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The Malliavin differentiability of a SDE plays a crucial role in the study ofdensity smoothness and ergodicity among others. For Gaussian driven SDEs thedifferentiability property is now well established.  In this paper, we consider the Malliavin differentiability for the Eulerscheme of such SDEs. We will focus on SDEs driven by fractional Brownianmotions (fBm), which is a very natural class of Gaussian processes.  We derive a uniform (in the step size $n$) path-wise upper-bound estimate forthe Euler scheme for stochastic differential equations driven by fBm with Hurstparameter $H>1/3$ and its Malliavin derivatives.

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