The Annals of Probability

A Stratonovich–Skorohod integral formula for Gaussian rough paths

Thomas Cass and Nengli Lim

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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

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

Received: May 2016
Revised: January 2018
First available in Project Euclid: 13 December 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H05: Stochastic integrals
Secondary: 60H07: Stochastic calculus of variations and the Malliavin calculus 60H10: Stochastic ordinary differential equations [See also 34F05]

Rough paths theory Malliavin calculus generalized Itô–Stratonovich correction formulas


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.

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