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

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 L¹-estimates are provided. Important tools used in the proof are a new version of Gundy’s decomposition of continuous-time martingales and weak L¹-bounds for a certain class of vector-valued continuous-time martingale transforms.