The data of the equation are the $\mathcal{F}_T$-measurable r.v. $\xi$, called the terminal condition, and the mapping $f:[0,T] \times \Omega \times {\mathbb{R}}^2 \longrightarrow {\mathbb{R}}$ which is a progressively measurable process and where according to the notations used in the literature we write $f(t,y,z)$ for $f(t,\omega,y,z)$. A solution to the BSDE is then a pair of predictable processes $(Y,Z)$, with appropriate integrability properties, such that Relation holds $\P-$a.s.

When dealing with applications, one needs to obtain regularity properties on the solution $(Y,Z)$, such as the Malliavin differentiability of the random variables $Y_t$, $Z_t$ at a given time $t$ in $[0,T]$. Note that for the $Z$ component this question needs to be clarified a little bit because of the definition of $Z$, cf. Theorem \[thm:yzd12\] for a precise statement. More precisely, one needs to answer the following question: $$\textrm{Which conditions on the data $\xi$ and $f$ in \eqref{eq:BSDE} ensure that $Y_t$, $Z_t$ are Malliavin differentiable?}$$ This question was first addressed in the paper [@pardouxpeng] in a Markovian setting, that is when $\xi:=g(X_T)$ and $f(t,\omega,y,z):=h(t,X_t(\omega),y,z)$ where $g:{\mathbb{R}}\to {\mathbb{R}}$ and $h:[0,T]\times {\mathbb{R}}^3 \to {\mathbb{R}}$ are regular enough deterministic functions and $X:=(X_t)_{t\in [0,T]}$ is the unique solution to a SDE of the form: $$X_t=X_0+\int_0^t \sigma(s,X_s) 