Burkholder-Gundy-Davis Inequality in Martingale Hardy Spaces with Variable Exponent

TL;DR

This paper extends classical martingale inequalities to variable exponent Lebesgue spaces, broadening the theoretical framework and providing new tools for stochastic analysis with variable exponents.

Contribution

It introduces variable exponent analogues of key martingale inequalities and explores their relations within martingale Hardy spaces, extending classical results.

Findings

Variable exponent Burkholder-Gundy-Davis inequality established

New relations between variable exponent martingale Hardy spaces identified

Extension of Dellacherie's theorem to variable exponent spaces

Abstract

In this paper, the classical Dellacherie's theorem about stochastic process is extended to variable exponent Lebesgue spaces. As its applications, we obtain variable exponent analogues of several famous inequalities in classical martingale theory, including convexity lemma, Burkholder-Gundy-Davis' inequality and Chevalier's inequality. Moreover, we investigate some other equivalent relations between variable exponent martingale Hardy spaces.