and Nengli Lim
title: '[A Stratonovich-Skorohod formula for Volterra Gaussian rough paths]{}'
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Introduction
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Lyons’ rough path theory is a framework for giving a path-wise interpretation to stochastic differential equations of the form $$\begin{aligned}
 \label{basic SDE}
\mathrm{d}Y_{t}=V(Y_{t})\circ\mathrm{d}X_{t},\quad
Y_{0}=y_{0},\end{aligned}$$ in particular for a broad class of continuous, vector-valued Gaussian processes $X$. A fundamental contribution of Lyons [@lyons98; @lcl2006] was to realize that this needs $X$ to be enriched to a *rough path* $\mathbf{X}$ whose components comprise not only $X$, but also the higher-order iterated integrals up to some finite degree. The model then ought to be rewritten as $$\begin{aligned}
 \label{basic SDE1}
\mathrm{d}Y_{t}=V(Y_{t})\circ\mathrm{d}\mathbf{X}_{t},\quad
Y_{0}=y_{0},\end{aligned}$$ to reflect the dependence of the solution on the enriched rough path. The existence of $\mathbf{X}$ for a given stochastic process cannot be taken for granted. Nevertheless, if $X$ is Gaussian then $\mathbf{X}$ can be constructed in a canonical way as an appropriate limit of iterated integrals of smooth (i.e. bounded variation) approximations to $X$ such as piecewise linear approximations, see [@fv2010a; @fv2010b]. For this, conditions are needed on the covariance function, but these are flexible enough to encompass a range of examples including fractional Brownian motion with Hurst index $H > \frac{1}{4}$.

In this paper we 