A Stratonovich-Skorohod integral formula for Gaussian rough paths

TL;DR

This paper derives a correction formula linking rough and Skorohod integrals for Gaussian processes, generalizing classical stochastic calculus results to rough paths like fractional Brownian motion with Hurst parameter greater than 1/3.

Contribution

It provides a closed-form correction formula for Gaussian rough paths, extending the Stratonovich-to-Itô conversion to a broader class of processes with finite p-variation.

Findings

Established convergence of Riemann-sum approximants of the Skorohod integral in L^2

Developed a novel characterization of the Cameron-Martin norm via Young-Stieltjes integrals

Derived the correction formula for Gaussian rough paths including fractional Brownian motion with H > 1/3

Abstract

Given a Gaussian process $X$, its canonical geometric rough path lift $\mathbf{X}$, and a solution $Y$ to the rough differential equation (RDE) $\mathrm{d}Y_{t} = V\left (Y_{t}\right ) \circ \mathrm{d} \mathbf{X}_t$, we present a closed-form correction formula for $\int Y \circ \mathrm{d} \mathbf{X} - \int Y \, \mathrm{d} X$, i.e. 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-It{\^o} 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 re-balancing of terms.