Analysis of asymptotic mean-square stability of a class of Runge–Kutta schemes for linear systems of stochastic differential equations

In this paper the linear asymptotic mean-square stability of class of diagonally drift-implicit Runge–Kutta schemes (DDISRK) for the weak solution of systems of stochastic differential equations (SDEs) is investigated. We provide explicit structure of the stability matrices of this class of Runge–Kutta schemes for general form of linear systems of SDEs. Then we apply this analysis to several particular linear test SDE systems, that can capture the dynamics of a relatively large subclass of general linear SDE systems, to provide more detailed descriptions of stability properties of DDISRK schemes. Based on this analysis we also propose some optimal parameters that improve asymptotic mean-square stability of some SDE systems with larger drift stiffness. Some comparisons and numerical and illustrative experiments are given that confirm the theoretical discussion.