Discretizing the fractional Levy area

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In this article, we give sharp bounds for the Euler- and trapezoidal discretization of the Levy 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. For H<3/4 the exact convergence rate is n⁻²ʰ⁺¹/², where n denotes the number of the discretization subintervals, while for H=3/4 it is n⁻¹ (log(n))¹/² and for H>3/4 the exact rate is n⁻¹. Moreover, the trapezoidal scheme has exact convergence rate n⁻²ʰ⁺¹/² for H>1/2. Finally, we also derive the asymptotic error distribution of the Euler scheme. For H lesser than 3/4 one obtains a Gaussian limit, while for H>3/4 the limit distribution is of Rosenblatt type.