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Book review

Author(s) Bichteler, K.
Gravereaux, J.-B.
Jacod, J.
Title Malliavin calculus for processes with jumps
Publisher Gordon and Breach
Year of publication 1987
Reviewed by Günter Vojta

The Malliavin calculus, named after the French mathematician Paul Malliavin (1978), is a new method in stochastic analysis particularly of Wiener processes (Brownian motion) based on a differential calculus on Wiener space. It represents a probabilistic technique for deriving deep old and new results for stochastic partial differential equations, e.g. on semigroups of transition probabilities. This calculus is rather intricate, and its genesis and real nature have only been understood recently.

This book gives a short introduction into the calculus and its extension to discontinuous processes, i.e. processes with jumps, e.g. random walks on Poisson spaces. It contains several new results on Poisson driven stochastic differential equations the emphasis being, however, on methods. The material is divided into four chapters. First the calculus is introduced and important results are given. There the techniques are discussed including problems of the stability for stochastic differential equations. Then follows a chapter on the approach of J.M. Bismut (1984) where the stochastic calculus of variations represents a basic topic. Finally, Malliavin's approach itself is extended to Wiener-Poisson space, Malliavin operators playing a central role. A list of notations, a short index, and a list of 29 references conclude the work.

The volume is a highly specialized research monograph. Unfortunately, the text is reproduced from a typescript where the often complicated formulae are sometimes not well legible. The monograph is vividly written and yields interesting and valuable information for experts in the theory of stochastic processes in mathematics and theoretical physics.