A-m.s.-stable numerical methods for solving stochastic differential equations in the Ito-sense

In this paper we investigate the asymptotic mean-square stability (m.s.-stability) of the family of numerical methods for solving SDE's in the Ito-sense generalizing Rosenbrock's type methods. The connection between the asymptotic m.s.-stability of the numerical method for solving SDE and the absolute stability of the corresponding Rosenbrock's type method are shown. Examples of A-m.s.-stable numerical methods are given.