form}
\begin{split}
\int_{0}^{T}\left\langle Y_{t}\circ\mathrm{d}\mathbf{X}_{t}\right\rangle - \delta^{X}\left(  Y\right)  
&= \frac{1}{2}\int_{0}^{T}\mathrm{tr}\left[V(Y_{t})\right]  \,\mathrm{d}R(t) \\
&\qquad+ \int_{[0,T]^{2}}\mathds{1}_{[0,t)}(s) \mathrm{tr}\left[  J_{t}^{\mathbf{X}}\left(  J_{s}^{\mathbf{X}}\right)^{-1} V(Y_{s})-V(Y_{t})\right]  \,\mathrm{d}R(s,t).
\end{split}\end{aligned}$$ Here, $J_{t}^{\mathbf{X}}$ denotes the Jacobian of the flow map $y_{0} \rightarrow Y_{t},$ and the second part of the correction term is a proper 2D Young-Stieltjes integral (see [@fv2010a; @fv2010b]) with respect to the covariance function of $X.$ When $X$ is standard Brownian motion, this last term vanishes since the integrand is zero on the diagonal and $\mathrm{d}R(s,t) = \delta_{\{s=t\}}\,\mathrm{d}s\,\mathrm{d}t$. This, together with the fact that $R(t)=t$, allows us to recover the classical Itô-Stratonovich conversion formula. This a rather basic tool in stochastic analysis and generalizations are likely to be important especially given the now widespread adoption of Gaussian models, e.g. most recently in mathematical finance [@bfg2016].

In [@cl2018], conditions need to be imposed in the proof of the formula (\[corr form\]) which limit the range of applications. An important assumption, for instance, is that the covariance function of $X$ has finite (two-parameter) $\rho$-variation for $\rho \in \left[ 1, \frac{3}{2} \right)$. This implies that the sample paths of $X$ will have finite $p$-variation, for some $p\in [2, 3)$, and this excludes interesting examples such as fractional Brownian motion with $H \in \left( 