Discretizing the fractional Lévy area

In this article, we give sharp bounds for the Euler discretization of the Lévy area associated to a dd-dimensional fractional Brownian motion. We show that there are three different regimes for the exact root mean square convergence rate of the Euler scheme, depending on the Hurst parameter H∈(1/4,1)H∈(1/4,1). For H<3/4H<3/4 the exact convergence rate is n−2H+1/2n−2H+1/2, where nn denotes the number of the discretization subintervals, while for H=3/4H=3/4 it is n−1log(n) and for H>3/4H>3/4 the exact rate is n−1n−1. Moreover, we also show that a trapezoidal scheme converges (at least) with the rate n−2H+1/2n−2H+1/2. Finally, we derive the asymptotic error distribution of the Euler scheme. For H≤3/4H≤3/4 one obtains a Gaussian limit, while for H>3/4H>3/4 the limit distribution is of Rosenblatt type.