Multistep methods for SDEs and their application to problems with small noise

In this article the numerical approximation of solutions of Itô stochastic differential equations is considered, in particular for equations with a small parameter $\epsilon$ in the noise coefficient. We construct stochastic linear multistep methods and develop the fundamental numerical analysis concerning their mean-square consistency, numerical stability in the mean-square sense and mean-square convergence. For the special case of two-step Maruyama schemes we derive conditions guaranteeing their mean-square consistency. Further, for the small noise case we obtain expansions of the local error in terms of the step size and the small parameter $\epsilon$. Simulation results using several explicit and implicit stochastic linear $k$-step schemes, $k=1,\;2$, illustrate the theoretical findings.