Mean-square stability analysis of numerical schemes for stochastic differential systems

For ordinary differential systems, the study of A-stability for a numerical method reduces to the scalar case by means of a transformation that uncouples the linear test system as well as the difference system provided by the method. For stochastic differential equations (SDEs), mean-square stability (MS-stability) has been successfully proposed as the generalization of A-stability, and numerical MS-stability has been analyzed for one-dimensional equations. However, unlike the deterministic case, the extension of this analysis to multi-dimensional systems is not straightforward. In this paper we give necessary and sufficient conditions for the MS-stability of multi-dimensional systems with one Wiener noise. The criterion presented does not depend on any norm. Based on the Routh–Hurwitz theorem, we offer a particular criterion of MS-stability for two-dimensional systems in terms of their coefficients. In addition, a counterpart criterion of MS-stability is given for numerical schemes applied to multi-dimensional systems. The MS-stability behavior of a stochastic numerical method is determined by the comparison of its stability region with the stability region of the system. As an application, the numerical MS-stability of θ$\theta$-methods applied to bi-dimensional systems is investigated.