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Euler scheme for SDEs driven by fractional Brownian motions: Malliavin differentiability and uniform upper-bound estimates

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  • León, Jorge A.
  • Liu, Yanghui
  • Tindel, Samy

Abstract

The Malliavin differentiability of a SDE plays a crucial role in the study of density smoothness and ergodicity among others. For Gaussian driven SDEs the differentiability issue is solved essentially in Cass et al., (2013). In this paper, we consider the Malliavin differentiability for the Euler scheme of such SDEs. We will focus on SDEs driven by fractional Brownian motions (fBm), which is a very natural class of Gaussian processes. We derive a uniform (in the step size n) path-wise upper-bound estimate for the Euler scheme for stochastic differential equations driven by fBm with Hurst parameter H>1/3 and its Malliavin derivatives.

Suggested Citation

  • León, Jorge A. & Liu, Yanghui & Tindel, Samy, 2024. "Euler scheme for SDEs driven by fractional Brownian motions: Malliavin differentiability and uniform upper-bound estimates," Stochastic Processes and their Applications, Elsevier, vol. 175(C).
    • Handle: RePEc:eee:spapps:v:175:y:2024:i:c:s0304414924001182
    DOI: 10.1016/j.spa.2024.104412
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    References listed on IDEAS

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    1. Benjamin Gess & Cheng Ouyang & Samy Tindel, 2020. "Density Bounds for Solutions to Differential Equations Driven by Gaussian Rough Paths," Journal of Theoretical Probability, Springer, vol. 33(2), pages 611-648, June.
    2. Nualart, David & Saussereau, Bruno, 2009. "Malliavin calculus for stochastic differential equations driven by a fractional Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 119(2), pages 391-409, February.
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