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Abstract: In this article, we give sharp bounds for the Euler- and trapezoidaldiscretization of the Levy area associated to a d-dimensional fractionalBrownian motion. We show that there are three different regimes for the exactroot mean-square convergence rate of the Euler scheme. For H<3-4 the exactconvergence rate is n^{-2H+1-2}, where n denotes the number of thediscretization subintervals, while for H=3-4 it is n^{-1} logn^{1-2} andfor H>3-4 the exact rate is n^{-1}. Moreover, the trapezoidal scheme has exactconvergence rate n^{-2H+1-2} for H>1-2. Finally, we also derive the asymptoticerror distribution of the Euler scheme. For H lesser than 3-4 one obtains aGaussian limit, while for H>3-4 the limit distribution is of Rosenblatt type.

Author: Andreas Neuenkirch, Samy Tindel IECN, Jérémie Unterberger IECN

Source: https://arxiv.org/

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