$M$. We also identify $\R^k$ with $\R^{1,k}$. Let now $\Omega:=C_0([0,T],\R^d)$ be the canonical Wiener space of continuous function $\omega:=(\omega^1,\ldots,\omega^d)^\top$ from $[0,T]$ to $\R^d$ such that $\omega(0)=(0,\ldots,0)^\top$. Let $W:=(W_t^1,\ldots,W_t^d)^\top_{t\in [0,T]}$ be the canonical Wiener process, that is, for any time $t$ in $[0,T]$, $W_t$ denotes the evaluation mapping: $W_t^i(\omega):=\omega_t^{i}$ for any element $\omega$ in $\Omega$ and $i$ in $\{1,\ldots,d\}$. We set $\F^o$ the natural filtration of $W$. Under the Wiener measure $\P_0$, the process $W$ is a standard Brownian motion and we denote by $\F:=(\mathcal F_t)_{t\in[0,T]}$ the usual augmentation (which is right-continuous and complete) of $\F^o$ under $\P_0$. Unless otherwise stated, all the expectations considered in this paper will have to be understood as expectations under $\P_0$, and all notions of measurability for elements of $\Omega$ will be with respect to the filtration $\F$ or the $\sigma$-field $\mathcal F_T$.

For any Hilbert space $\mathcal K$, for any $p\geq 1$ and for any $t\in [0,T]$, we set $L^p([t,T];\mathcal K)$ to be following space $$L^p([t,T];\mathcal K):=\left\{f:[t,T]\longrightarrow\mathcal K, \text{ Borel-measurable, s.t. }\int_t^T \| f(s)\|^p_{\mathcal{K}}ds <+\infty \right\},$$ where the norm ${\left\|\cdot\right\|}_{\Kc}$ is the one canonically induced by the inner product on $\Kc$. We denote, for simplicity, by $\h:=L^2([0,T];\R^{ d})$ and by $\langle \cdot, \cdot \rangle_\h$ 