Stochastic Stability of Differential Equations

verfasst von: Rafail Khasminskii

Verlag: Springer Berlin Heidelberg

Buchreihe : Stochastic Modelling and Applied Probability

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

Einloggen, um Zugang zu erhalten

Über dieses Buch

Since the publication of the first edition of the present volume in 1980, the stochastic stability of differential equations has become a very popular subject of research in mathematics and engineering. To date exact formulas for the Lyapunov exponent, the criteria for the moment and almost sure stability, and for the existence of stationary and periodic solutions of stochastic differential equations have been widely used in the literature. In this updated volume readers will find important new results on the moment Lyapunov exponent, stability index and some other fields, obtained after publication of the first edition, and a significantly expanded bibliography.

This volume provides a solid foundation for students in graduate courses in mathematics and its applications. It is also useful for those researchers who would like to learn more about this subject, to start their research in this area or to study the properties of concrete mechanical systems subjected to random perturbations.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Boundedness in Probability and Stability of Stochastic Processes Defined by Differential Equations

Abstract

Conditions for non-explosion, boundedness in probability and stability in probability of stochastic processes defined by the system of ODE with random coefficients are proven in this chapter.

Rafail Khasminskii

Chapter 2. Stationary and Periodic Solutions of Differential Equations

Abstract

A stochastic process ξ(t)=ξ(t,ω) (∞<t<∞) with values in ℝ^l is said to be stationary (in the strict sense) if for every finite sequence of numbers t ₁,…,t _n the joint distribution of the random variables ξ(t ₁+h),…,(t _n+h) is independent of h. If we replace the arbitrary number h by a multiple of a fixed number θ, h=kθ (k=±1,±2,…), we get the definition of a periodic stochastic process with period θ, or a θ-periodic stochastic process. Stationary and periodic stochastic processes constitute a mathematical idealization of physical noise acting on linear and nonlinear devices functioning in a medium with unvarying or periodically varying properties. Sufficient conditions for the existence of stationary and periodic solutions for ODE with random coefficients are given in this chapter.

Rafail Khasminskii

Chapter 3. Markov Processes and Stochastic Differential Equations

Abstract

Properties of the Markov processes defined by the Ito stochastic differential equations (SDE) are studied. In particular conditions for regularity (non-explosion), existence of the stationary and periodic solutions of SDE’s are given in the terms of the existence of Lyapunov functions with suitable properties.

Rafail Khasminskii

Chapter 4. Ergodic Properties of Solutions of Stochastic Equations

Abstract

A time-homogeneous Markov process is called positive recurrent if the recurrence time for any non-empty open set from any initial point has the finite expectation. The main result of this chapter is the assertion: any positive recurrent diffusion Markov process has the stationary initial distribution. Under some additional assumptions the ergodicity of this process is proven. The law of large numbers and the limiting behavior the solution of the Cauchy problem for the parabolic equation are also studied.

Rafail Khasminskii

Chapter 5. Stability of Stochastic Differential Equations

Abstract

In Chap. 1 we studied problems of stability under random perturbations of the parameters. We noted there that no significant results can be expected unless the random perturbations possess sufficiently favorable mixing properties. Fortunately, in practical applications one may often assume that the “noise” has a “short memory interval.” The natural limiting case of such noise is of course white noise. Thus it is very important to study the stability of solutions of Itô equations since this is equivalent to the study of stability of systems perturbed by white noise. Generalization of well known results on stability and instability for the deterministic ODE in terms of the Lyapunov functions are proven for SDE. Conditions for stability and instability of moments are also proven.

Rafail Khasminskii

Chapter 6. Systems of Linear Stochastic Equations

Abstract

In this chapter we shall study a linear homogeneous system of equations whose coefficients are perturbed by Gaussian white noise \(\dot{\eta}_{i}^{j}(t)\). Necessary and sufficient conditions for the stability and instability for such type of systems are proven. In particular, exact formula for the Lyapunov exponent is found for the system with constant coefficients. The stabilization problem of unstable deterministic systems by additive white noise. Is also considered.

Rafail Khasminskii

Chapter 7. Some Special Problems in the Theory of Stability of SDE’s

Abstract

Many problems concerning the stability of a nonlinear stochastic system can be reduced to problems about a linear system, obtained from the original system by dropping terms of higher than first order in x. This circumstance makes the study of stability for linear SDE’s especially important. Theorems on stability and instability in the first (linear) approximation are proven. Stability under damped random perturbations and applications to the stochastic approximation method are also given.

Rafail Khasminskii

Chapter 8. Stabilization of Controlled Stochastic Systems

Abstract

As mentioned in the preface, the stability theory of SDEs was developed mainly to meet the needs of stabilization of moving systems subjected to random perturbations. In this chapter we shall consider some problems concerning the stabilization of controlled stochastic systems. The results achieved to date in this field are rather sparse, despite the fact that the basic formulations of the problems and the fundamental equations have been known for some time. The only results of any significance are those pertaining to linear systems and employing quadratic control criteria. We devote to them the exposition which now follows, based on the material of Chaps. 5 through 7.

Rafail Khasminskii

Backmatter

Titel: Stochastic Stability of Differential Equations
verfasst von: Rafail Khasminskii
Verlag: Springer Berlin Heidelberg
Electronic ISBN: 978-3-642-23280-0
Print ISBN: 978-3-642-23279-4
DOI: https://doi.org/10.1007/978-3-642-23280-0