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 multidimensional Holder path with exponent \beta> 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 prove that by using L_2 estimates of iterated integrals, the criterion and the speed of convergence for the stochastic Taylor expansion can be improved using Borel-Cantelli type arguments when H\in (1/2, 3/4).