Pointwise properties of martingales with values in Banach function spaces

TL;DR

This paper investigates local martingales in UMD Banach function spaces, establishing their representation as martingale fields and deriving a new Burkholder--Davis--Gundy inequality.

Contribution

It introduces a novel representation of martingales as fields and presents a new inequality extending classical results to Banach function spaces.

Findings

Martingales have a version as martingale fields in UMD Banach spaces.

A new Burkholder--Davis--Gundy type inequality is established.

Results extend classical martingale inequalities to Banach function spaces.

Abstract

In this paper we consider local martingales with values in a UMD Banach function space. We prove that such martingales have a version which is a martingale field. Moreover, a new Burkholder--Davis--Gundy type inequality is obtained.