A Stratonovich–Skorohod integral formula for Gaussian rough paths

Abstract

Given a Gaussian process $X$, its canonical geometric rough path lift $\mathbf{X}$, and a solution $Y$ to the rough differential equation (RDE) $\mathrm{d}Y_{t}=V(Y_{t})\circ\mathrm{d}\mathbf{X}_{t}$, we present a closed-form correction formula for $\int Y\circ\mathrm{d}\mathbf{X}-\int Y\,\mathrm{d}X$, that is, the difference between the rough and Skorohod integrals of $Y$ with respect to $X$. When $X$ is standard Brownian motion, we recover the classical Stratonovich-to-Itô conversion formula, which we generalize to Gaussian rough paths with finite $p$-variation, $p<3$, and satisfying an additional natural condition. This encompasses many familiar examples, including fractional Brownian motion with $H>\frac{1}{3}$. To prove the formula, we first show that the Riemann-sum approximants of the Skorohod integral converge in $L^{2}(\Omega)$ by using a novel characterization of the Cameron–Martin norm in terms of higher-dimensional Young–Stieltjes integrals. Next, we append the approximants of the Skorohod integral with a suitable compensation term without altering the limit, and the formula is finally obtained after a rebalancing of terms.