A Stratonovich-Skorohod integral formula for Gaussian rough paths

Abstract

Given a Gaussian process X, its canonical geometric rough path lift X, and a solution Y to the rough differential equation (RDE) dYt=V(Yt)∘dXt, we present a closed-form correction formula for ∫Y∘dX−∫YdX, 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>13. To prove the formula, we first show that the Riemann-sum approximants of the Skorohod integral converge in L2(Ω) 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.