Martingale decompositions and weak differential subordination in UMD Banach spaces

Journal article (2019)

Authors

I.S. Yaroslavtsev Analysis - EEMCS
Research Group
Analysis (EEMCS) (TU Delft)
Copyright: © 2019 I.S. Yaroslavtsev
DOI: https://doi.org/10.3150/18-BEJ1031
UMD Banach spaces 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 Weak differential subordination Yoeurp decomposition
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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

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