Riemann-Skorohod and Stratonovich integrals for Gaussian processes
Yanghui Liu
TL;DR
The paper develops a rigorous connection between Skorohod-type and Stratonovich-type integrals for Gaussian processes under finite covariance variation ($\rho$-variation) and diagonal variation ($\rho'$-variation). By combining Malliavin calculus with rough path techniques, it defines a Stratonovich-type integral via compensated Taylor expansions and a Skorohod-type integral via Skorohod-Riemann sums, and proves a conversion formula that adds a half-diagonal Young integral to the Skorohod integral. The main results hold in both one- and multi-dimensional settings, do not require a geometric rough path, and show that the Skorohod integral is the limit of a $[\rho]$-th order Skorohod-Riemann sum. Under the conditions $\rho<3/2$ or $\frac{1}{2\rho}+\frac{1}{\rho'}>1$, the relevant sums converge and the conversion formula is explicit, with the diagonal term interpreted as a Young integral. Altogether, the work unifies rough-path and Malliavin approaches for Gaussian processes and extends conversion results beyond prior assumptions, including non-autonomous integrands and reduced expansion order.
Abstract
In this paper we consider Skorohod and Stratonovich-type integrals in a general setting of Gaussian processes. We show that a conversion formula holds when the covariance functions of the Gaussian process are of finite $ρ$-variation for $ρ\geq 1$ and that the diagonals of covariance functions are of finite $ρ'$-variation for $ρ'\geq 1$ such that $\frac{1}{ρ'}+\frac{1}{2ρ}>1$. The difference between the two types of integrals is identified with a Young integral. We also show that the Skorohod integral is the limit of a $[ρ]$-th order Skorohod-Riemann sum.