Stochastic Volterra integro-differential equations: stability and numerical methods

We consider the reliability of some numerical methods in preserving the stability properties of the linear stochastic functional differential equation ẋ(t) = αx(t) + β ∫ t 0 x(s)ds+ σx(t− τ )Ẇ (t), where α, β, σ, τ ≥ 0 are real constants, and W (t) is a standard Wiener process. We adopt the shorthand notation of ẋ(t) to represent the differential dx(t) etc. Our choice of test equation is a stochastic perturbation of the classical deterministic Brunner & Lambert test equation for σ = 0 and so our investigation may be thought of as an extension of their work. Sufficient conditions for the asymptotic mean square stability of solutions to both the differential equation and discrete analogues are derived using the general method of Lyapunov functionals construction proposed by Kolmanovskii & Shaikhet which has previously been successfully employed for deterministic and stochastic differential and difference equations with delay. The areas of the regions of asymptotic stability for each θ method, indicated by the sufficient conditions for the discrete system, are shown to be equal and we show that an upper bound can be put on the time-step parameter for the numerical method fo 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. Donetsk State Academy of Management, Ukraine, [email protected], collaboration supported by NATO, grant ref. PST.EV.979727 Mathematics Department, University College Chester, England, [email protected] 1