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

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.