Improved linear multi-step methods for stochastic ordinary differential equations

We consider linear multi-step methods for stochastic ordinary differential equations and study their convergence properties for problems with small noise or additive noise. We present schemes where the drift part is approximated by well-known methods for deterministic ordinary differential equations. Previously, we considered Maruyama-type schemes, where only the increments of the driving Wiener process are used to discretize the diffusion part. Here, we suggest to improve the discretization of the diffusion part by taking into account also mixed classical-stochastic integrals. We show that the relation of the applied step-sizes to the smallness of the noise is essential to decide whether the new methods are worth to be used. Simulation results illustrate the theoretical findings.