Bernoulli

  • Bernoulli
  • Volume 25, Number 3 (2019), 1659-1689.

Martingale decompositions and weak differential subordination in UMD Banach spaces

Ivan S. Yaroslavtsev

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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 $p\in(1,\infty)$, any $X$-valued $L^{p}$-martingale $M$ has a unique decomposition $M=M^{d}+M^{c}$ such that $M^{d}$ is a purely discontinuous martingale, $M^{c}$ is a continuous martingale, $M^{c}_{0}=0$ and \[\mathbb{E}\big\|M^{d}_{\infty}\big\|^{p}+\mathbb{E}\big\|M^{c}_{\infty}\big\|^{p}\leq c_{p,X}\mathbb{E}\|M_{\infty}\|^{p}.\] 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 $p\in(1,\infty)$ and for all $X$-valued martingales $M$ and $N$ such that $N$ is weakly differentially subordinated to $M$, one has the estimate $\mathbb{E}\|N_{\infty}\|^{p}\leq C_{p,X}\mathbb{E}\|M_{\infty}\|^{p}$.

Article information

Source
Bernoulli, Volume 25, Number 3 (2019), 1659-1689.

Dates
Received: June 2017
Revised: February 2018
First available in Project Euclid: 12 June 2019

Permanent link to this document
https://projecteuclid.org/euclid.bj/1560326423

Digital Object Identifier
doi:10.3150/18-BEJ1031

Mathematical Reviews number (MathSciNet)
MR3961226

Zentralblatt MATH identifier
07066235

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

Citation

Yaroslavtsev, Ivan S. Martingale decompositions and weak differential subordination in UMD Banach spaces. Bernoulli 25 (2019), no. 3, 1659--1689. doi:10.3150/18-BEJ1031. https://projecteuclid.org/euclid.bj/1560326423


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Supplemental materials

  • Some proofs. Recall that throughout the paper many technical proofs have been omitted. The reader can find those proofs in the supplementary file.