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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

$d𝐗\left(t\right)=𝐃𝐗\left(t\right)\phantom{\rule{0.166667em}{0ex}}dt+𝐁𝐗\left(t\right)\phantom{\rule{0.166667em}{0ex}}dW\left(t\right),\phantom{\rule{1.em}{0ex}}𝐗\left(0\right)=1,$

where

$𝐃=\left[\begin{array}{cc}{\lambda }_{1}& 0\\ 0& {\lambda }_{2}\end{array}\right],\phantom{\rule{2.em}{0ex}}𝐁=\left[\begin{array}{cc}{\alpha }_{1}& {\beta }_{1}\\ {\beta }_{2}& {\alpha }_{2}\end{array}\right],$

and $W\left(t\right)$ 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 Stochastic differential and integral equations 60H10 Stochastic ordinary differential equations 65L06 Multistep, Runge-Kutta, and extrapolation methods 65L20 Stability and convergence of numerical methods for ODE 60H35 Computational methods for stochastic equations