A Stratonovich-Skorohod integral formula for Volterra Gaussian rough paths

TL;DR

This paper extends the Stratonovich-Skorohod integral formula to Volterra Gaussian rough paths with p-variation between 3 and 4, including fractional Brownian motion with H > 1/4, using approximation and compensation techniques.

Contribution

It introduces a generalized integral formula for Gaussian processes with higher p-variation, expanding previous results to a broader class of Volterra Gaussian processes.

Findings

Convergence of Riemann-sum approximants in L^2(Ω)

Extension of integral formula to p-variation 3 to 4

Recovery of Itô formulas for commutative vector fields

Abstract

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.