Asymptotic expansion of stochastic flows

Summary We study the asymptotic expansion in small time of the solution of a stochastic differential equation. We obtain a universal and explicit formula in terms of Lie brackets and iterated stochastic Stratonovich integrals. This formula contains the results of Doss 6], Sussmann 15], Fliess and Normand-Cyrot 7], Krener and Lobry 10], Yamato 17] and Kunita 11] in the nilpotent case, and extends to general diffusions the representation given by Ben Arous 3] for invariant diffusions on a Lie group. The main tool is an asymptotic expansion for deterministic ordinary differential equations, given by Strichartz 14].