$Y_t$ in $L^2$ such that each variable $F^n$ is Malliavin differentiable with derivative $DF^n$ and such that $DF^n$ converges, with respect to a suitable norm, to $DY_t$. As a consequence, in the approach described above, one believes that this sequence $(F^n)_n$ can be chosen to be the Picard iteration $(Y^n)_n$. Once again, this idea looks very natural, according to the same type of proofs for SDEs, but then one sees that in the BSDE framework this intuitive idea leads to pretty heavy assumptions. We elaborate a little bit more on this point in Section \[section:discu\].

Regarding the discussion above, one could think of trying to find a sequence of processes known to approximate the Malliavin derivative of $Y$ (and $Z$) when $Y$ is Malliavin differentiable. This approximation is provided by the well-known interpretation of the Malliavin derivative as a Gâteaux derivative in the directions of the Cameron-Martin space. More precisely, a necessary condition for $Y_t$ to belong to $\D^{1,2}$, is that for any absolutely continuous function $h$ starting from $0$ at $0$ with derivative denoted $\dot{h}$, the difference quotient $\varepsilon^{-1}(Y_t(\omega+\varepsilon h)-Y_t(\omega))$ converges, in a sense to be made precise, as $\varepsilon$ goes to $0$ to $\langle D Y_t,\dot{h}\rangle_{L^2([0,T])}$. This fact 