Skorohod and rough integration with respect to the non-commutative fractional Brownian motion

TL;DR

This paper compares pathwise and Skorohod-type stochastic integrals for non-commutative fractional Brownian motion, establishing an Itô-Stratonovich correction formula within non-commutative Malliavin calculus for Hurst index H > 1/4.

Contribution

It introduces a Skorohod interpretation of stochastic integration for NC-fBm and proves a correction formula, expanding the understanding of non-commutative stochastic calculus.

Findings

Skorohod and pathwise integrals are comparable for NC-fBm with H > 1/4

Established non-commutative Malliavin calculus tools

Derived an Itô-Stratonovich correction formula for NC-fBm

Abstract

We pursue our investigations, initiated in [8], about stochastic integration with respect to the non-commutative fractional Brownian motion (NC-fBm). Our main objective in this paper is to compare the pathwise constructions of [8] with a Skorohod-type interpretation of the integral. As a first step, we provide details on the basic tools and properties associated with non-commutative Malliavin calculus, by mimicking the presentation of Nualart's celebrated treatise [14]. Then we check that, just as in the classical (commutative) situation, Skorohod integration can indeed be considered in the presence of the NC-fBm, at least for a Hurst index H > 1 4.This finally puts us in a position to state and prove the desired comparison result, which can be regarded as an It{\^o}-Stratonovich correction formula for the NC-fBm.