The maximum rate of convergence for the approximation of the fractional Lévy area at a single point

In this note we study the approximation of the fractional Lévy area with Hurst parameter H>1/2H>1/2, considering the mean square error at a single point as error criterion. 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. This rate is n−2H+1/2n−2H+1/2, where nn denotes the number of evaluations of the fractional Brownian motion, and is obtained by a trapezoidal rule.