Discretizing the fractional Levy area

Neuenkirch, Andreas ; Tindel, Samy ; Unterberger, Jérémie

URL: https://arxiv.org/abs/0902.0497
Document Type: Working paper
Year of publication: 2009
Place of publication: Ithaca, NY
Publishing house: Cornell University
Publication language: English
Institution: School of Business Informatics and Mathematics > Wirtschaftsmathematik II: Stochastische Numerik (Neuenkirch 2013-)
Subject: 510 Mathematics
Abstract: 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.

Dieser Datensatz wurde nicht während einer Tätigkeit an der Universität Mannheim veröffentlicht, dies ist eine Externe Publikation.

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