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Journal of Theoretical Probability

, Volume 28, Issue 1, pp 396–422 | Cite as

Rate of Convergence for Discretization of Integrals with Respect to Fractional Brownian Motion

  • Ehsan Azmoodeh
  • Lauri ViitasaariEmail author
Article

Abstract

In this article, a uniform discretization of stochastic integrals \(\int _{0}^{1} f^{\prime }_-(B_t)\mathrm d B_t\), where \(B\) denotes the fractional Brownian motion with Hurst parameter \(H \in (\frac{1}{2},1)\), is considered for a large class of convex functions \(f\). In Azmoodeh et al. (Stat Decis 27:129–143, 2010), for any convex function \(f\), the almost sure convergence of uniform discretization to such stochastic integral is proved. Here, we prove \(L^r\)-convergence of uniform discretization to stochastic integral. In addition, we obtain a rate of convergence. It turns out that the rate of convergence can be brought arbitrarily close to \(H - \frac{1}{2}\).

Keywords

Fractional Brownian motion Stochastic integral Discretization Rate of convergence 

Mathematics Subject Classification (2010)

60G22 60H05 41A25 

Notes

Acknowledgments

The authors thank Esko Valkeila for discussions and comments which improved the paper. The authors also thank anonymous referee for useful comments and remarks. Ehsan Azmoodeh thanks the Magnus Ehrnrooth foundation for financial support. Lauri Viitasaari thanks the Finnish Doctoral Programme in Stochastics and Statistics for financial support.

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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.de la Technologie et de la Communication, Faculté des SciencesUniversity of LuxembourgLuxembourg CityLuxembourg
  2. 2.Department of Mathematics and Systems AnalysisAalto University School of ScienceEspooFinland

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