## The Annals of Probability

### 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.

#### Article information

Source
Ann. Probab., Volume 47, Number 1 (2019), 1-60.

Dates
Revised: January 2018
First available in Project Euclid: 13 December 2018

https://projecteuclid.org/euclid.aop/1544691617

Digital Object Identifier
doi:10.1214/18-AOP1254

Mathematical Reviews number (MathSciNet)
MR3909965

Zentralblatt MATH identifier
07036333

#### Citation

Cass, Thomas; Lim, Nengli. A Stratonovich–Skorohod integral formula for Gaussian rough paths. Ann. Probab. 47 (2019), no. 1, 1--60. doi:10.1214/18-AOP1254. https://projecteuclid.org/euclid.aop/1544691617

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