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Mean-square stability of numerical schemes for stochastic differential systems. (English) Zbl 1035.65009
Criteria are derived for establishing mean-square (MS)-stability of the system of stochastic differential equations $$d{\bold X}(t)= {\bold D}{\bold X}(t)\,dt+{\bold B}{\bold X}(t)\,dW(t),\quad{\bold X}(0)= 1,$$ where $${\bold D}= \left[\matrix\lambda_1 & 0\\ 0 &\lambda_2\endmatrix\right],\qquad{\bold B}= \left[\matrix \alpha_1 &\beta_1\\ \beta_2 &\alpha_2\endmatrix\right],$$ and $W(t)$ is a Wiener process. This leads to criteria under which the Euler-Maruyama method for approximating the solution of the system will be numerically MS-stable, and to the identification of its region of MS-stability. Results of numerical experiments are presented which affirm the accuracy of the criteria.

65C30Stochastic differential and integral equations
60H10Stochastic ordinary differential equations
65L06Multistep, Runge-Kutta, and extrapolation methods
65L20Stability and convergence of numerical methods for ODE
60H35Computational methods for stochastic equations