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Acta Applicandae

Book review

Author(s) Rozovskii, B.L.
Title Stochastic evolution systems. Linear theory and applications to non-linear filtering
Publisher Kluwer
Year of publication 1990
Reviewed by Jordan Stoyanov

When talking about "Stochastic Differential Equations" (SDE) we usually mean ordinary SDE as a well developed branch of the contemporary mathematics (K. Ito, T. Gihman, J. Doob, A. Skorohod, F.-A. Meyer, and others) with effective applications to many fields. However a new necessity arose in the middle of 1970s. The analysis of important but quite complicated phenomena (from statistical physics, quantum mechanics, biology, control theory, etc.) needed to develop the theory of stochastic partial differential equations (stochastic PDE).

Even the theory of linear stochastic POE, to which the present book is devoted, was created by the efforts of several outstanding mathematicians in all over the world and not instantaneously. Among the main contributors are N.V. Krylov, E. Pardoux, H. Kunita, R.F. Curtain, J.-M. Bismut, M. Zakai, N.U. Ahmed, G. Da Prato, J. Zabczyk and B.L. Rozovskii himself. The author's interest and necessity to study the stochastic evolution systems came from filtering theory of diffusion processes. Actually the author has done much more in the present book. He has developed with many details the theory of stochastic integration and the corresponding stochastic evolution systems in Hilbert space (Ch. 2 and Ch. 3). Due to their importance the 2nd order Ito's parabolic equations are treated in a separate chapter as is done with the Ito' PDE related to diffusion processes (Ch. 4 and Ch. 5). Stochastic PDE giving the solution to filtering and related estimation problems are considered in Ch. 6. Then we arrive at Ch. 7 which is a short but nice introduction to the Malliavin calculus and the stochastic interpretation of famous Hörmander's results on 2nd order eliptic-parabolic PDE. A representative list of references and an Index are included.

This boot is not a translation of the Russian original (1983). In particular, Ch. 7 is entirely new and written especially for this English edition. Several extensions, additions and improvements are also incorporated. Behind the great value of this systematically and well written book, we see how big is the advantage, for the reader of course, when the author takes the initiative to work additionally with the only goal to make his book corresponding to the last achievements in the field. Thus the author has deserved our thanks!

There is no doubt that many specialists will meet with an interest the book under review. Definitely each good Library has to buy the book, but I suspect that many individuals will do this, too.