A Stratonovich-Skorohod integral formula for Volterra Gaussian rough paths

Given a solution \(Y\) to a rough differential equation (RDE), a recent result [8] extends the classical It\"{o}-Stratonovich formula and provides a closed-form expression for \(\int Y \circ \mathrm{d} \mathbf{X} - \int Y \, \mathrm{d} X\), i.e. the difference between the rough and Skorohod integrals of \(Y\) with respect to \(X\), where \(X\) is a Gaussian process with finite \(p\)-variation less than 3. In this paper, we extend this result to Gaussian processes with finite \(p\)-variation such that \(3 \leq p < 4\). The constraint this time is that we restrict ourselves to Volterra Gaussian processes with kernels satisfying a natural condition, which however still allows the result to encompass many standard examples, including fractional Brownian motion with \(H > \frac{1}{4}\). Analogously to [8], we first show that the Riemann-sum approximants of the Skorohod integral converge in \(L^2(\Omega)\) by adopting a suitable characterization of the Cameron-Martin norm, before appending the approximants with higher-level compensation terms without altering the limit. Lastly, the formula is obtained after a re-balancing of terms, and we also show how to recover the standard It\"{o} formulas in the case where the vector fields of the RDE governing \(Y\) are commutative.