Long ago （or so it seems today）, 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…un cours de six mois sur les definitions. Que peut on y faire?' I have thought up as intuitive a picture of the subject as I can, written it down at speed, and refused to be lured back by piety （or even by wit!） to cancel half a line. 'First' intuition, which is what you need when you are learning the subject, is raw, rough and ready; and, as you have guessed, I make the excuse that it demands a compatible style and lack of polish. Note that I wrote 'first intuition'. Consider an example. Meyer's concept of a right process is exactly right for Markov process theory, but the concept is the result of a long evolution. To understand it properly, you need a highly developed intuition, and that takes time to acquire. The difficulty with the best advanced literature is that its authors have too much intuition; never make the mistake of thinking otherwise.
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目录
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 Rd 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
