On pathwise counterparts of Doob's maximal inequalities

TL;DR

This paper develops pathwise versions of Doob's maximal inequalities for submartingales and supermartingales, providing deterministic bounds that complement classical probabilistic results.

Contribution

It introduces novel pathwise inequalities that serve as deterministic counterparts to traditional probabilistic maximal inequalities for martingales.

Findings

Pathwise inequalities for submartingales and supermartingales

Deterministic bounds complement probabilistic maximal inequalities

Framework applicable to various stochastic processes

Abstract

In this paper, we present pathwise counterparts of Doob's maximal inequalities (on the probability of exceeding a level) for submartingales and supermartingales.