Two-Scale Stochastic Systems: Asymptotic Analysis and Control

by Yuri Kabanov, Sergei Pergamenshchikov

Overview

Two-scale systems described by singularly perturbed SDEs have been the subject of ample literature. However, this new monograph develops subjects that were rarely addressed and could be given the collective description "Shastic Tikhonov-Levinson theory and its applications." The book provides a mathematical apparatus designed to analyze the dynamic behaviour of a randomly perturbed system with fast and slow variables. In contrast to the deterministic Tikhonov-Levinson theory, the basic model is described in a more realistic way by shastic differential equations. This leads to a number of new theoretical questions but simultaneously allows us to treat in a unified way a surprisingly wide spectrum of applications like fast modulations, approximate filtering, and shastic approximation.Two-scale systems described by singularly perturbed SDEs have been the subject of ample literature. However, this new monograph develops subjects that were rarely addressed and could be given the collective description "Shastic Tikhonov-Levinson theory and its applications." The book provides a mathematical apparatus designed to analyze the dynamic behaviour of a randomly perturbed system with fast and slow variables. In contrast to the deterministic Tikhonov-Levinson theory, the basic model is described in a more realistic way by shastic differential equations. This leads to a number of new theoretical questions but simultaneously allows us to treat in a unified way a surprisingly wide spectrum of applications like fast modulations, approximate filtering, and shastic approximation.

Product Details

ISBN-13:	9783642084676
Publisher:	Springer Berlin Heidelberg
Publication date:	12/06/2010
Series:	Stochastic Modelling and Applied Probability , #49
Edition description:	Softcover reprint of hardcover 1st ed. 2003
Pages:	266
Product dimensions:	6.10(w) x 9.25(h) x 0.02(d)

Table of Contents

0 Warm-up.- 1 Toolbox: Moment Bounds for Solutions of Stable SDEs.- 2 The Tikhonov Theory for SDEs.- 3 Large Deviations.- 4 Uniform Expansions for Two-Scale Systems.- 5 Two-Scale Optimal Control Problems.- 6 Applications.- A.1 Basic Facts About SDEs.- A.1.1 Existence and Uniqueness of Strong Solutions for SDEs with Random Coefficients.- A.1.2 Existence and Uniqueness with a Lyapunov Function.- A.1.3 Moment Bounds for Linear SDEs.- A.1.4 The Novikov Condition.- A.2 Exponential Bounds for Fundamental Matrices.- A.2.1 Uniform Bound in the Time-Homogeneous Case.- A.2.2 Nonhomogeneous Case.- A.2.3 Models with Singular Perturbations.- A.3 Total Variation Distance and Hellinger Processes.- A.3.1 Total Variation Distance and Hellinger Integrals.- A.3.2 The Hellinger Processes.- A.3.3 Example: Diffusion-Type Processes.- A.4 Hausdorff Metric.- A.5 Measurable Selection.- A.5.1 Aumann Theorem.- A.5.2 Filippov Implicit Function Lemma.- A.5.3 Measurable Version of the Carathéodory Theorem.- A.6.1 Notations and Preliminaries.- A.6.2 Integration of Shastic Kernels.- A.6.3 Distributions of Integrals.- A.6.4 Compactness of the Limit of Attainability Sets.- A.6.5 Supports of Conditional Distributions.- A.7 The Komlós Theorem.- Historical Notes.- References.

Editorial Reviews

From the reviews:

"The book presents Tikhonov theorems for shastic systems with two time scales when a parameter decreases to zero. … The topic of the book is relevant for the research areas of shastics and of control and system theory. … The strength of the book is in the analysis of Tikhonov theorems … . The text is very well structured, runs very smooth, and the exposition allows detailed scrutiny of all proofs. The book is expected to become a basic reference on two-scale shastic systems." (J. H. van Schuppen, Nieuw Archief voor Wiskunde, Vol. 6 (2), 2005)

"This research monograph needs to be placed on your shelves … . The monograph is organized by seven chapters and a valuable appendix. … The book is written for a graduated mathematically oriented readership who is supposed to be familiar with basic concepts of shastic differential equations and related issues." (Henri Schurz, Zentralblatt MATH, Vol. 1033 (8), 2004)

"This very well-crafted monograph examines the coupled shastic and singularly perturbed shastic dynamics … . the monograph displays in a very attractive manner the general scene of singularly perturbed shastic differential equations. Telling examples are provided along with enlightening explanations, comparisons and detailed background material needed to grasp the theory. … To sum up, a mixture of modern techniques concerning singularly perturbed shastic differential equations, along with classical arguments, is presented in a rigorous, clear and attractive manner." (Zvi Artstein, Mathematical Reviews, 2004 c)

"Two time-scale problems arise in many problems of practical interest. The essential ingredient in these problems is a decomposition of the dynamics into ‘fast’ and ‘slow’ modes. … This book deals, inter alia with ‘averaging methods’ where the coefficients of the limiting slow dynamics are obtained by averaging the fast dynamics. Indeed, the idea of‘averaging’ has found wide-spread application in related problems. The book is uncompromisingly mathematical but clearly and elegantly written." (G. C. Goodwin, Short Book Reviews, Vol. 23 (3), 2003)

From the Publisher