will assume that $X=\left(  X^{1},...,X^{d}\right)  $ has i.i.d components, each centered with covariance function $R$, and that $X$ is defined on a probability space $\left(  \Omega,\mathcal{F},\mathbb{P}\right)
$. For simplicity we assume that $\mathcal{F}$ is generated by $X.$ The process $X$ then gives rise an isonormal Gaussian process w.r.t. the Hilbert $\mathcal{H}_{1}^{d}=\oplus_{i=1}^{d}\mathcal{H}_{1}^{\left(  i\right)  }$ where, for all $i=1,...,d,$ $\mathcal{H}_{1}^{\left(  i\right)  } = \mathcal{H}_{1}$ and $\mathcal{H}_{1}$ is the completion of the real vector space $$\begin{aligned}
\text{span}\left\{  \mathds{1}_{[0,t)}(\cdot):\bigg\vert\;t\in[0,T]\right\}\end{aligned}$$ endowed with the inner-product $\left\langle \mathds{1}_{[0,t)}(\cdot), \mathds{1}_{[0,s)}(\cdot) \right\rangle_{\mathcal{H}_1} = R(t,s)$. The solution $Y$ to (\[basic SDE1\]) can also be viewed as a Wiener functional on $\left(\Omega,\mathcal{F},\mathbb{P}\right) $, and its properties can then be studied using Malliavin calculus. A number of recent works have opened up the interplay between Lyons’ and Malliavin’s calculi, see e.g. [@cfv2009], [@cf2011], [@hp2013] and [@chlt2015]. In particular, in a recent paper [@cl2018] the authors have proven a conversion formula for the difference between the rough path integral of $Y$ w.r.t $\mathbf{X}$ and the Skorohod integral $\delta^{X}$ of $Y$ (i.e. the $L^{2}\left( \Omega\right)$ adjoint of the Malliavin derivative operator). In more detail this result shows, for the case where $Y$ and $X$ are both $\mathbb{R}^{d}$-valued, the following almost sure identity $$\begin{aligned}
 \label{corr 