Numerical methods for ordinary differential systems: the initial value problem
Numerical methods for ordinary differential systems: the initial value problem
Stability Analysis of Numerical Schemes for Stochastic Differential Equations
SIAM Journal on Numerical Analysis
Scientific Computing with Ordinary Differential Equations
Scientific Computing with Ordinary Differential Equations
Mean-Square and Asymptotic Stability of the Stochastic Theta Method
SIAM Journal on Numerical Analysis
SIAM Journal on Numerical Analysis
Mathematics and Computers in Simulation
Asymptotic mean-square stability of two-step Maruyama schemes for stochastic differential equations
Journal of Computational and Applied Mathematics
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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 @q-methods applied to bi-dimensional systems is investigated.