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

Jorge A. León; Yanghui Liu; Samy Tindel

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arXiv · 2305.10365

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

Jorge A. León·Yanghui Liu·Samy Tindel

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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 property is now well established. 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.

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Jorge A. León, Yanghui Liu, Samy Tindel. 2023-05-17. Euler scheme for SDEs driven by fractional Brownian motions: Malliavin differentiability and uniform upper-bound estimates. https://arxiv.org/abs/2305.10365

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