On the rough-paths approach to non-commutative stochastic calculus

Aurélien Deya (1) , René Schott (1, 2)

1 IECL - Institut Élie Cartan de Lorraine
2 LORIA - Laboratoire Lorrain de Recherche en Informatique et ses Applications

We study different possibilities to apply the principles of rough paths theory in a non-commutative probability setting. First, we extend previous results obtained by Capitaine, Donati-Martin and Victoir in Lyons' original formulation of rough paths theory. Then we settle the bases of an alternative non-commutative integration procedure, in the spirit of Gubinelli's controlled paths theory, and which allows us to revisit the constructions of Biane and Speicher in the free Brownian case. New approximation results are also derived from the strategy.

Mots clés: approximation results; rough paths theory; free Brownian motion; non-commutative stochastic calculus

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

Aurélien Deya, René Schott. On the rough-paths approach to non-commutative stochastic calculus. Journal of Functional Analysis, 2013, 265 (4), pp.594-628. ⟨10.1016/j.jfa.2013.04.008⟩. ⟨hal-00781383v2⟩