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We show that applying any deterministic B-series method of order $p_d$ with arandom step size to single integrand SDEs gives a numerical method convergingin the mean-square and weak sense with order $\lfloor p_d/2\rfloor$.As anapplication, we derive high order energy-preserving methods for stochasticPoisson systems as well as further geometric numerical schemes for this wideclass of Stratonovich SDEs.

High order numerical integrators for single integrand Stratonovich SDEs