Burkholder's submartingales from a stochastic calculus perspective

TL;DR

This paper offers a simplified proof and generalizations of a recent characterization of certain continuous submartingales and supermartingales using elementary stochastic calculus, with applications to Burkholder-Davis-Gundy inequalities.

Contribution

It introduces a straightforward proof and broader generalizations of a key result relating submartingales to Brownian motion and its maximum, enhancing understanding and applications.

Findings

Explicit expressions for constants in Burkholder-Davis-Gundy inequalities

Connection established between submartingales and balayage formulae

Generalizations of submartingale characterizations

Abstract

We provide a simple proof, as well as several generalizations, of a recent result by Davis and Suh, characterizing a class of continuous submartingales and supermartingales that can be expressed in terms of a squared Brownian motion and of some appropriate powers of its maximum. Our techniques involve elementary stochastic calculus, as well as the Doob-Meyer decomposition of continuous submartingales. These results can be used to obtain an explicit expression of the constants appearing in the Burkholder-Davis-Gundy inequalities. A connection with some balayage formulae is also established.