Discretizing the fractional Lévy area

Abstract

In this article, we give sharp bounds for the Euler discretization of the Lévy area associated to a d -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 ) . For H < 3 / 4 the exact convergence rate is n − 2 H + 1 / 2 , where n denotes the number of the discretization subintervals, while for H = 3 / 4 it is n − 1 log ( n ) and for H > 3 / 4 the exact rate is n − 1 . Moreover, we also show that a trapezoidal scheme converges (at least) with the rate n − 2 H + 1 / 2 . Finally, we derive the asymptotic error distribution of the Euler scheme. For H ≤ 3 / 4 one obtains a Gaussian limit, while for H > 3 / 4 the limit distribution is of Rosenblatt type.