On the martingale decompositions of Gundy, Meyer, and Yoeurp in infinite dimensions

Ivan Yaroslavtsev

doi:10.1214/18-AIHP940

On the martingale decompositions of Gundy, Meyer, and Yoeurp in infinite dimensions

Ivan Yaroslavtsev^*

^*Corresponding author for this work

Analysis

Research output: Contribution to journal › Article › Scientific › peer-review

3 Citations (Scopus)

Abstract

We show that the canonical decomposition (comprising both the Meyer–Yoeurp and the Yoeurp decompositions) of a general X-valued local martingale is possible if and only if X has the UMD property. More precisely, X is a UMD Banach space if and only if for any X-valued local martingale M there exist a continuous local martingale M^c, a purely discontinuous quasi-left continuous local martingale M^q, and a purely discontinuous local martingale M^a with accessible jumps such that M = M^c + M^q + M^a. The corresponding weak L¹-estimates are provided. Important tools used in the proof are a new version of Gundy’s decomposition of continuous-time martingales and weak L¹-bounds for a certain class of vector-valued continuous-time martingale transforms.

Original language	English
Pages (from-to)	1988-2018
Number of pages	31
Journal	Annales de l'institut Henri Poincare (B) Probability and Statistics
Volume	55
Issue number	4
DOIs	https://doi.org/10.1214/18-AIHP940
Publication status	Published - 2019

Keywords

Canonical decomposition
Continuous-time martingales
Gundy’s decomposition
Meyer–Yoeurp decomposition
UMD spaces
Weak differential subordination
Weak estimates
Yoeurp decomposition

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abstract = "We show that the canonical decomposition (comprising both the Meyer–Yoeurp and the Yoeurp decompositions) of a general X-valued local martingale is possible if and only if X has the UMD property. More precisely, X is a UMD Banach space if and only if for any X-valued local martingale M there exist a continuous local martingale Mc, a purely discontinuous quasi-left continuous local martingale Mq, and a purely discontinuous local martingale Ma with accessible jumps such that M = Mc + Mq + Ma. The corresponding weak L1-estimates are provided. Important tools used in the proof are a new version of Gundy{\textquoteright}s decomposition of continuous-time martingales and weak L1-bounds for a certain class of vector-valued continuous-time martingale transforms.",

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T1 - On the martingale decompositions of Gundy, Meyer, and Yoeurp in infinite dimensions

AU - Yaroslavtsev, Ivan

PY - 2019

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N2 - We show that the canonical decomposition (comprising both the Meyer–Yoeurp and the Yoeurp decompositions) of a general X-valued local martingale is possible if and only if X has the UMD property. More precisely, X is a UMD Banach space if and only if for any X-valued local martingale M there exist a continuous local martingale Mc, a purely discontinuous quasi-left continuous local martingale Mq, and a purely discontinuous local martingale Ma with accessible jumps such that M = Mc + Mq + Ma. The corresponding weak L1-estimates are provided. Important tools used in the proof are a new version of Gundy’s decomposition of continuous-time martingales and weak L1-bounds for a certain class of vector-valued continuous-time martingale transforms.

AB - We show that the canonical decomposition (comprising both the Meyer–Yoeurp and the Yoeurp decompositions) of a general X-valued local martingale is possible if and only if X has the UMD property. More precisely, X is a UMD Banach space if and only if for any X-valued local martingale M there exist a continuous local martingale Mc, a purely discontinuous quasi-left continuous local martingale Mq, and a purely discontinuous local martingale Ma with accessible jumps such that M = Mc + Mq + Ma. The corresponding weak L1-estimates are provided. Important tools used in the proof are a new version of Gundy’s decomposition of continuous-time martingales and weak L1-bounds for a certain class of vector-valued continuous-time martingale transforms.

KW - Canonical decomposition

KW - Continuous-time martingales

KW - Gundy’s decomposition

KW - Meyer–Yoeurp decomposition

KW - UMD spaces

KW - Weak differential subordination

KW - Weak estimates

KW - Yoeurp decomposition

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JO - Annales de l'institut Henri Poincare (B) Probability and Statistics

JF - Annales de l'institut Henri Poincare (B) Probability and Statistics

IS - 4

ER -

On the martingale decompositions of Gundy, Meyer, and Yoeurp in infinite dimensions

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Keywords

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