This book is devoted to unstable solutions of stochastic differential equations (SDEs). Despite the huge interest in the theory of SDEs, this book is the first to present a systematic study of the instability and asymptotic behavior of the corresponding unstable stochastic systems. The limit theorems contained in the book are not merely of purely mathematical value; rather, they also have practical value. Instability or violations of stability are noted in many phenomena, and the authors attempt to apply mathematical and stochastic methods to deal with them. The main goals include exploration of Brownian motion in environments with anomalies and study of the motion of the Brownian particle in layered media. A fairly wide class of continuous Markov processes is obtained in the limit. It includes Markov processes with discontinuous transition densities, processes that are not solutions of any Itô's SDEs, and the Bessel diffusion process. The book is self-contained, with presentation of definitions and auxiliary results in an Appendix. It will be of value for specialists in stochastic analysis and SDEs, as well as for researchers in other fields who deal with unstable systems and practitioners who apply stochastic models to describe phenomena of instability.

Autorentext

Prof. Grigorij Kulinich received his PhD in probability and statistics from Kyiv University in 1968 and completed his postdoctoral degree in probability and statistics (Habilitation) in 1981. His research work focuses mainly on asymptotic problems of stochastic differential equations with nonregular dependence on parameter, theory of stochastic differential equations, and theory of stochastic processes. He is the author of more than 150 published papers.

Prof. Yuliya Mishura received her PhD in probability and statistics from Kyiv University in 1978 and completed her postdoctoral degree in probability and statistics (Habilitation) in 1990. She is currently a professor at Taras Shevchenko National University of Kyiv. She is the author/coauthor of more than 270 research papers and 9 books. Her research interests include theory and statistics of stochastic processes, stochastic differential equations, fractional processes, stochastic analysis, and financial mathematics.

Dr. Svitlana Kushnirenko is an Associate Professor in the Department of General Mathematics, Taras Shevchenko National University of Kyiv, where she also completed her PhD in probability and statistics in 2006. Her research interests include theory of stochastic differential equations and stochastic analysis. She is the author of 20 papers.

Inhalt

Introduction to Unstable Processes and Their Asymptotic Behavior.- Convergence of Unstable Solutions of SDEs to Homogeneous Markov Processes with Discontinuous Transition Density.- Asymptotic Analysis of Equations with Ergodic and Stochastically Unstable Solutions.- Asymptotic Behavior of Integral Functionals of Stochastically Unstable Solutions.- Asymptotic Behavior of Homogeneous Additive Functionals Defined on the Solutions of Itô SDEs with Non-regular Dependence on a Parameter.- Asymptotic Behavior of Homogeneous Additive Functionals of the Solutions to Inhomogeneous Itô SDEs with Non-regular Dependence on a Parameter.- A Selected Facts and Auxiliary Results.- References.