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Delay-dependent stability analysis of numerical methods for stochastic delay differential equations. (English) Zbl 1246.65013
The mean square asymptotic stability condition in terms of the coefficients of the linear stochastic delay differential equation (SDDE) has been given by J. A. D. Appleby, X. Mao and M. Riedle [Proc. Am. Math. Soc. 137, No. 1, 339–348 (2009; Zbl 1156.60045)], of which the deterministic part involves no delay. In the present paper, the mean square asymptotic stability of the numerical algorithm of a discretization approximation, so called $\theta$ method, of the above SDDE is studied, where the $\theta$ method uses the sum of weight $\theta$ and weight $1-\theta$ of the forward and backward discretization approximation respectively for the deterministic integral. The asymptotic stability condition is then obtained by analyzing the characteristic equation of the difference equation of the mean square sequence, It is proven that the backward Euler method preserves this property, while the Euler-Maruyama method preserves the instability property. Examples are illustrated.
##### MSC:
 65C30 Stochastic differential and integral equations 60H35 Computational methods for stochastic equations 60H10 Stochastic ordinary differential equations 34K50 Stochastic functional-differential equations 34K06 Linear functional-differential equations 65L20 Stability and convergence of numerical methods for ODE 65L12 Finite difference methods for ODE (numerical methods)