Flots et series de Taylor stochastiques

Summary We study the expansion of the solution of a stochastic differential equation as an (infinite) sum of iterated stochastic (Stratonovitch) integrals. This enables us to give a universal and explicit formula for any invariant diffusion on a Lie group in terms of Lie brackets, as well as a universal and explicit formula for the brownian motion on a Riemannian manifold in terms of derivatives of the curvature tensor. The first of these formulae contains, and extends to the non nilpotent case, the results of Doss 6], Sussmann 17], Yamato 18], Fliess and Normand-Cyrot 7], Krener and Lobry 19] and Kunita 11] on the representation of solutions of stochastic differential equations.