Vector-valued decoupling and the Burkholder-Davis-Gundy inequality

TL;DR

This paper establishes p-independence of a one-sided decoupling inequality for Banach space-valued tangent martingales and derives Burkholder-Davis-Gundy type estimates for stochastic integrals in UMD and related spaces.

Contribution

It proves p-independence of the one-sided decoupling inequality and extends Burkholder-Davis-Gundy estimates to broader classes of Banach spaces.

Findings

p-independence of the one-sided decoupling inequality

Optimal constants for various Banach spaces

Burkholder-Davis-Gundy estimates for stochastic integrals

Abstract

Let X be a Banach space. We prove p-independence of the one-sided decoupling inequality for X-valued tangent martingales as introduced by Kwapien and Woyczynski. It is known that a Banach space X satisfies the two-sided decoupling inequality if and only if X is a UMD Banach space. The one-sided decoupling inequality is a weaker property, including e.g. the space L^1. We provide information on the optimal constants for various spaces, and give a upper estimate of order p in general. In the second part of our paper we derive Burkholder-Davis-Gundy type estimates for p-th moments, p in (0,infty), of X-valued stochastic integrals, provided X is a UMD Banach space or a space in which the one-sided decoupling inequality holds.