Stochastically stable one-step approximations of solutions of stochastic ordinary differential equations

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A series of numerical methods (schemes) for solving stochastic ordinary differential equations (SODEs) are given. Under appropriate conditions the numerical solutions converge in mean square sense to the true solution. There is a general belief, supported by theorems for the Euler–Maruyama and Milstein methods that the above statement implies the stochastic stability, i.e., the continuous dependence on initial values of the numerical solutions. In this paper this assertion is rigorously proved for methods widely used, but not investigated from this aspect.