• 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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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

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

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

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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


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.

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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.