Skorohod and Stratonovich integrals for controlled processes

TL;DR

This paper explores the relationship between Skorohod and Stratonovich integrals for controlled processes driven by Gaussian rough paths, using rough paths theory and Malliavin calculus to analyze their connection.

Contribution

It establishes conditions under which the Skorohod and Stratonovich integrals coincide for controlled processes driven by Gaussian rough paths, combining rough paths theory and Malliavin calculus.

Findings

Derived conditions linking Skorohod and Stratonovich integrals

Used rough paths and Malliavin calculus to analyze integral relationships

Provided a framework for understanding integrals of controlled processes

Abstract

Given a continuous Gaussian process $x$ which gives rise to a $p$-geometric rough path for $p\in (2,3)$, and a general continuous process $y$ controlled by $x$, under proper conditions we establish the relationship between the Skorohod integral $\int_0^t y_s {\mathrm{d}}^\diamond x_s$ and the Stratonovich integral $\int_0^t y_s {\mathrm{d}} {\mathbf x}_s$. Our strategy is to employ the tools from rough paths theory and Malliavin calculus to analyze discrete sums of the integrals.