\mathcal{K}^* \left( \phi\mathds{1}_{[0,a)}\right) (s)=\mathds{1}_{[0,a)}(s)\left(  \phi(s)K(a,s)+\int_{s}^{a}\left[ \phi(r)-\phi(s)\right] K(\mathrm{d}r,s)\right).\end{aligned}$$

Malliavin calculus {#MDsection}
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We will primarily work with the following Hilbert space which is isomorphic to $\mathcal{H}^{d}$.

Let $\mathcal{H}_{1}^{d}$ denote the completion of the linear span of $$\begin{aligned}
\left\{  \mathds{1}_{[0,t)}^{(u)}(\cdot):=\mathds{1}_{[0,t)}(\cdot
)\,e_{u}\;\bigg\vert\;t\in\lbrack0,T],\;u=1,\ldots,d\right\}\end{aligned}$$ (cf. [@amn2001], [@nualart2006]) with respect to the inner-product given by $$\begin{aligned}
\left\langle \mathds{1}_{[0,t)}^{(u)}(\cdot),\mathds{1}_{[0,s)}^{(v)}
(\cdot)\right\rangle _{\mathcal{H}_{1}^{d}}=\delta_{uv}\,R(t,s),\end{aligned}$$ where $\delta$ denotes the Kronecker delta. Furthermore, let $\Phi:\mathcal{H}_{1}^{d}\rightarrow\mathcal{H}^{d}$ denote the Hilbert space isomorphism obtained from extending the map $\mathds{1}_{[0,t)}^{(u)}(\cdot) \mapsto R(t,\cdot)^{(u)}$, $t \in [0, T],\;u=1,\ldots,d$.

We record some basic properties about the Malliavin calculus. For simplicity, we assume here that $d=1;$ the case of $d \in \mathbb{N}$ case needs only minor modifications. First we recall that the map $\mathds{1}_{\left[ 0, t\right) } (\cdot) \mapsto X_{t}$ extends to a unique linear isometry $I$ from $\mathcal{H}_{1}$ to $L^{2}\left(  \Omega\right) $. It follows that $I\left(  h\right)  $ is a mean-zero Gaussian random variable with variance $\left\Vert h\right\Vert_{\mathcal{H}_{1}}^{2}$.

Given a smooth function $f: \mathbb{R}^{n}\rightarrow\mathbb{R}$ of at most polynomial growth, the Malliavin derivative $\mathcal{D}F$ of the functional $F=f\left(  I\left(  h_{1}\right)  ,\ldots,I\left(  h_{n}\right)  \right) $ is the $\mathcal{H}_{1}$-valued random variable given by $$\begin{aligned}
\mathcal{D}F:=\sum_{i=1}^{n}\frac{\partial f}{\partial x_{i}}\left(  I\left(
h_{1}\right)  ,\ldots,I\left(  h_{n}\right)  \right)  h_{i}.\end{aligned}$$ We let $\mathbb{D}^{1,2}$ denote the Hilbert space that arises 