Asymptotic analysis of unstable solutions of stochastic differential equations (Q1986987)

SDEs (stochastic differential equations) is one of the main topics of modern probability theory and its applications. Usually we deal with a stochastic process, say \(X_t, \ t \geq 0\), obtained as a solution of a specific SDE and derive a series of `nice' properties of its distributions and trajectories. One of the fundamental questions of interest is: \ what is \ \(\lim_{t \to \infty}X_t.\) There are many results showing that, under appropriate conditions, \(\lim_{t \to \infty}X_t\) tends to zero, or belongs to a bounded domain in which case we say that the solution \(X_t, \ t \geq 0\) is stable. Of course, the stability property is specified in any concrete case. Thus, if the SDE is such that in one or another sense the limit \(\lim_{t \to \infty}X_t\) is unbounded, we say the solution \(X_t, t \geq 0\) is unstable. Studying unstable SDEs is not less important and not easier to deal with than studying stable SDEs. The present book is the first systematic account of most models, problems, results and ideas available in the literature. One of the co-authors, Prof. G. Kulinich was the first who started studying unstable SDEs. The topic was suggested to him by A.V. Skorokhod in 1965. The material is well structured and distributed in six chapters. The main goal is to analyze appropriate integral functionals of unstable stochastic processes related to diverse sort of SDEs and establish limit theorems. The Brownian motion and Itô type of SDEs are essentially involved. A large number of results with specified kind of convergence is presented together with their proofs and illustrative examples. The limiting objects always have a simple structure, e.g., a constant, a proper random variable or a specific `easier' stochastic process. All limits are described in detail. Thus, to start with an unstable stochastic process and transform it into something easier and tractable is a successful way of a `domestication' of unstable stochastic processes. Important is to see `domesticated' limiting objects, all easy to work with. The book ends with an appendix containing basic notions and results used intensively in the text and references of both theoretical and applied nature. There is no index. Besides their theoretical value, many of the results in this book are related to specific practical problems. Specific indications are given in the text. Some ideas and techniques exploited here can eventually be used or extended for studying other classes of unstable stochastic processes. The book will be of interest to anybody working in stochastic analysis and its applications, from master and PhD students to professional researchers. Applied scientists can also benefit from this book by seeing efficient methods to deal with unstable processes.