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2014 A maximal inequality for supermartingales
Bruce Hajek
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Electron. Commun. Probab. 19: 1-10 (2014). DOI: 10.1214/ECP.v19-3237

Abstract

A tight upper bound is given on the distribution of the maximum of a supermartingale. Specifically, it is shown that if $Y$ is a semimartingale with initial value zero and quadratic variation process $[Y,Y]$ such that $Y + [Y,Y]$ is a supermartingale, then the probability the maximum of $Y$ is greater than or equal to a positive constant $a$ is less than or equal to$1/(1+a).$ The proof makes use of the semimartingale calculus and is inspired by dynamic programming.

Citation

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Bruce Hajek. "A maximal inequality for supermartingales." Electron. Commun. Probab. 19 1 - 10, 2014. https://doi.org/10.1214/ECP.v19-3237

Information

Accepted: 14 August 2014; Published: 2014
First available in Project Euclid: 7 June 2016

zbMATH: 1314.60089
MathSciNet: MR3254734
Digital Object Identifier: 10.1214/ECP.v19-3237

Subjects:
Primary: 60G40
Secondary: 60G42, 60G44, 93E20

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