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**Stochastic flows and stochastic differential equations.**
*(English)*
Zbl 0743.60052

Cambridge Studies in Advanced Mathematics. 24. Cambridge etc.: Cambridge University Press. xiv, 346 p. (1990).

The monograph is the first comprehensive treatment of the subject and is written by the expert in the field. Large parts are based on earlier work by the author [e.g., Ecole d’été de probabilités de Saint Flour XII-1982, Lect. Notes Math. 1097, 143-303 (1984; Zbl 0554.60066) and Lectures on stochastic flows and applications (1986; Zbl 0625.60073)].

The first two chapters (70 pages) out of six cover background material like basics about random fields and stochastic integration with respect to continuous semimartingales. Chapter 3 contains the theory of stochastic integration with respect to continuous semimartingales with spatial parameters which is fundamental for the following chapters. The central part of the book is Chapter 4 (104 pages) which covers the theory of stochastic flows on Euclidean space and on manifolds with a special treatment of Brownian flows. One key point is the relation between stochastic differential equations (with respect to spatial semimartingales) and flows. Chapter 5 contains limit theorems on stochastic flows of various kinds. Finally in Chapter 6 results of previous chapters are applied to study certain stochastic partial differential equations — appearing e.g. in nonlinear filtering.

The book is carefully written and contains all background material needed — omitting some proofs of well-known results e.g. about martingales. The monograph is an ideal reference for anyone interested in the subject who has some basic knowledge about stochastic processes.

The first two chapters (70 pages) out of six cover background material like basics about random fields and stochastic integration with respect to continuous semimartingales. Chapter 3 contains the theory of stochastic integration with respect to continuous semimartingales with spatial parameters which is fundamental for the following chapters. The central part of the book is Chapter 4 (104 pages) which covers the theory of stochastic flows on Euclidean space and on manifolds with a special treatment of Brownian flows. One key point is the relation between stochastic differential equations (with respect to spatial semimartingales) and flows. Chapter 5 contains limit theorems on stochastic flows of various kinds. Finally in Chapter 6 results of previous chapters are applied to study certain stochastic partial differential equations — appearing e.g. in nonlinear filtering.

The book is carefully written and contains all background material needed — omitting some proofs of well-known results e.g. about martingales. The monograph is an ideal reference for anyone interested in the subject who has some basic knowledge about stochastic processes.

Reviewer: M.Scheutzow (Berlin)