Chung wrote on page 196 of his book[1]:'One wonders if the present theory of stochastic processes is not still too difficult for applications.'Advances in the theory since that time have been phenomenal,but these have been accompanied by an increase in the technical difficulty of the subject so bewildering as to give a quaint charm to Chung's use of the word 'still'.Meyer writes in the preface to his definitive account of stochastic integral theory:'...il faut...

Some Frequently Used Notation

CHAPTERⅠ.BROWNIAN MOTION

1.INTRODUCTION

　1.What is Brownian motion,and why study it

　2.Brownian motion as a martingale

　3.Brownian motion as a Gaussian process

　4.Brownian motion as a Markov process

　5.Brownian motion as a diffusion and martingale

2.BASICS ABOUT BROWNIAN MOTION

　6.Existence and uniqueness of Brownian motion

　7.Skorokhod embedding

　8.Donsker's Invariance Principle

　9.Exponential martingales and first-passage distributions

　10.Some sample-path properties

　11.Quadratic variation

　12.The strong Markov property

　13.Reflection

　14.Reflecting Brownian motion and local time

　15.Kolmogorov's test

　16.Brownian exponential martingales and the Law of the Iterated Logarithm

3.BROWNIAN MOTION IN HIGHER DIMENSIONS

　17.Some martingales for Brownian motion

　18.Recurrence and transience in higher dimensions

　19.Some applications of Brownian motion to complex analysis

　20.Windings of planar Brownian motion

　21.Multiple points,cone points,cut points

　22.Potential theory of Brownian motion in IRd(d≥3)

　23.Brownian motion and physical diffusion

4.GAUSSIAN PROCESSES AND LEVY PROCESSES

　Gaussian processes

　　24.Existence results for Gaussian processes

　　25.Continuity results

　　26.Isotropic random flows

　　27.Dynkin's Isomorphism Theorem

　Levy processes

　　28.Levy processes

　　29.Fluctuation theory and Wiener-Hopf factorisation

　　30.Local time of Levy processes

CHAPTERⅡ.SOME CLASSICAL THEORY

　1.BASIC MEASURE THEORY

　　Measurability and measure

　　　1.Measurable spaces;a-algebras;n-systems;d-systems

　　　2.Measurable functions

　　　3.Monotone-Class Theorems

　　　4.Measures;the uniqueness lemma;almost everywhere;a.e.(u,∑)

　　　5.Caratheodory's Extension Theorem

　　　6.Inner and outer u-measures;completion

　　Integration

　　　7.Definition of the integral ∫ f du

　　　8.Convergence theorems

　　　9.The Radon-Nikodym Theorem;absolute continuity;<< notation;equivalent measures

　　　10.Inequalities;and spaces(p≥1)

　　Product structures

　　　11.Product a-algebras

　　　12.Product measure;Fubini's Theorem

　　　13.Exercises

　2.BASIC PROBABILITY THEORY

　　Probability and expectation

　　　14.Probability triple;almost surely(a.s.);a.s.(P),a.s.(P,F)

　　　15.lim sup En:First Borel-Cantelli Lemma

　　　16.Law of random variable;distribution function:joint law

　　　17.Expectation:E(X;F)

　　　18.Inequalities:Markov,Jensen,Schwarz,Tchebychev

　　　19.Modes of convergence of random variables

　　Uniform integrability and L1 convergence

　　　20.Uniform integrability

　　　21.L1 convergence

　　Independence

　　　22.Independence of a-algebras and of random variables

　　　23.Existence of families of independent variables

　　　24.Exercises

3.STOCHASTIC PROCESSES

4.DISCRETE-PARAMETER MARTINGALE THEORY

5.CONTINUOUS-PARAMETER SUPERMARTINGALES

CHAPTERⅢ.MARKOV PROCESSES

1.TRANSITION FUNCTIONS AND RESOLVENTS

2.FELLER-DYNKIN PROCESSES

3.ADDITIVE FUNCTIONALS

4.APPROACH TO RAY PROCESSES:

5.RAY PROCESSES

6.APPLICATIONS

References for Volumes 1 and 2

Index to Volumes 1 and 2