Continuous Martingales and Brownian Motion

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Springer Science & Business Media, Mar 9, 2013 - Mathematics - 602 pages
From the reviews: "This is a magnificent book! Its purpose is to describe in considerable detail a variety of techniques used by probabilists in the investigation of problems concerning Brownian motion. The great strength of Revuz and Yor is the enormous variety of calculations carried out both in the main text and also (by implication) in the exercises. ... This is THE book for a capable graduate student starting out on research in probability: the effect of working through it is as if the authors are sitting beside one, enthusiastically explaining the theory, presenting further developments as exercises, and throwing out challenging remarks about areas awaiting further research..."
Bull.L.M.S. 24, 4 (1992) Since the first edition in 1991, an impressive variety of advances has been made in relation to the material of this book, and these are reflected in the successive editions.
 

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Contents

Preliminaries
1
2 Representation Theorem for Additive Functionals
2
Introduction
15
3 Canonical Processes and Gaussian Processes
33
Notes and Comments
48
3 Optional Stopping Theorem
68
2 Feller Processes
88
3 Strong Markov Property
102
Notes and Comments
362
2 Existence and Uniqueness in the Case of Lipschitz Coefficients
375
3 The Case of Hlder Coefficients in Dimension One
388
Additive Functionals of Brownian Motion
401
3 Ergodic Theorems for Additive Functionals
422
4 Asymptotic Results for the Planar Brownian Motion
430
Notes and Comments
436
2 RayKnight Theorems
454

Notes and Comments
117
2 Stochastic Integrals
137
3 Its Formula and First Applications
146
4 BurkholderDavisGundy Inequalities
160
Notes and Comments
176
2 Conformal Martingales and Planar Brownian Motion
189
4 Integral Representations
209
2 The Local Time of Brownian Motion
239
4 First Order Calculus
260
Notes and Comments
277
2 Diffusions and It Processes
294
4 Time Reversal and Applications 3 13
321
2 Application of Girsanovs Theorem to the Study of Wieners Space
338
3 Bessel Bridges
463
Notes and Comments
469
2 The Excursion Process of Brownian Motion
480
3 Excursions Straddling a Given Time
488
Notes and Comments
511
2 Asymptotic Behavior of Additive Functionals of Brownian Motion
522
3 Asymptotic Properties of Planar Brownian Motion
531
Notes and Comments
541
4 Hausdorff Measures and Dimension
547
Index of Notation
595
Catalogue
605
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