2023/08/18 by Jekel, David A., Kemp, Todd A., Nikitopoulos, Evangelos A.
#46L54 #60H05 (Primary) 46L52 (Secondary) #FOS: Mathematics #Operator Algebras (math.OA) #Probability (math.PR)
paper · doi:10.48550/arxiv.2308.09856
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) for continuous-time noncommutative martingales and a noncommutative Itô's formula for "adapted C2 maps," including trace ∗-polynomial maps and operator functions associated to the noncommutative C2 scalar functions ℝ → ℂ introduced by Nikitopoulos, as well as the more general multivariate tracial noncommutative C2 functions introduced by Jekel, Li, and Shlyakhtenko.