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Martingale decompositions in UMD Banach spaces Yaroslavtsev, Ivan
Description
In this talk we present the Meyer-Yoeurp decomposition 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 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 $\mathbb E \|M^d_{\infty}\|^p + \mathbb E \|M^c_{\infty}\|^p\leq c_p \mathbb E \|M_{\infty}\|^p$. An analogous assertion is shown for the Yoeurp decomposition of a purely discontinuous martingale into a sum of a quasi-left continuous martingale and a martingale with accessible jumps. Meyer-Yoeurp and Yoeurp decompositions play a significant role in stochastic integration theory for càdlàg martingales, For instance one can show sharp estimates for an $L^p$-norm of an $L^q$-valued stochastic integral with respect to a general local martingale. An important tool for obtaining these estimates are the recently proven Burkholder-Rosenthal-type inequalities for discrete $L^q$-valued martingales. This talk is partially based on joint work with Sjoerd Dirksen (RWTH Aachen University).
Item Metadata
Title |
Martingale decompositions in UMD Banach spaces
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Creator | |
Publisher |
Banff International Research Station for Mathematical Innovation and Discovery
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Date Issued |
2017-06-01T15:49
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Description |
In this talk we present the Meyer-Yoeurp decomposition 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 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 $\mathbb E \|M^d_{\infty}\|^p + \mathbb E \|M^c_{\infty}\|^p\leq c_p \mathbb E \|M_{\infty}\|^p$. An analogous assertion is shown for the Yoeurp decomposition of a purely discontinuous martingale into a sum of a quasi-left continuous martingale and a martingale with accessible jumps.
Meyer-Yoeurp and Yoeurp decompositions play a significant role in stochastic integration theory for càdlàg martingales, For instance one can show sharp estimates for an $L^p$-norm of an $L^q$-valued stochastic integral with respect to a general local martingale. An important tool for obtaining these estimates are the recently proven Burkholder-Rosenthal-type inequalities for discrete $L^q$-valued martingales.
This talk is partially based on joint work with Sjoerd Dirksen (RWTH Aachen University).
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Extent |
32 minutes
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Subject | |
Type | |
File Format |
video/mp4
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Language |
eng
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Notes |
Author affiliation: Delft University of Technology
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Series | |
Date Available |
2017-11-28
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Provider |
Vancouver : University of British Columbia Library
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Rights |
Attribution-NonCommercial-NoDerivatives 4.0 International
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DOI |
10.14288/1.0360767
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URI | |
Affiliation | |
Peer Review Status |
Unreviewed
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Scholarly Level |
Graduate
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Rights URI | |
Aggregated Source Repository |
DSpace
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Rights
Attribution-NonCommercial-NoDerivatives 4.0 International