Advances in Difference Equations
Volume 2006 (2006), Article ID 73897, 22 pages
doi:10.1155/ADE/2006/73897

Reliability of difference analogues to preserve stability properties of stochastic Volterra integro-differential equations

Leonid E. Shaikhet1 and Jason A. Roberts2

1Department of Higher Mathematics, Donetsk State University of Management, Donetsk 83015, Ukraine
2Mathematics Department, University of Chester, Chester CH14BJ, United Kingdom

Received 2 August 2004; Revised 16 January 2005; Accepted 10 April 2005

Copyright © 2006 Leonid E. Shaikhet and Jason A. Roberts. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We consider the reliability of some numerical methods in preserving the stability properties of the linear stochastic functional differential equation dx(t)=(αx(t)+β∫0tx(s)ds)dt+σx(t−τ)dW(t), where α,β,σ,τ≥0 are real constants, and W(t) is a standard Wiener process. The areas of the regions of asymptotic stability for the class of methods considered, indicated by the sufficient conditions for the discrete system, are shown to be equal in size to each other and we show that an upper bound can be put on the time-step parameter for the numerical method for which the system is asymptotically mean-square stable. We illustrate our results by means of numerical experiments and various stability diagrams. We examine the extent to which the continuous system can tolerate stochastic perturbations before losing its stability properties and we illustrate how one may accurately choose a numerical method to preserve the stability properties of the original problem in the numerical solution. Our numerical experiments also indicate that the quality of the sufficient conditions is very high.