space of bounded mapping which are infinitely continuously differentiable with bounded derivatives. For any $F$ in $\mathcal S$ of the form , the Malliavin derivative $\nabla F$ of $F$ is defined as the following $H$-valued random variable: $$\label{eq:DF}
\nabla F:=\sum_{i=1}^n f_{x_i}(W(h_1),\ldots,W(h_n)) h_i,$$ where $f_{x_i}:=\frac{df}{dx_i}$. It is then customary to identify $\nabla F$ with the stochastic process $(\nabla_t F)_{t\in [0,T]}$. More precisely, we define for any $(t,\omega)\in [0,T]\times \Omega,$ $$\nabla_t F(\omega):=\sum_{i=1}^n f_{x_i}\left(W(h_1)(\omega),\ldots,W(h_n)(\omega)\right) h_i(t).$$ Denote then by $\mathbb{D}^{1,p}$ the closure of $\mathcal{S}$ with respect to the Malliavin-Sobolev semi-norm $\|\cdot\|_{1,p}$, defined as: $$\|F\|_{1,p}:=\left(\E\left[|F|^p\right] + \E\left[\|\nabla F\|_{H}^p\right]\right)^{1/p}.$$ We set $\D^{1,\infty}:= \bigcap_{p\geq 2} \D^{1,p}$. In order to link our notations with the ones of the related papers [@pardouxpeng; @EPQ] we make use of the notation $DF$ to represent the derivative of $\nabla F$ as: $$\nabla_t F=\int_0^t D_s F ds, \quad t\in [0,T].$$

We denote by $\delta: L^p(H)\longrightarrow L^p(\R)$ the adjoint operator of $\nabla$ by the following duality relationship: $$\E[F\delta(u)]=\E[\langle \nabla F, u\rangle_{H}],\quad \forall u\in \text{dom}(\delta), \quad \textrm{where}$$ $$\text{dom}(\delta):=\left\{u\in L^p(H), \;\exists c_u>0, \; |\E[\langle \nabla F,u\rangle_{H}]| \leq c_u\| F\|_{L^p(\R)}, \; \forall F\in \D^{1,p} \right\}.$$ $\delta$ is also known under the name of Skorohod (or divergence) operator. Recall that any element $u$ of the form $u:=G h$ 