Revuz D., Yor M. Continuous Martingales and Brownian Motion

Third Edition. — Springer, 1999. Corr. 3rd printing 2005. — 602 p. — ISBN: 978-3642084003.
Series: Grundlehren der mathematischen Wissenschaften (Book 293).The book describes 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. The third edition contains some additional exercises related to recent advances in the subject. It is a valuable update of this basic reference book, which has been very helpful for researchers and students.Preliminaries.
Basic Notation, Monotone Class Theorem, Completion, Functions of Finite Variation and Stieltjes Integrals,
Weak Convergence in Metric Spaces, Gaussian and Other Random Variables.Examples of Stochastic Processes, Brownian Motion, Local Properties of Brownian Paths,
Canonical Processes and Gaussian Processes, Filtrations and Stopping Times.
Martingales.
Definitions, Maximal Inequalities and Applications,
Convergence and Regularization Theorems, Optional Stopping Theorem.
Markov Processes.
Basic Definitions, Feller Processes, Strong Markov Property, Summary of Results on Lévy Processes.
Stochastic Integration.
Quadratic Variations, Stochastic Integrals, Itô's Formula and First Applications,
Burkholder-Davis-Gundy Inequalities, Predictable Processes.
Representation of Martingales.
Continuous Martingales as Time-changed Brownian Motions,
Conformal Martingales and Planar Brownian Motion, Brownian Martingales, Integral Representations.
Local Times.
Definition and First Properties, The Local Time of Brownian Motion, The Three-Dimensional Bessel Process,
First Order Calculus, The Skorokhod Stopping Problem.
Generators and Time Reversal.
Infinitesimal Generators, Diffusions and Itô Processes, Linear Continuous Markov Processes,
Time Reversal and Applications.
Girsanov's Theorem and First Applications.
Girsanov's Theorem, Application of Girsanov's Theorem to the Study of Wiener's Space,
Functionals and Transformations of Diffusion Processes.
Stochastic Differential Equations.
Formal Definitions and Uniqueness, Existence and Uniqueness in the Case of Lipschitz Coefficients,
The Case of Hölder Coefficients in Dimension One.
Additive Functionals of Brownian Motion.
General Definitions, Representation Theorem for Additive Functionals of Linear Brownian Motion,
Ergodic Theorems for Additive Functionals, Asymptotic Results for the Planar Brownian Motion.
Bessel Processes and Ray-Knight Theorems.
Bessel Processes, Ray-Knight Theorems, Bessel Bridges.
Excursions.
Prerequisites on Poisson Point Processes, The Excursion Process of Brownian Motion,
Excursions Straddling a Given Time, Descriptions of Itô's Measure and Applications.
Limit Theorems in Distribution.
Convergence in Distribution, Asymptotic Behavior of Additive Functionals of Brownian Motion,
Asymptotic Properties of Planar Brownian Motion.
Appendix.
Gronwall's Lemma, Distributions, Convex Functions, Hausdorff Measures and Dimension,
Ergodic Theory, Probabilities on Function Spaces, Bessel Functions, Sturm-Liouville Equation.