Stochastic differential equations. An introduction with applications (Q1059928)

This book is an excellent elementary introduction. The author has completely succeeded to follow his program mentioned in the preface. ''These notes are an attempt to approach the subject from the nonexpert point of view: Not knowing anything about a subject to start with, what would I like to know first of all? My answer would be: 1) In what situations does the subject arise? 2) What are its essential features? 3) What are the applications and the connections to other fields? I would not be so interested in the proof of the most general case, but rather in an easier proof of a special case, which may give just as much of the basic idea in the argument. And I would be willing to believe some basic results without proof in order to have time for some more basic applications.'' The book starts with some concrete examples where stochastic differential equations arise. After developing the necessary machinery the examples are taken up again and solved. The prerequisites to read the book is some probability theory, although some topics, e.g. conditional expectations are reviewed in the appendix, but on the other hand the reader has to know what a Markov process is, at least on an intuitive level. The presentation is always broad enough to make it easy to follow but nevertheless rigor. Stochastic integrals are defined for Wiener processes and not in an abstract martingale setting. Since many applications are presented on this level the book really fills a gap in the existing literature. In order to indicate which aspects of the theory and applications are covered we quote the main headlines: Stochastic integrals (Ito and Stratonovich) and the Ito formula, stochastic differential equations - existence and uniqueness theorems; the filtering problem (innovation process, Kalman-Bucy-filter), Ito diffusion processes, Dynkin formula, Feynman-Kac formula, random time change, the Cameron-Martin-Girsanov transformation, application to partial differential equations, the Dirichlet problem, the Poisson problem, application to optimal stopping, applications to stochastic control, the Hamilton-Jacobi-Bellman equation.