Brownian motion and stochastic calculus / Ioannis Karatzas, Steven E. Shreve

Bibliogr. p. [447]-458. Index

Chapter 1 presents basics of martingale theory in continuous time, including stopping times, fundamental inequalities, convergence theorems, optional sampling, the Doob-Meyer decomposition, and a treatment of square-integrable martingales. Chapter 2 is heavy on Brownian motion. It starts with three constructions of Brownian motion, and ends with the Lévy modulus of continuity. In between there is a discussion of the Markov property, the strong Markov property, Brownian filtration, and some useful computations involving passage times and last exit times.
In Chapter 3, the authors present stochastic integration with respect to continuous martingales; Itô's formula, Girsanov's theorem, an indispensable result in stochastic control, and filtering theory are detailed. The remaining sections are reserved for local time and related results. Chapter 4 is about solutions of partial differential equations using Brownian motion. More specifically, such solutions are expressed in terms of certain relevant functionals of Brownian motion. Among other things a detailed treatment of the Dirichlet problem is given. Chapter 5 gives an extensive treatment of stochastic differential equations. Final sections are devoted to applications in economics. Chapter 6 begins with Lévy's approach to local time. Pertinent results of D. Williams , H. Taylor and Ray-Knight are discussed in detail.
A number of good problems in the heart of the subject matter are given in each chapter. Solutions to some problems are supplied. The notes at the end of each chapter are very important. These notes are more than comments on what the authors have done. They indicate some of the topics which have been omitted. This book is a good text for a course covering the topics reviewed above—a valuable book for every graduate student studying stochastic processes, and for those who are interested in pure and applied probability. The authors have done a good job. (MathSciNet)