Exponential stability of a kind of stochastic delay difference equations

Xiaohua Ding

doi:10.1155/ddns/2006/94656

Abstract

We present a Razumilchin-type theorem for stochastic delay difference equation, and use it to investigate the mean square exponential stability of a kind of nonautonomous stochastic difference equation which may also be viewed as an approximation of a nonautonomous stochastic delay integrodifferential equations (SDIDEs), and of a difference equation arises from some of the earliest mathematical models of the macroeconomic trade cycle with the environmental noise.

Highlights

The problem of stability of stochastic difference equation has been investigated in a number of papers
We present a Razumilchin-type theorem for stochastic delay difference equation, and use it to investigate the mean square exponential stability of a kind of nonautonomous stochastic difference equation which may be viewed as an approximation of a nonautonomous stochastic delay integrodifferential equations (SDIDEs), and of a difference equation arises from some of the earliest mathematical models of the macroeconomic “trade cycle” with the environmental noise
Is satisfied, the inequality (3.6) holds, with γ = min{log(1 + λ),log q/4}

Summary

Introduction

The problem of stability of stochastic difference equation has been investigated in a number of papers. Very few results on the Razumilchin-type theorem for stochastic delay difference equation have been published. We present a Razumilchin-type theorem for stochastic delay difference equation, and use it to investigate the mean square exponential stability of a kind of nonautonomous stochastic difference equation. −1, 0}) is the initial segment to be Ᏺ0-measurable. Μn are independent N(0,1)-distributed Gaussian random variables. Among all the sequences {Xn}n∈N of the random variables, we distinguish those for which Xn are Ᏺn-measurable for all n ∈ N.

Main result

Application

Examples