Martingale decompositions and weak differential subordination in UMD Banach spaces

Ivan S. Yaroslavtsev*

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

6 Citations (Scopus)
62 Downloads (Pure)

Abstract

In this paper, we consider Meyer–Yoeurp decompositions for UMD Banach space-valued martingales. Namely, we prove that X is a UMD Banach space if and only if for any fixed p ∈ (1, ∞), any X-valued Lp-martingale M has a unique decomposition M = Md + Mc such that Md is a purely discontinuous martingale, Mc is a continuous martingale, M0 c = 0 and EM d p + EM c p ≤ cp,XEM p. An analogous assertion is shown for the Yoeurp decomposition of a purely discontinuous martingales into a sum of a quasi-left continuous martingale and a martingale with accessible jumps. As an application, we show that X is a UMD Banach space if and only if for any fixed p ∈ (1, ∞) and for all X-valued martingales M and N such that N is weakly differentially subordinated to M, one has the estimate EN p ≤ Cp,XEM p

Original languageEnglish
Pages (from-to)1659-1689
Number of pages31
JournalBernoulli
Volume25
Issue number3
DOIs
Publication statusPublished - 2019

Keywords

  • Accessible jumps
  • Brownian representation
  • Burkholder function
  • Canonical decomposition of martingales
  • Continuous martingales
  • Differential subordination
  • Meyer–Yoeurp decomposition
  • Purely discontinuous martingales
  • Quasi-left continuous
  • Stochastic integration
  • UMD Banach spaces
  • Weak differential subordination
  • Yoeurp decomposition

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