A matrix Burkholder-Davis-Gundy inequality

TL;DR

This paper establishes a matrix-valued Burkholder-Davis-Gundy inequality, extending classical stochastic integral inequalities to the non-commutative matrix setting using Freedman's inequality.

Contribution

It introduces a non-commutative version of the Burkholder-Davis-Gundy inequality for matrix stochastic integrals, extending existing inequalities to a broader matrix-valued context.

Findings

Proves a spectral norm inequality for matrix stochastic integrals.

Extends non-commutative Khintchine inequality to stochastic integrals.

Uses Freedman's inequality for matrix martingales in the proof.

Abstract

We prove an inequality for the spectral norm of matrix valued stochastic integrals. This inequality can be seen either as a non-commutative version of the Burkholder-Davis-Gundy inequality or as an extension of the non-commutative Khintchine inequality of Lust-Piquard to stochastic integrals. The proof relies on a version of Freedman's inequality for matrix valued martingales.