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Discretizing the fractional Lévy area

2009; Elsevier BV; Volume: 120; Issue: 2 Linguagem: Inglês

10.1016/j.spa.2009.10.007

1879-209X

Andreas Neuenkirch, Samy Tindel, Jérémie Unterberger,

Probability and Risk Models

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 rate is n−1. Moreover, we also show that a trapezoidal scheme converges (at least) with the rate n−2H+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.