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A martingale approach to noncommutative stochastic calculus

David A. Jekel, Todd A. Kemp, Evangelos A. Nikitopoulos

TL;DR

This work advances noncommutative stochastic calculus by adopting a classical martingale framework to derive a universal theory for stochastic integration, quadratic variation, BDG-type inequalities, and Itô’s formula in NC probability. It introduces $L^2$-decomposable processes as the NC counterpart of semimartingales and constructs stochastic integrals against them, together with a notion of NC quadratic covariation that supports a robust Itô product rule. The main contributions include continuous-time NC BDG inequalities for martingales, a noncommutative Itô formula for adapted $C^2$ maps (extending trace polynomials and tracial $C^k$ maps), and a unifying perspective that recovers known NC Itô formulas as special cases, including those for $q$-Brownian and matrix Brownian motions. The results provide new tools for NC stochastic differential equations and have potential applications to random matrix theory and free probability, offering a universal calculus that parallels the classical theory while accommodating noncommutativity through trace-polynomial and MOI-based constructions.

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 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.

A martingale approach to noncommutative stochastic calculus

TL;DR

This work advances noncommutative stochastic calculus by adopting a classical martingale framework to derive a universal theory for stochastic integration, quadratic variation, BDG-type inequalities, and Itô’s formula in NC probability. It introduces -decomposable processes as the NC counterpart of semimartingales and constructs stochastic integrals against them, together with a notion of NC quadratic covariation that supports a robust Itô product rule. The main contributions include continuous-time NC BDG inequalities for martingales, a noncommutative Itô formula for adapted maps (extending trace polynomials and tracial maps), and a unifying perspective that recovers known NC Itô formulas as special cases, including those for -Brownian and matrix Brownian motions. The results provide new tools for NC stochastic differential equations and have potential applications to random matrix theory and free probability, offering a universal calculus that parallels the classical theory while accommodating noncommutativity through trace-polynomial and MOI-based constructions.

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 -Brownian motions and classical matrix-valued Brownian motions. As applications, we obtain Burkholder--Davis--Gundy inequalities (with ) for continuous-time noncommutative martingales and a noncommutative Itô's formula for "adapted maps," including trace -polynomial maps and operator functions associated to the noncommutative scalar functions introduced by Nikitopoulos, as well as the more general multivariate tracial noncommutative functions introduced by Jekel, Li, and Shlyakhtenko.
Paper Structure (27 sections, 68 theorems, 207 equations)

This paper contains 27 sections, 68 theorems, 207 equations.

Table of Contents

  1. Introduction
  2. Philosophy of the approach
  3. Summary of main results
  4. Preliminaries
  5. Noncommutative probability
  6. Multilinear maps on
  7. Trace -polynomials
  8. Noncommutative processes
  9. Filtrations and adaptedness
  10. Decomposable processes
  11. Stochastic integrals
  12. Stieltjes integrals
  13. Integration against -decomposable processes
  14. Tools to calculate stochastic integrals
  15. Quadratic covariation
  16. ...and 12 more sections

Key Result

Theorem 1.2

If $X$ is a semimartingale and $H$ is an adapted continuous process, then there exists a unique-up-to-indistinguishability semimartingale $\int_0^{\boldsymbol{\cdot}} H_s \,\mathrm{d} X_s$ such that for all $t \geq 0$, The limit above is a limit in probability as $|\Pi| \to 0$. We call $\int_0^{\boldsymbol{\cdot}} H_s \,\mathrm{d} X_s$ the stochastic integral of $H$ with respect to $X$.

Theorems & Definitions (142)

  • Theorem 1.2: Stochastic integral
  • Theorem 1.2: Stochastic integral
  • Remark 1.3
  • Theorem 1.4: Quadratic covariation
  • Theorem 1.5: Itô's formula
  • Example 1.6
  • Theorem 1.7: Noncommutative stochastic integral
  • Example 1.8
  • Theorem 1.9: Noncommutative quadratic covariation
  • Remark 1.10
  • ...and 132 more