dW_s +\int_0^t b(s,X_s) ds, \quad t\in [0,T],$$ with regular enough coefficients $\sigma, b:[0,T]\times {\mathbb{R}}\longrightarrow {\mathbb{R}}$. In that framework, Pardoux and Peng proved in [@pardouxpeng Proposition 2.2] that, under (essentially) the following conditions:

-   $g$ is continuously differentiable with bounded derivative.

-   $h$ is continuously differentiable in $(x,y,z)$ with bounded derivatives uniformly in time,

$Y_t$ is Malliavin differentiable at any time $t$, with a similar statement for $Z$, and the Malliavin derivatives of $Y$ and $Z$ provide a solution to an explicit linear BSDE. To be more precise, in [@pardouxpeng] the authors make one assumption for the whole paper which is stronger than (PP1)-(PP2) above. However a careful reading of the proof of [@pardouxpeng Proposition 2.2] enables one to conclude that Conditions (PP1)-(PP2) are sufficient to obtain the Malliavin differentiability of the solution. Assumptions (PP1)-(PP2) look pretty intuitive since they basically require the Malliavin differentiability of the terminal condition $\xi$ and of the generator $f$ once the component $(y,z)$ are frozen, i.e., of the process $(t,\omega) \longmapsto f(t,\omega,y,z)$ for given $(y,z)$. Hence, it is natural to expect that the latter conditions can be easily generalized to the non-Markovian framework. Unfortunately, the first result in that direction, given by El Karoui, Peng 