A Maximal Inequality for Supermartingales

Details A Maximal Inequality for Supermartingales A Maximal Inequality for Supermartingales

2009-11-23

arxiv.org/abs/0911.4444v1

Mathematics

Probability

8 pages, no figures

Scientific paper

A tight upper bound is given involving 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 is inspired by dynamic programming. Complements and extensions are also given.

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