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arXiv · 1712.00401

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

Ivan Yaroslavtsev
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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 $M^c$, 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 $L^1$-estimates are provided. Important tools used in the proof are a new version of Gundy's decomposition of continuous-time martingales and weak $L^1$-bounds for a certain class of vector-valued continuous-time martingale transforms.

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Ivan Yaroslavtsev. 2017-12-01. On the martingale decompositions of Gundy, Meyer, and Yoeurp in infinite dimensions. https://arxiv.org/abs/1712.00401

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