Martingale decompositions and weak differential subordination in UMD Banach spaces

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 ∈ (1, ∞), 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, M₀ ^c = 0 and EM_∞ ^{d p} + EM_∞ ^{c p} ≤ c_p,XEM_∞ ^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 ∈ (1, ∞) and for all X-valued martingales M and N such that N is weakly differentially subordinated to M, one has the estimate EN_∞ ^p ≤ C_p,XEM_∞ ^p