A Stratonovich-Skorohod integral formula for Gaussian rough paths

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Title: A Stratonovich-Skorohod integral formula for Gaussian rough paths
Author(s): Cass, T
Lim, N
Item Type: Journal Article
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.
Date of Acceptance: 8-Jan-2018
URI: http://hdl.handle.net/10044/1/56747
ISSN: 0091-1798
Publisher: Institute of Mathematical Statistics
Journal / Book Title: Annals of Probability
Copyright Statement: This article is under embargo until publication
Sponsor/Funder: Engineering & Physical Science Research Council (EPSRC)
Funder's Grant Number: EP/M00516X/1
Keywords: math.PR
math.PR
math.PR
math.PR
0104 Statistics
Statistics & Probability
Embargo Date: publication subject to indefinite embargo
Appears in Collections:Financial Mathematics
Mathematics
Faculty of Natural Sciences



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