A Stratonovich-Skorohod integral formula for Volterra Gaussian rough paths

Abstract

Given a solution $Y$ to a rough differential equation (RDE), a recent result [8] extends the classical Itö-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 $3\le p<4$. 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 Itö formulas in the case where the vector fields of the RDE governing $Y$ are commutative.