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A Stratonovich-Skorohod integral formula for Gaussian rough paths

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Title: A Stratonovich-Skorohod integral formula for Gaussian rough paths
Authors: Cass, T
Lim, N
Item Type: Journal Article
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.
Issue Date: 1-Jan-2019
Date of Acceptance: 8-Jan-2018
URI: http://hdl.handle.net/10044/1/56747
DOI: https://dx.doi.org/10.1214/18-AOP1254
ISSN: 0091-1798
Publisher: Institute of Mathematical Statistics
Start Page: 1
End Page: 60
Journal / Book Title: Annals of Probability
Volume: 47
Issue: 1
Copyright Statement: © Institute of Mathematical Statistics, 2019
Sponsor/Funder: Engineering & Physical Science Research Council (EPSRC)
Funder's Grant Number: EP/M00516X/1
Keywords: math.PR
0104 Statistics
Statistics & Probability
Publication Status: Published
Online Publication Date: 2018-12-13
Appears in Collections:Financial Mathematics
Faculty of Natural Sciences