Bernoulli

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

Martingale decompositions and weak differential subordination in UMD Banach spaces

Ivan S. Yaroslavtsev

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

https://projecteuclid.org/euclid.bj/1560326423

Digital Object Identifier
doi:10.3150/18-BEJ1031

Mathematical Reviews number (MathSciNet)
MR3961226

Zentralblatt MATH identifier
07066235

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.