A martingale approach to noncommutative stochastic calculus

Abstract

We present a new approach to noncommutative stochastic calculus that is, like the classical theory, based primarily on the martingale property. Using this approach, we introduce a general theory of stochastic integration and quadratic (co)variation for a certain class of noncommutative processes, analogous to semimartingales, that includes both the q-Brownian motions and classical matrix-valued Brownian motions. As applications, we obtain Burkholder–Davis–Gundy inequalities (with $p\ge 2$) for continuous-time noncommutative martingales and a noncommutative Itô's formula for “adapted $C^2$ maps,” including trace ⁎-polynomial maps and operator functions associated to the noncommutative $C^2$ scalar functions $R\to C$ introduced by Nikitopoulos, as well as the more general multivariate tracial noncommutative $C^2$ functions introduced by Jekel, Li, and Shlyakhtenko.