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On the signature and cubature of the fractional Brownian motion for H>12

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  • Passeggeri, Riccardo

Abstract

We present several results concerning the fractional Brownian motion (fBm) for H>1∕2. First, we show that the rate of convergence of the expected signature of the linear piecewise approximation of the fBm to its exact value is given by 2H. Second, we show that, for the 2k-th term in the signature, the coefficient of the rate of convergence is uniformly bounded by Ãk(2k−1)(k−1)!2k. Third, we show that the 2k-th term of the expected signature is bounded by 1k!2k. Finally, we develop the general cubature method for the fBm for H>1∕2 for small times and provide a numerical example.

Suggested Citation

  • Passeggeri, Riccardo, 2020. "On the signature and cubature of the fractional Brownian motion for H>12," Stochastic Processes and their Applications, Elsevier, vol. 130(3), pages 1226-1257.
    • Handle: RePEc:eee:spapps:v:130:y:2020:i:3:p:1226-1257
    DOI: 10.1016/j.spa.2019.04.013
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    References listed on IDEAS

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    1. Baudoin, Fabrice & Coutin, Laure, 2007. "Operators associated with a stochastic differential equation driven by fractional Brownian motions," Stochastic Processes and their Applications, Elsevier, vol. 117(5), pages 550-574, May.
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    Cited by:

    1. Hocquet, Antoine & Vogler, Alexander, 2023. "An application of the multiplicative Sewing Lemma to the high order weak approximation of stochastic differential equations," Stochastic Processes and their Applications, Elsevier, vol. 165(C), pages 183-217.

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