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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 $d{\mathbf X}(t)= {\mathbf D}{\mathbf X}(t)\,dt+{\mathbf B}{\mathbf X}(t)\,dW(t),\quad{\mathbf X}(0)= 1,$ where ${\mathbf D}= \left[\begin{matrix}\lambda_1 & 0\\ 0 &\lambda_2\end{matrix}\right],\qquad{\mathbf B}= \left[\begin{matrix} \alpha_1 &\beta_1\\ \beta_2 &\alpha_2\end{matrix}\right],$ 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.

##### MSC:
 65C30 Numerical solutions to stochastic differential and integral equations 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations 65L20 Stability and convergence of numerical methods for ordinary differential equations 60H35 Computational methods for stochastic equations (aspects of stochastic analysis)