擴散 馬爾可夫過程和鞅 第1卷

內(nèi)容概要

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.

書籍目錄

Some Frequently Used NotationCHAPTERⅠ. BROWNIAN MOTION1. 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 martingale2. 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 Logarithm3. 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 diffusion4. 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 processesCHAPTERⅡ. 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;

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