High order numerical integrators for single integrand Stratonovich SDEs

We show that applying any deterministic B-series method of order pdwith a random step size to single integrand SDEs gives a numerical method converging in the mean-square and weak sense with order [Pd/2]. As an application, we derive high order energy-preserving methods for stochastic Poisson systems as well as further geometric numerical schemes for this wide class of Stratonovich SDEs