Rough paths and symmetric-Stratonovich integrals driven by singular covariance gaussian processes

TL;DR

This paper establishes a connection between rough path integrals and symmetric-Stratonovich integrals for Gaussian processes, showing their equivalence under certain conditions and analyzing convergence rates of numerical schemes.

Contribution

It proves the equality of rough path and symmetric-Stratonovich integrals driven by Gaussian processes under mild regularity, and demonstrates solutions of rough differential equations are Stratonovich SDEs.

Findings

Equality between rough path and symmetric-Stratonovich integrals for Gaussian processes.

Almost sure convergence rates of Stratonovich schemes to rough path integrals.

Applicability to a broad class of Gaussian processes with specific covariance regularity.

Abstract

We examine the relation between a stochastic version of the rough path integral with the symmetric-Stratonovich integral in the sense of regularization. Under mild regularity conditions in the sense of Malliavin calculus, we establish equality between stochastic rough path and symmetric-Stratonovich integrals driven by a class of Gaussian processes. As a by-product, we show that solutions of multi-dimensional rough differential equations driven by a large class of Gaussian rough paths they are actually solutions to Stratonovich stochastic differential equations. We obtain almost sure convergence rates of the first-order Stratonovich scheme to rough paths integrals in the sense of Gubinelli. In case the time-increment of the Malliavin derivative of the integrands is regular enough, the rates are essentially sharp. The framework applies to a large class of Gaussian processes whose the second-order derivative of the covariance function is a sigma-finite non-positive measure on ${\mathbb R}^2$ + off diagonal.