Martingale decompositions in UMD Banach spaces

Abstract

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).Non UBCUnreviewedAuthor affiliation: Delft University of TechnologyGraduat