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

Journal article (2019)

Authors

I.S. Yaroslavtsev Analysis - EEMCS
Research Group
Analysis (EEMCS) (TU Delft)
UMD spaces Meyer–Yoeurp decomposition Weak differential subordination Yoeurp decomposition Canonical decomposition Continuous-time martingales Gundy’s decomposition Weak estimates
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Published Date

2019

Language

English

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Faculty

Electrical Engineering, Mathematics and Computer Science

Department

Delft Institute of Applied Mathematics

Research Group

Analysis

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 Mc, a purely discontinuous quasi-left continuous local martingale Mq, and a purely discontinuous local martingale Ma with accessible jumps such that M = Mc + Mq + Ma. 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.