A Maximal Inequality for Supermartingales

Bruce Hajek

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arXiv · 0911.4444

A Maximal Inequality for Supermartingales

Bruce Hajek

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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.

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Bruce Hajek. 2009-11-23. A Maximal Inequality for Supermartingales. https://arxiv.org/abs/0911.4444

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