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

In the paper we study sharp maximal inequalities for martingales and non-negative submartingales: if $f$, $g$ are martingales satisfying $$\mathrm{\mid }\mathrm{d}{g}_n\mathrm{\mid }\le \mathrm{\mid }\mathrm{d}{f}_n\mathrm{\mid },n=0,1,2,\mathrm{.}\mathrm{.}\mathrm{.},$$ almost surely, then $$\parallel \underset{n\ge 0}{\sup }\mathrm{\mid }{g}_n\mathrm{\mid }{\parallel }_p\le p\mathrm{\parallel }f{\mathrm{\parallel }}_p,p\ge 2,$$ and the inequality is sharp. Furthermore, if $\alpha \in [0,1]$, $f$ is a non-negative submartingale and $g$ satisfies $$\mathrm{\mid }\mathrm{d}{g}_n\mathrm{\mid }\le \mathrm{\mid }\mathrm{d}{f}_n\mathrm{\mid }and\mathrm{\mid }\mathbb{E}(\mathrm{d}{g}_{n+1}\mathrm{\mid }{\mathcal{F}}_n)\mathrm{\mid }\le \alpha \mathbb{E}(\mathrm{d}{f}_{n+1}\mathrm{\mid }{\mathcal{F}}_n),n=0,1,2,\mathrm{.}\mathrm{.}\mathrm{.},$$ almost surely, then $$\parallel \underset{n\ge 0}{\sup }\mathrm{\mid }{g}_n\mathrm{\mid }{\parallel }_p\le (\alpha +1)p\mathrm{\parallel }f{\mathrm{\parallel }}_p,p\ge 2,$$ 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.