Stochastic integrals. Reprint of the 1969 edition, with errata. (English) Zbl 1101.60002

McKean, Henry P.

Providence, RI: AMS Chelsea Publishing (ISBN 0-8218-3887-3/hbk). xiii, 141 p. (2005).

The AMS is excited to bring this volume, originally published in 1969 (Zbl 0191.46603), back into print. This well-written book has been used for many years to learn about stochastic integrals. The author starts with the presentation of Brownian motion, then deals with stochastic integrals and differentials, including the famous Itô’s lemma. The rest of the book is devoted to various topics of stochastic integral equations and stochastic integral equations on smooth manifolds.
E. B. Dynkin wrote about the original edition in Mathematical Reviews: “This little book is a brilliant introduction to an important boundary field between the theory of probability and differential equations.” These words continue to ring true today. This classic book is ideal for supplementary reading or independent study. It is suitable for graduate students and researchers interested in probability, stochastic processes, and their applications. (Publisher’s description)
The book is stucturized as follows. Chapter 1 is devoted to the theory of Brownian motion. Chapter 2 deals with stochastic integrals and differentials, Itô’s change-of-variable formula, and random time-change in stochastic integrals. Chapter 3 deals with stochastic differential equations. Feller’s test for explosion, relationships between measures corresponding to the solution of stochastic differential equations and Cameron-Martin’s formula, Brownian local time, Skorokhod’s stochastic integral equation for Brownian motion on the half-line with the reflecting barrier, as well as examples of singular equations are treated. In Chapter 4 stochastic differential equations on smooth manifolds are investigated. Khasminskij’s test for explosions and relationships between explosions are considered. In the last part Brownian motion on the Lie group is constructed starting from Brownian motion on the Lie algebra and by using Malyutov-Dynkin results about Brownian motion with obigue reflection. The book contains many examples and applications.

Reviewer: Pavel Gapeev (Berlin)

MSC:

60-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to probability theory
60H05	Stochastic integrals
60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)
60J60	Diffusion processes
60J65	Brownian motion

Keywords:

Brownian motion; stochastic integral equation; Itô’s formula; stochastic integral equations on smooth manifolds; Brownian motion on the Lie algebra

Citations:

Zbl 0191.46603