On the martingale decompositions of Gundy, Meyer, and Yoeurp in infinite dimensions

TL;DR

This paper establishes that the canonical martingale decomposition in infinite-dimensional Banach spaces is possible if and only if the space has the UMD property, linking geometric space properties with martingale theory.

Contribution

It proves the equivalence between the UMD property of Banach spaces and the existence of canonical martingale decompositions for vector-valued local martingales.

Findings

Canonical decomposition exists iff the space is UMD

Provides weak L^1-estimates for martingale transforms

Introduces a new Gundy's decomposition for continuous-time martingales

Abstract

We show that the canonical decomposition (comprising both the Meyer-Yoeurp and the Yoeurp decompositions) of a general $X$-valued local martingale is possible if and only if $X$ has the UMD property. More precisely, $X$ is a UMD Banach space if and only if for any $X$-valued local martingale $M$ there exist a continuous local martingale $M^c$, a purely discontinuous quasi-left continuous local martingale $M^q$, and a purely discontinuous local martingale $M^a$ with accessible jumps such that $M = M^c + M^q + M^a$. The corresponding weak $L^1$-estimates are provided. Important tools used in the proof are a new version of Gundy's decomposition of continuous-time martingales and weak $L^1$-bounds for a certain class of vector-valued continuous-time martingale transforms.