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The Strong Convergence and Numerical Stability of Multistep Approximations of Solutions of Stochastic Ordinary Differential Equations

Abstract Stochastic ordinary differential equations (SODEs) model physical phenomena driven by stochastic processes. As for deterministic differential equations, various numerical schemes have been proposed for SODEs. One of the key statements says that stability and consistency imply convergence of approximations to the exact solution for well-posed initial value problems. In this work, we state and prove, correspondingly, generalized stochastic versions of the forementioned approximation principle for multistep methods. Theorems on numerical stability in the mean-square sense are proven, too.