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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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  • Andreas Neuenkirch

    (Johann Wolfgang Goethe-Universität Frankfurt)

  • Ivan Nourdin

    (Université Pierre et Marie Curie Paris 6)

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

Suggested Citation

  • Andreas Neuenkirch & Ivan Nourdin, 2007. "Exact Rate of Convergence of Some Approximation Schemes Associated to SDEs Driven by a Fractional Brownian Motion," Journal of Theoretical Probability, Springer, vol. 20(4), pages 871-899, December.
    • Handle: RePEc:spr:jotpro:v:20:y:2007:i:4:d:10.1007_s10959-007-0083-0
    DOI: 10.1007/s10959-007-0083-0
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    References listed on IDEAS

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    1. Benassi, Albert & Cohen, Serge & Istas, Jacques & Jaffard, Stéphane, 1998. "Identification of filtered white noises," Stochastic Processes and their Applications, Elsevier, vol. 75(1), pages 31-49, June.
    2. Breuer, Péter & Major, Péter, 1983. "Central limit theorems for non-linear functionals of Gaussian fields," Journal of Multivariate Analysis, Elsevier, vol. 13(3), pages 425-441, September.
    3. Nourdin, Ivan & Simon, Thomas, 2006. "On the absolute continuity of one-dimensional SDEs driven by a fractional Brownian motion," Statistics & Probability Letters, Elsevier, vol. 76(9), pages 907-912, May.
    4. Coeurjolly, Jean-Francois, 2000. "Simulation and identification of the fractional Brownian motion: a bibliographical and comparative study," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 5(i07).
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    Cited by:

    1. Ivan Nourdin & David Nualart, 2010. "Central Limit Theorems for Multiple Skorokhod Integrals," Journal of Theoretical Probability, Springer, vol. 23(1), pages 39-64, March.
    2. Orimar Sauri, 2024. "Asymptotic Error Distribution of the Euler Scheme for Fractional Stochastic Delay Differential Equations with Additive Noise," Papers 2402.08513, arXiv.org.
    3. Nicholas Ma & David Nualart, 2020. "Rate of Convergence for the Weighted Hermite Variations of the Fractional Brownian Motion," Journal of Theoretical Probability, Springer, vol. 33(4), pages 1919-1947, December.
    4. Héctor Araya & Jorge A. León & Soledad Torres, 2020. "Numerical Scheme for Stochastic Differential Equations Driven by Fractional Brownian Motion with $$ 1/4," Journal of Theoretical Probability, Springer, vol. 33(3), pages 1211-1237, September.
    5. Kento Ueda, 2025. "Error Distribution for One-Dimensional Stochastic Differential Equations Driven By Fractional Brownian Motion," Journal of Theoretical Probability, Springer, vol. 38(1), pages 1-61, March.
    6. Kęstutis Kubilius, 2024. "Approximation of the Fractional SDEs with Stochastic Forcing," Mathematics, MDPI, vol. 12(24), pages 1-22, December.
    7. Nobuaki Naganuma, 2015. "Asymptotic Error Distributions of the Crank–Nicholson Scheme for SDEs Driven by Fractional Brownian Motion," Journal of Theoretical Probability, Springer, vol. 28(3), pages 1082-1124, September.

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