Since the first edition of this book (1991), the interest for Brownian motion and related stochastic processes has not abated in the least. This is probably due to the fact that Brownian motion is in the intersection of many fundamental classes of processes. It is a continuous martingale, a Gaussian process, a Markov process or more specifically a process with independent increments; it can actually be defined, up to simple transformations, as the real-valued, centered process with stationary independent increments and continuous paths.
After a first chapter where Brownian motion is introduced, each of the following ones is devoted to a new technique or notion and to some of its applications to Brownian motion. Among these techniques, two are of paramount importance: stochastic calculus, the use of which pervades the whole book, and the powerful excursion theory, both of which are introduced in a self-contained fashion and with a minimum of apparatus.
Chapter 0. Preliminaries
§1. Basic Notation
§2. Monotone Class Theorem
§3. Completion
§4. Functions of Finite Variation and Stieltjes Integrals
§5. Weak Convergence in Metric Spaces
§6. Gaussian and Other Random Variables
Chapter I. Introduction
§1. Examples of Stochastic Processes. Brownian Motion
§2. Local Properties of Brownian Paths
§3. Canonical Processes and Gaussian Processes
§4. Filtrations and Stopping Times
Notes and Comments
Chapter II. Martingales
§1. Definitions, Maximal Inequalities and Applications
§2. Convergence and Regularization Theorems
§3. Optional Stopping Theorem
Notes and Comments
Chapter III. Markov Processes
§1. Basic Definitions
§2. Feller Processes
§3. Strong Markov Property
§4. Summary of Results on Levy Processes
Notes and Comments
Chapter IV. Stochastic Integration
§1. Quadratic Variations
§2. Stochastic Integrals
§3. Ito's Formula and First Applications
§4. Burkholder-Davis-Gundy Inequalities
§5. Predictable Processes
Notes and Comments
Chapter V. Representation of Martingales
§1. Continuous Martingales as Time-changed Brownian Motions
§2. Conformal Martingales and Planar Brownian Motion
§3. Brownian Martingales
§4. Integral Representations
Notes and Comments
Chapter VI. Local Times
§1. Definition and First Properties
§2. The Local Time of Brownian Motion
§3. The Three-Dimensional Bessel Process
§4. First Order Calculus
§5. The Skorokhod Stopping Problem
Notes and Comments
Chapter VII. Generators and Time Reversal
§1. Infinitesimal Generators.
§2. Diffusions and Ito Processes
§3. Linear Continuous Markov Processes
§4. Time Reversal and Applications
Notes and Comments
Chapter VIII. Girsanov's Theorem and First Applications
§1. Girsanov's Theorem
§2. Application of Girsanov's Theorem to the Study of Wiener's Space
§3. Functionals and Transformations of Diffusion Processes
Notes and Comments
Chapter IX. Stochastic Differential Equations
§1. Formal Definitions and Uniqueness
§2. Existence and Uniqueness in the Case of Lipschitz Coefficients
§3. The Case of Holder Coefficients in Dimension One
Notes and Comments
Chapter X. Additive Functionals of Brownian Motion
§1. General Definitions
§2. Representation Theorem for Additive Functionals of Linear Brownian Motion
§3. Ergodic Theorems for Additive Functionals
§4. Asymptotic Results for the Planar Brownian Motion
Notes and Comments
Chapter XI. Bessel Processes and Ray-Knight Theorems
§1. Bessel Processes
§2. Ray-Knight Theorems
§3. Bessel Bridges
Notes and Comments
Chapter XII. Excursions
§1. Prerequisites on Poisson Point Processes
§2. The Excursion Process of Brownian Motion
§3. Excursions Straddling a Given Time
§4. Descriptions of Ito's Measure and Applications
Notes and Comments
Chapter XIII. Limit Theorems in Distribution
§1. Convergence in Distribution
§2. Asymptotic Behavior of Additive Functionals of Brownian Motion
§3. Asymptotic Properties of Planar Brownian Motion
Notes and Comments
Appendix
§1. Gronwall's Lemma
§2. Distributions
§3. Convex Functions
§4. Hausdorff Measures and Dimension
§5. Ergodic Theory
§6. Probabilities on Function Spaces
§7. Bessel Functions
§8. Sturm-Liouville Equation
Bibliography
Index of Notation
Index of Terms
Catalogue
