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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(t)=𝐃𝐗(t)dt+𝐁𝐗(t)dW(t),𝐗(0)=1,

where

𝐃=λ 1 00λ 2 ,𝐁=α 1 β 1 β 2 α 2 ,

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:
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