On stochastic calculus with respect to q-Brownian motion

Aurélien Deya (1) , René Schott (2)

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

Following the approach and the terminology introduced in [A. Deya and R. Schott, On the rough paths approach to non-commutative stochastic calculus, J. Funct. Anal., 2013], we construct a product Lévy area above the $q$-Brownian motion (for $q\in [0,1)$) and use this object to study differential equations driven by the process. We also provide a detailled comparison between the resulting "rough" integral and the stochastic "Itô" integral exhibited by Donati-Martin in [C. Donati-Martin, Stochastic integration with respect to $q$ Brownian motion, Probab. Theory Related Fields, 2003].

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

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

Aurélien Deya, René Schott. On stochastic calculus with respect to q-Brownian motion. Journal of Functional Analysis, 2018, 274 (4), pp.1047-1075. ⟨10.1016/j.jfa.2017.08.019⟩. ⟨hal-01416950v3⟩