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 --...

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Bibliographic Details
Published in:arXiv.org 2023-08
Main Authors: Jekel, David A, Kemp, Todd A, Nikitopoulos, Evangelos A
Format: Article
Language:English
Subjects:
Functions (mathematics)
Martingales
Mathematical analysis
Operators (mathematics)
Polynomials
Online Access:Get full text
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Summary: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 \(n \times n\) 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} 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-Shlyakhtenko.
ISSN:2331-8422