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Stochastic calculus in manifolds. With an appendix by P.A. Meyer. (English) Zbl 0697.60060
Universitext. Berlin etc.: Springer-Verlag. x, 151 p. DM 48.00 (1989).
Occasionally in mathematics someone writes a book that introduces a mildly sophisticated reader to a subject in a way that clarifies the issues, improves the notation, and reveals an elegance and beauty of the subject. This is such a book. It is not a comprehensive treatment of stochastic differential geometry (it does not treat, for example, Bismut’s proof of the index theorem), but rather it is an introduction to manifold-valued martingales and their related calculus, a subject developed primarily by L. Schwartz, P. A. Meyer, and the author himself. It explores the relationship of second order geometry to stochastic processes. The prerequisites to read the book are much more a knowledge of conventional stochastic calculus (for continuous processes) than that of differential geometry. The book is clearly written and the proofs are carefully thought out. Often a result for the theory in \({\mathbb{R}}^ n\) is recalled, and then it is explained how (or how not) the analogous theorem extends to manifolds. One small example out of many in the book is its new treatment of the non-confluence of martingales:
It is shown that if a manifold is “small enough”, a martingale can be recovered from its final value and its (entire) filtration. This contrasts with the \({\mathbb{R}}^ n\) case, where the value of a martingale at time t can of course be computed with only the knowledge at the terminal time a and the \(\sigma\)-field \({\mathcal F}_ t\) (and not the whole filtration \({\mathcal F}_ s\), \(t\leq s\leq a).\)
A special emphasis is given to Brownian motion, but continuous semimartingales are treated in detail, along with their Itô and Stratonovich calculi, as well as more special topics such as parallel transport.
A surprising feature of the book is the appendix by P. A. Meyer, entitled “A short presentation of stochastic calculus”. This is an overview of stochastic calculus giving the key ideas and concepts, with an emphasis on recent developments. Besides differential geometry applications, the author mentions (for example) predictable representations and the martingale structure equations. Semimartingales are presented as “good integrators”. This is not a pedagogic treatment of stochastic integration, but it is worth reading by anyone, novice and expert alike. However it has little to do with Emery’s book, and it may be confusing that they are published together: Meyer treats the general right continuous case (Emery needs and wants only the continuous - and simpler - case), and much of what Meyer discusses is not related to the contents of Emery’s book.
Finally I call the reader’s attention to a minor mistake in Emery’s book: the first few lines of the last paragraph on p. 88, concerning the imbedding of a manifold N in \({\mathbb{R}}^ p\) being diffeomorphic to \(N\times {\mathbb{R}}^ q(q=p-\dim N)\), is false, and the proof of Theorem 6.41 is corrected in Emery’s recent paper “On two transfer principles in stochastic differential geometry” in Sémin. Probabilités XXIV, 407- 441 (1990). Also, the remark following Theorem 6.41 on p. 87 contains an error: Y must tend to \(\{\infty \}\) in the Aleksandrov compactification.
Reviewer: P.Protter

60Hxx Stochastic analysis
58J65 Diffusion processes and stochastic analysis on manifolds
60-02 Research exposition (monographs, survey articles) pertaining to probability theory