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

Aurélien Deya (1) , René Schott (2)

1 IECL - Institut Élie Cartan de Lorraine
2 LORIA - NSS - Department of Networks, Systems and Services

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 Itô-Stratonovich correction formula for the NC-fBm.

Mots clés: non-commutative fractional Brownian motion; non-commutative stochastic calculus; Malliavin calculus; Itô-Stratonovich formula

Domaines: Probabilités [math.PR]; Algèbres d'opérateurs [math.OA]

Aurélien Deya, René Schott. Skorohod and rough integration with respect to the non-commutative fractional Brownian motion. ALEA : Latin American Journal of Probability and Mathematical Statistics, inPress, ⟨10.48550/arXiv.1909.06270⟩. ⟨hal-02286188v2⟩