is well-known that by definition of $\P_0$, any $\Fc_t$-measurable random variable admits a $\Fc_t^o$-measurable version. Therefore, there exists some measurable map $\varphi:\Omega \to {\mathbb{R}}$, such that $$F=\varphi(W_{\cdot\wedge t}),\ \mathbb P_0-a.s.$$ Hence, we deduce by Lemma \[lemma:UstunelZakai\] that for $\P_0-a.e.$ $\omega\in\Omega$ $$F{\circ \tau_{h}}(\omega)=\varphi(W_{\cdot\wedge t}(\omega)){\circ \tau_{h}}=\varphi(W_{\cdot\wedge t}{\circ \tau_{h}}(\omega))=\varphi(\omega(\cdot\wedge t)+h(\cdot \wedge t))=F\circ \tau_{\widetilde{h^t}}(\omega).$$

We conclude this section with the following lemma which might be known. However since we did not find it in the literature we provide a proof in order to make this paper self-contained.

\[lemma:shift2\] Let $Z\in\H_{d}^2$ and $h$ in $H$. It holds that $$\int_0^T Z_s \cdot dW_s {\circ \tau_{h}} = \int_0^T Z_s{\circ \tau_{h}} \cdot dW_s+\int_0^T Z_s {\circ \tau_{h}} \cdot \dot{h}(s)ds, \ \P_0-a.s.$$

Let $\mathfrak{S}$ be the class of simple processes $X$ of the form $$X_t:=(X_t^1,\ldots,X_t^d)^\top, \quad X_t^j=\sum_{i=0}^{n_j} \lambda_{i}^j\mathbf{1}_{(t_i^j,t_{i+1}^j]}(t),\; j\in \{1,\ldots,d\}$$ where for any $j\in \{ 1, \ldots,d\}$, $n_j\in\N^*$, $t_0^j=0<t_1^j<...<t_n^j=T$ and where for any $0\leq i\leq n_j$, $(\lambda_i^j)_{i=1,...n_j}$ are $\mathcal{F}_{t_i^j}$-measurable and in $L^2(\R)$.

We start by proving the result for $Z$ in $\mathfrak{S}$ and then we prove the result for any element $Z$ in $\H_{ d}^2$ using a density argument. Let $Z\in \mathfrak{S}$ with the decomposition $$Z_s= (Z_s^1,\ldots,Z_s^d)^\top, \quad Z_s^j:=\sum_{i=0}^{n_j} \lambda_i^j\mathbf{1}_{(t_i^j,t_{i+1}^j]}(s), \ s\in[0,T], \ j\in \{1,\ldots, d \}.$$ Then, for any 