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
method, of the above SDDE is studied, where the
method uses the sum of weight
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.