by taking advantage of the assumption that $\rho \in \left[1, \frac{3}{2} \right)$, but the less regular cases need a new argument that exploits the structure of the Volterra kernel. To handle the Malliavin derivative $\mathcal{D}Y$, we need a similar result that identifies a class of two-parameter functions as a subset of $\mathcal{H}_{1}\otimes\mathcal{H}_{1}$.

(ii) For the examples considered in this paper, the Gaussian rough path $\mathbf{X}$ will consist of iterated integrals up to degree three; i.e. $\mathbf{X=}\left(  1,X,\mathbf{X}^{2},\mathbf{X}^{3}\right)  .$ This contrasts with the result in [@cl2018], where only the case $\mathbf{X=} \left(  1,X,\mathbf{X}^{2}\right)$ needs to be considered. This increases the complexity of the the arguments significantly; indeed, the rough integral in the left side of (\[corr form\]) is now well approximated locally by terms up to third-order $$\begin{aligned}
    \int_{s}^{t}\left\langle Y_{t}\circ\mathrm{d}\mathbf{X}_{t} \right\rangle
    \simeq\left\langle Y_{s},X_{s,t}\right\rangle +V\left(  Y_{s}\right) \mathbf{X}_{s,t}^{2} + V^2(Y_{s}) \left(
    \mathbf{X}_{s,t}^{3}\right).\end{aligned}$$ A key step in [@cl2018] is the proof that the second-order terms in this approximation satisfy $$\begin{aligned}
    \lim_{\left\Vert \pi(n)\right\Vert \rightarrow 0}\left\Vert \sum_{i:\pi(n)=\left\{  t_{i}^{n}\right\}}V(Y_{t_{i}^{n}})\left(  \mathbf{X}_{t_{i}^{n}, t_{i+1}^{n}}^{2}-\frac{1}{2}\sigma^{2}\left(  t_{i}^{n},t_{i+1}^{n}\right) \mathcal{I}_{d}\right) \right\Vert_{L^{2} (\Omega)} = 0.\end{aligned}$$ For the present work we need to address the same problem for the third order terms, namely the existence of an $L^{2}(\Omega)$-limit for sums of terms of the form $$\begin{aligned}
    