A STRATONOVICH-SKOROHOD INTEGRAL FORMULA FOR GAUSSIAN ROUGH PATHS

• 成果类型: Article
• 署名作者: Cass, Thomas; Lim, Nengli
• 署名单位: Imperial College London; Singapore University of Technology & Design
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/18-AOP1254
• 发表日期: 2019
• 页码: 1-60
• 关键词: differential-equations driven; stochastic integration; calculus; Respect

摘要

Given a Gaussian process X, its canonical geometric rough path lift X, and a solution Y to the rough differential equation (RDE) dY(t) = V (Y-t) circle dX(t), we present a closed-form correction formula for integral Y circle dX - integral Y dX, 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-Ito 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 > 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.