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 $\ast$-polynomial maps and operator functions associated to the noncommutative $C^2$ scalar functions $\mathbb{R}\to \mathbb{C}$ introduced by Nikitopoulos, as well as the more general multivariate tracial noncommutative $C^2$ functions introduced by Jekel, Li, and Shlyakhtenko.

75 pages; updated to agree with the published version

References in corpus (8)

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• Non-commutative martingale inequalities
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