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Stochastic evolution systems. Linear theory and applications to non- linear filtering. Transl. from the Russian by A. Yarkho. (English) Zbl 0724.60070
Mathematics and Its Applications (Soviet Series), 35. Dordrecht etc.: Kluwer Academic Publishers. xviii, 315 p. Dfl. 210.00; $ 129.00; £73.00 (1990).
The main difference between the original Russian version (1983; Zbl 0525.60063) and the English translation (apart from the language) is the addition of Chapter 7 on hypoellipticity of Itô’s second order parabolic equations. After a thorough introduction to the Malliavin calculus, the results are applied to Itô’s parabolic equations and to the filtering equation in particular. It is shown that under an appropriate Hörmander type condition the filtering equation has an a.s. smooth solution.
In spite of the considerable number of typographical errors the monograph provides a highly competent and readable treatment of the subject.

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60-02 Research exposition (monographs, survey articles) pertaining to probability theory
60G35 Signal detection and filtering (aspects of stochastic processes)
60H07 Stochastic calculus of variations and the Malliavin calculus
93E11 Filtering in stochastic control theory