Optimal pointwise approximation of stochastic differential equations driven by fractional Brownian motion

Abstract

We study the approximation of stochastic differential equations driven by a fractional Brownian motion with Hurst parameter H > 1 / 2 . For the mean-square error at a single point we derive the optimal rate of convergence that can be achieved by arbitrary approximation methods that are based on an equidistant discretization of the driving fractional Brownian motion. We find that there are mainly two cases: either the solution can be approximated perfectly or the best possible rate of convergence is n − H − 1 / 2 , where n denotes the number of evaluations of the fractional Brownian motion. In addition, we present an implementable approximation scheme that obtains the optimal rate of convergence in the latter case.