On stochastic calculus with respect to q-Brownian motion

Aurélien Deya (1) , René Schott (1)

1 Probabilités et statistiques

We pursue the investigations initiated by Donati-Martin [8] regarding stochastic calculus with respect to the q-Brownian motion, and essentially extend the previous results along two directions: (i) We develop a robust $L^\infty$-integration theory based on rough-paths principles and apply it to the study of q-Bm-driven differential equations; (ii) We provide a comprehensive description of the multiplication properties in the q-Wiener chaos. Our presentation follows a probabilistic pattern, in the sense that it only leans on the law of the process and not on its particular construction. Besides, our formulation puts the stress on the rich combinatorics behind non-commutative processes, in the spirit of the machinery developed by Nica and Speicher in [14].

Mots clés: Non-commutative stochastic calculus; q-Brownian motion; rough paths theory; q-Wiener chaoses

Domaines: Probabilités [math.PR]

Aurélien Deya, René Schott. On stochastic calculus with respect to q-Brownian motion. 2016. ⟨hal-01416950v1⟩