Optimal approximation of SDE's with additive fractional noise

Abstract

We study pathwise approximation of scalar stochastic differential equations with additive fractional Brownian noise of Hurst parameter $H>\frac{1}{2}$, considering the mean square $L^2$-error criterion. By means of the Malliavin calculus we derive the exact rate of convergence of the Euler scheme, also for non-equidistant discretizations. Moreover, we establish a sharp lower error bound that holds for arbitrary methods, which use a fixed number of bounded linear functionals of the driving fractional Brownian motion. The Euler scheme based on a discretization, which reflects the local smoothness properties of the equation, matches this lower error bound up to the factor 1.39.