On the maximal inequalities of Burkholder, Davis and Gundy

TL;DR

This paper provides a stochastic calculus-based proof of the maximal inequalities for real and Hilbert-space-valued local martingales, with some original contributions in the infinite-dimensional case.

Contribution

It offers a new proof approach for the maximal inequalities of Burkholder, Davis, and Gundy, including original parts for the infinite-dimensional setting.

Findings

Proof using stochastic calculus for real and Hilbert-space martingales

Original methods introduced for infinite-dimensional case

Clarification of the inequalities' applicability in different settings

Abstract

We give a proof of the maximal inequalities of Burkholder, Davis and Gundy for real as well as Hilbert-space-valued local martingales using almost only stochastic calculus. Some parts of the exposition, especially in the infinite dimensional case, appear to be original.