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

Leonid E Shaikhet,Jason A Roberts

doi:10.1155/ade/2006/73897

Leonid E Shaikhet, Jason A Roberts

Open Access

https://doi.org/10.1155/ade/2006/73897

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Abstract

We consider the reliability of some numerical methods in preserving the stability properties of the linear stochastic functional differential equation , 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.

Highlights

Volterra integro-differential equations arise in the modelling of hereditary systems such as population growth, pollution, financial markets and mechanical systems
For example—will a population decline to dangerously low levels? Could a small change in the environmental conditions have drastic consequences on the long-term survival of the population? There is a growing body of works devoted to such investigations. Analytical solutions to such problems are generally unavailable and numerical methods are adopted for obtaining approximate solutions
A large number of the numerical methods are developed from existing numerical methods for systems of ordinary differential

Summary

Introduction

Volterra integro-differential equations arise in the modelling of hereditary systems (i.e., systems where the past influences the present) such as population growth, pollution, financial markets and mechanical systems (see, e.g., [1, 4]). Many real-world phenomena are subject to random noise or perturbations (e.g., freak weather conditions may adversely affect the supports of a bridge, possibly changing the long-term integrity of the structure) It is a natural extension of the deterministic work carried out by ourselves and others to consider the stability of stochastic systems and of numerical solutions to such systems. The conditions for asymptotic mean square stability are obtained here by virtue of Kolmanovskii and Shaikhet’s general method of Lyapunov functionals construction ([17,18,19,20,21,22,23, 29, 31–33]) which is applicable for both differential and difference equations, both for deterministic and stochastic systems with delay. The trivial solution of (1.3) is asymptotically mean square stable

Reliability to preserve stability properties

Stability of difference analogues to the integro-differential equation