Sharp maximal inequalities for the moments of martingales and non-negative submartingales

TL;DR

This paper establishes sharp maximal inequalities for martingales and non-negative submartingales, providing optimal bounds for their moments and extending classical results with new inequalities and applications to stochastic integrals.

Contribution

The paper introduces new sharp maximal inequalities for martingales and submartingales, extending classical results and providing optimal bounds with applications to stochastic calculus.

Findings

Proved sharp bounds for martingale maximal functions for p≥2.

Extended inequalities to non-negative submartingales with additional conditions.

Applied results to stochastic integrals and Itô processes.

Abstract

In the paper we study sharp maximal inequalities for martingales and non-negative submartingales: if $f$, $g$ are martingales satisfying \[|\mathrm{d}g_n|\leq|\mathrm{d}f_n|,\qquad n=0,1,2,...,\] almost surely, then \[\Bigl\|\sup_{n\geq0}|g_n|\Bigr\|_p\leq p\|f\|_p,\qquad p\geq2,\] and the inequality is sharp. Furthermore, if $\alpha\in[0,1]$, $f$ is a non-negative submartingale and $g$ satisfies \[|\mathrm{d}g_n|\leq|\mathrm{d}f_n|\quad and\quad |\mathbb{E}(\mathrm{d}g_{n+1}|\mathcal {F}_n)|\leq\alpha\mathbb{E}(\mathrm{d}f_{n+1}|\mathcal{F}_n),\qquad n=0,1,2,...,\] almost surely, then \[\Bigl\|\sup_{n\geq0}|g_n|\Bigr\|_p\leq(\alpha+1)p\|f\|_p,\qquad p\geq2,\] and the inequality is sharp. As an application, we establish related estimates for stochastic integrals and It\^{o} processes. The inequalities strengthen the earlier classical results of Burkholder and Choi.