Martingale decompositions and weak differential subordination in UMD Banach spaces

Yaroslavtsev, I.S.

doi:10.3150/18-BEJ1031

Martingale decompositions and weak differential subordination in UMD Banach spaces

Title

Martingale decompositions and weak differential subordination in UMD Banach spaces

Author

Yaroslavtsev, I.S. (TU Delft Analysis)

Date

2019

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