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November 2011 Sharp maximal inequalities for the moments of martingales and non-negative submartingales
Adam Osȩkowski
Bernoulli 17(4): 1327-1343 (November 2011). DOI: 10.3150/10-BEJ314

Abstract

In the paper we study sharp maximal inequalities for martingales and non-negative submartingales: if $f$, $g$ are martingales satisfying \[ |\mathrm{d}g_n|\leq|\mathrm{d}f_n|,\qquad n=0, 1, 2, \ldots, \] 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 \[ |\mathrm{d}g_n|\leq|\mathrm{d}f_n|\quad \mbox{and}\quad |\mathbb{E}(\mathrm{d}g_{n+1}|\mathcal {F}_n)|\leq\alpha\mathbb{E} (\mathrm{d}f_{n+1}|\mathcal{F}_n),\qquad n=0, 1, 2, \ldots, \] 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 Itô processes. The inequalities strengthen the earlier classical results of Burkholder and Choi.

Citation

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Adam Osȩkowski. "Sharp maximal inequalities for the moments of martingales and non-negative submartingales." Bernoulli 17 (4) 1327 - 1343, November 2011. https://doi.org/10.3150/10-BEJ314

Information

Published: November 2011
First available in Project Euclid: 4 November 2011

MathSciNet: MR2854774
Digital Object Identifier: 10.3150/10-BEJ314

Keywords: Differential subordination , martingale , maximal function , maximal inequality , submartingale

Rights: Copyright © 2011 Bernoulli Society for Mathematical Statistics and Probability

Vol.17 • No. 4 • November 2011
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