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

Ivan Yaroslavtsev

Research output: Contribution to journalArticleScientificpeer-review

3 Citations (Scopus)

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.

Original languageEnglish
Pages (from-to)1988-2018
Number of pages31
JournalAnnales de l'institut Henri Poincare (B) Probability and Statistics
Volume55
Issue number4
DOIs
Publication statusPublished - 2019

Keywords

  • Canonical decomposition
  • Continuous-time martingales
  • Gundy’s decomposition
  • Meyer–Yoeurp decomposition
  • UMD spaces
  • Weak differential subordination
  • Weak estimates
  • Yoeurp decomposition

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