivalent:

00 1) {x,),=~ is uniformly integrable.

2) ,... E lx,( < oo; and VE > 0,36 > 0 such that VB E 5, as P(B) < 6

SUP,,^,^, ... E 1xnI IB < E.

Furthermore, i f there exists an x E L1(S1,5, P ) such that as n -+ oo, x, --, x, i n probability; then the following statement is also equivalent to 1): 3) xn + x , i n L1(S1,5,P).


Proof. Since 1) ==+ 3) is already proved in Lemma 22, we will show that 3) * 2) ==+ 1) *2).

1) ==+ 2): Take a Xo large enough such that sup,,,,,,... E lx, 1 Ilx,l>Xo < 1. Then

SUP,,^,^, ... E 1xnI I XO + 1. On the other hand, for any B E 5 since

E 1xnl IB = E 1xn1 I { ~ X , ~ > X ) ~ B + E 1xn1 I{1 x , I<A)nB -

< sup, =I,,, ... E IxnI I{lxnl>~) + XP(B) = I: + QB. Hence VE > 0 one can take a A, > 0 large enough such that I? < 5 , then let 6, = &. For this 6, > 0 one has that VB E S,P(B) < 6, j SUP,,^,^, ... E IxnI IB < E.

2) =+ 1): VE > 0 Take 6 > 0 such that 2) holds. Since P(IxnI > A) I S~~n=1,2, ... E IxnI.

Hence one can take an N large enough such that as X > N, P(lx,l > A) < 6, Vn = 1,2,. . . .

Thus by 2) as X > N, E Ixn( Ilxnl>X < E, Vn = 1,2,. . . . 3) & 2): Take an No large enough such that as n > No, EIx-x,l < 1.

Thus ,... E IxnI I maxi1 + E 1x1, E 1x11,. . . , E 1 ~ ~ ~ 1 ) < co.

On the other hand, observe that E lxn(IB < E 1% - X I + E 1x1 IB. Hence VE > 0, one can take an N, large enough such that as n > N,, EIx-xnl < f.

Then take a 6 > 0 small enough such that VB E 5, as P(B) < 6, maxn=l ,... ,N= {E lxnl IB) < E, and E 1x1 IB < ,512.

Thus as P(B) < 6, E lx,l IB < E, Vn = 1,2, - - . . rn Now let us use uniform integrability as a tool to study the martingales.

Theorem 25 If {x~):=~ is a submartingale such that { ~ , f ) z = ~ is uniformly integrable, then x, = limn,, x, exists, a.s., and

E[xwl5n] 2 X n , Vn, i.e. {x , ) ,~~ ,~ , , ,,., ,w is also a submartingale, and we call it a right-closed submartingale.

This theorem actually tells us that a uniformly integrable submartingale is a right-closed submartingale.

Proof. By uniform integrability one has that supn=o,l,2 ,... Ex; < 00.

Hence applying Theorem 20 one has that x, = limn,, x, exists, a.s. Now by the submartingale property { ~ , f ) ~ = ~ is also a submartingale (Example 3). Hence for any a > 0, and B E 5,, as m 2 n,

SB [(-a) xn]dP < SB [(-a) x,]dP. Letting m -+ co by the uniform integrability of {x,+):=~ one has that

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