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We present a technique, based on so-called word series, to write down in asystematic way expansions of the strong and weak local errors of splittingalgorithms for the integration of Stratonovich stochastic differentialequations. Those expansions immediately lead to the corresponding orderconditions. Word series are similar to, but simpler than, the B-series used toanalyze Runge-Kutta and other one-step integrators. The suggested approachmakes it unnecessary to use the Baker-Campbell-Hausdorff formula. As anapplication, we compare two splitting algorithms recently considered byLeimkuhler and Matthews to integrate the Langevin equations. The word seriesmethod bears out clearly reasons for the advantages of one algorithm over theother.

A technique for studying strong and weak local errors of splitting stochastic integrators