Lectures on stochastic differential equations and Malliavin calculus. (English) Zbl 0546.60054

Watanabe, S.

Lectures on Mathematics and Physics. Mathematics, 73. Tata Institute of Fundamental Research. Berlin etc.: Springer-Verlag. V, 118 p. DM 20.00; $ 7.30 (1984).

This book consists of lecture notes of a course given by the author in 1983 at the Indian Institute of Science in Bangalore. It has two main chapters. The first one develops the Malliavin calculus following the lines of I. Shigekawa [J. Math. Kyoto Univ. 20, 263-289 (1980; Zbl 0476.28008)] and P. A. Meyer [Séminaire de probabilités XVI, Lect. Notes Math. 920, 95-132 (1982; Zbl 0481.60041)].
The subsections of this chapters are: Abstract Wiener space, Ornstein- Uhlenbeck operators and semigroups, Sobolev spaces over the Wiener space, Composites of Wiener functionals and Schwartz distributions, The smoothness of probability laws.
The second chapter applies this theory to stochastic differential equations. It deals with solutions of stochastic differential equations as Wiener functionals, Existence of moments for a class of Wiener functionals, Regularity of transition probabilities.
There are plenty notes of reference on related treatments of the subject.

Reviewer: M.Breger

MSC:

60Hxx	Stochastic analysis
60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)
60-02	Research exposition (monographs, survey articles) pertaining to probability theory
60J60	Diffusion processes

Keywords:

Malliavin calculus; Ornstein-Uhlenbeck operators and semigroups; Wiener functionals

Citations:

Zbl 0476.28008; Zbl 0481.60041