Input-to-State Stability of Linear Stochastic Functional Differential Equations

Ramazan Kadiev,Arcady Ponosov

doi:10.1155/2016/8901563

Abstract

The purpose of the paper is to show how asymptotic properties, first of all stochastic Lyapunov stability, of linear stochastic functional differential equations can be studied via the property of solvability of the equation in certain pairs of spaces of stochastic processes, the property which we call input-to-state stability with respect to these spaces. Input-to-state stability and hence the desired asymptotic properties can be effectively verified by means of a special regularization, also known as “theW-method” in the literature. How this framework provides verifiable conditions of different kinds of stochastic stability is shown.

Highlights

This review paper is aimed at describing a general framework for analysis of asymptotic properties of linear stochastic functional differential equation driven by a semimartingale
In the present paper we considered the property of p-stability (1 ≤ p < ∞) for various linear stochastic equations
We showed how to obtain efficient stability results formulated in terms of the parameters of the equations in question

Summary

Introduction

This review paper is aimed at describing a general framework for analysis of asymptotic properties of linear stochastic functional differential equation driven by a semimartingale. The core idea of the method is an alternative description of asymptotic properties in terms of solvability of the equation in certain pairs of spaces of stochastic processes on the semiaxis. (a) Systems of linear ordinary (i.e., nondelay) stochastic differential equations driven by an arbitrary semimartingale (in particular, systems of ordinary Itoequations). (b) Systems of linear stochastic differential equations with discrete delays driven by a semimartingale (in particular, systems of Itoequations with discrete delays). (c) Systems of linear stochastic differential equations with distributed delays driven by a semimartingale (in particular, systems of Itoequations with distributed delays). If λ(t) is nonrandom (or equivalently if Ω contains only one point), we obtain the deterministic versions of all the above classes of equations

Input-to-State Stability

ISS with respect to Weighted Spaces and Bohl-Perron Type Theorems

Lyapunov Stability of Linear Hereditary Itô Equations

Itô Type Linear Functional Difference Equations

Conclusion