S_{ n}^p}:=\mathbb E\left[\underset{0\leq t\leq T}{\sup} \|Y_t\|^p\right]<+\infty.$$ We denote by $\mathbb H_{n,d}^p$ the space of $\mathbb R^{n \times d}$-valued and $\F$-predictable processes $Z$ such that $${\left\|Z\right\|}^p_{\mathbb H_{n,d}^p}:=\mathbb E\left[\left(\int_0^T \sum_{j=1}^n \|Z_t^j\|^2 dt\right)^{\frac p2}\right]<+\infty.$$ We set $\mathbb S^p:=\mathbb S_{1}^p$ and $\H_{d}^p:=\H_{1,d}^p$.

Some elements of analysis on the Wiener space {#section:Wiener}
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One of the main tool that we will use throughout this paper is the shift operator along directions in the Cameron-Martin space. More precisely, for any $h\in H$, we define the following shift operator $\tau_{h}:\Omega\longrightarrow\Omega$ by $$\tau_{h}(\omega):=\omega + h:=(\omega^1+h^1,\ldots,\omega^d+h^d)^\top.$$ Note that the fact that $h$ belongs to $H$ ensures that $\tau_h$ is a measurable shift on the Wiener space. In fact, one can be a bit more precise, since according to [@ustunelzakai Lemma B.2.1] for any $\mathcal{F}_T$-measurable r.v. $F$ the mapping $h \longmapsto F {\circ \tau_{h}}$ is continuous in probability from $H$ to $L^0(\R^{ d})$, the space of real-valued and $\Fc_T$-measurable random variables, see Lemma \[lemma:ctsH\] below. Taking $F={\rm Id}$, one gets that $\tau_h$ is a continuous mapping on $\Omega$ for any $h$ in $H$. We list below some other properties of such shifts.

\[lemma:UstunelZakai\] Let $X$ and $Y$ be two $\mathcal F_T$-measurable random variables. If $X=Y$, $\P_0-$a.s., then for any $h$ 