A Maximal Inequality for Supermartingales

TL;DR

This paper establishes a precise upper bound on the probability that the maximum of a certain class of supermartingales exceeds a threshold, using semimartingale calculus and dynamic programming techniques.

Contribution

It provides a new, tight inequality for the distribution of supermartingale maxima, extending the understanding of their probabilistic behavior.

Findings

The probability that the maximum exceeds a threshold is bounded by 1/(1+a).

The result applies to semimartingales with specific supermartingale properties.

The proof employs semimartingale calculus and dynamic programming methods.

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.