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Exact Rate of Convergence of Some Approximation Schemes Associated to SDEs Driven by a Fractional Brownian Motion

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Abstract

In this article, we derive the exact rate of convergence of some approximation schemes associated to scalar stochastic differential equations driven by a fractional Brownian motion with Hurst index H. We consider two cases. If H>1/2, the exact rate of convergence of the Euler scheme is determined. We show that the error of the Euler scheme converges almost surely to a random variable, which in particular depends on the Malliavin derivative of the solution. This result extends those contained in J. Complex. 22(4), 459–474, 2006 and C.R. Acad. Sci. Paris, Ser. I 340(8), 611–614, 2005. When 1/6<H<1/2, the exact rate of convergence of the Crank-Nicholson scheme is determined for a particular equation. Here we show convergence in law of the error to a random variable, which depends on the solution of the equation and an independent Gaussian random variable.

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Authors and Affiliations

  1. Fachbereich Mathematik, Johann Wolfgang Goethe-Universität Frankfurt, Robert-Mayer-Straße 10, 60325, Frankfurt am Main, Germany

    Andreas Neuenkirch

  2. LPMA, Université Pierre et Marie Curie Paris 6, Boîte courrier 188, 4 Place Jussieu, 75252, Paris Cedex 5, France

    Ivan Nourdin

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  1. Andreas Neuenkirch
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  2. Ivan Nourdin
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Correspondence to Andreas Neuenkirch.

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Neuenkirch, A., Nourdin, I. Exact Rate of Convergence of Some Approximation Schemes Associated to SDEs Driven by a Fractional Brownian Motion. J Theor Probab 20, 871–899 (2007). https://doi.org/10.1007/s10959-007-0083-0

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  • DOI: https://doi.org/10.1007/s10959-007-0083-0

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