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Random perturbations of differential functional equations. (Sluchajnye vozmushcheniya differentsial’no-funktsional’nykh uravnenij.) (Russian) Zbl 0725.34092

Riga: Zinatne (ISBN 5-7966-0171-9). 421 p. (1989).
This is an interesting book on stochastic retarded functional- differential equations with bounded delay. The book contains four chapters. In Chapter 1, the basic concepts of FDE and SFDE of Itô type are introduced. The existence, uniqueness and the Markov property of solutions of the initial value problem of SFDE with bounded delay are discussed. In Chapter 2 the formulation of stability problems for SRFDEs are introduced, and various applications of the Lyapunov functional are given. Chapter 3 deals with the mean square stability of the linear SDEs with and without delay. The theory of solution operator is also discussed. In Chapter 4 the averaging method for SRFDE is presented. The book is written clearly in classical mathematical style. For deep understanding, adequate knowledge of the general theory of FDE and SDE is necessary. The bibliography contains 157 items. Even though the results on SRFDEs obtained outside the USSR have not been noted adequately in this book, the reviewer believes that this is a useful work. A translation of it into English would be welcome.

MSC:

34K50 Stochastic functional-differential equations
34F05 Ordinary differential equations and systems with randomness
34C29 Averaging method for ordinary differential equations
34K20 Stability theory of functional-differential equations
60H99 Stochastic analysis
93E15 Stochastic stability in control theory