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

Анотація

We consider the reliability of some numerical methods in preserving the stability proper-ties of the linear stochastic functional differential equation dx(t) = (αx(t) +β ∫ t0 x(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 consid-ered, 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 pa-rameter 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 accu-rately choose a numerical method to preserve the stability properties of the original prob-lem in the numerical solution. Our numerical experiments also indicate that the quality of the sufficient conditions is very high. Copyright © 2006 L. E. Shaikhet and J. A. Roberts. This is an open access article distrib-uted under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1.

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