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

TL;DR

This paper develops a Taylor expansion method for solutions of differential equations driven by fractional Brownian motions, providing convergence criteria and improved estimates for Hurst parameters greater than 1/2.

Contribution

It introduces a convergence criterion for the Taylor expansion of solutions driven by fractional Brownian motions and enhances convergence estimates for certain Hurst parameters.

Findings

Established a convergence criterion for the Taylor expansion.

Applied deterministic results to stochastic differential equations with fractional Brownian motion.

Improved convergence speed estimates using Borel-Cantelli arguments for H in (1/2, 3/4).

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).