A martingale approach to noncommutative stochastic calculus

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 ≥ 2 [jls-end-space/]) for continuous-time noncommutative martingales and a noncommutative Itô's formula for “adapted C²maps,” including trace ⁎-polynomial maps and operator functions associated to the noncommutative C²scalar functions R → C introduced by Nikitopoulos, as well as the more general multivariate tracial noncommutative C²functions introduced by Jekel, Li, and Shlyakhtenko.