Taylor expansion for the solution of a stochastic differential equation driven by fractional Brownian motions

Abstract

We study the Taylor expansion for the solution of a differential equation driven by a multi-dimensional Hölder path with exponent β> 1/2. We derive a convergence criterion that enables us to write the solution as an infinite sum of iterated integrals on a nonempty interval. We apply our deterministic results to stochastic differential equations driven by fractional Brownian motions with Hurst parameter H > 1/2. We also study the convergence in L 2 of the stochastic Taylor expansion by using L 2 estimates of iterated integrals and Borel-Cantelli type arguments.