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. In previous work, we considered Maruyama-type schemes, where only the increments of the driving Wiener process are used to discretize the diffusion part. Here, we suggest the improvement of the discretization of the diffusion part by also taking into account mixed classical-stochastic integrals. We show that the relation of the applied step sizes to the smallness of the noise is essential in deciding whether the new methods are worthwhile. Simulation results illustrate the theoretical findings.