A martingale approach to noncommutative stochastic calculus

TL;DR

This paper develops a martingale-based framework for noncommutative stochastic calculus, introducing stochastic integration, quadratic variation, and Itô's formula for noncommutative processes including q-Brownian motions and matrix-valued Brownian motions.

Contribution

It presents a novel martingale approach to noncommutative stochastic calculus, establishing fundamental tools like inequalities and Itô's formula for a broad class of noncommutative processes.

Findings

Established Burkholder--Davis--Gundy inequalities for noncommutative martingales

Developed a noncommutative Itô's formula for operator functions

Introduced a general theory of stochastic integration and quadratic variation in noncommutative setting

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 $p \geq 2$) for continuous-time noncommutative martingales and a noncommutative It\^{o}'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.