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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.

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)