Maximal Inequalities and Some Applications

TL;DR

This paper reviews maximal inequalities for various stochastic processes, exploring their theoretical properties and connections to analysis, with applications across different classes of processes and stochastic calculus.

Contribution

It provides a comprehensive survey of maximal inequalities for multiple classes of stochastic processes, including new insights into their relations with harmonic analysis tools.

Findings

Maximal inequalities hold for martingales, Lévy processes, and solutions to SDEs.

Connections between probabilistic maximal estimates and Hardy-Littlewood maximal functions are established.

The survey unifies various maximal inequalities under a common framework.

Abstract

A maximal inequality is an inequality which involves the (absolute) supremum $\sup_{s\leq t}|X_{s}|$ or the running maximum $\sup_{s\leq t}X_{s}$ of a stochastic process $(X_t)_{t\geq 0}$. We discuss maximal inequalities for several classes of stochastic processes with values in an Euclidean space: Martingales, L\'evy processes, L\'evy-type - including Feller processes, (compound) pseudo Poisson processes, stable-like processes and solutions to SDEs driven by a L\'evy process -, strong Markov processes and Gaussian processes. Using the Burkholder-Davis-Gundy inequalities we als discuss some relations between maximal estimates in probability and the Hardy-Littlewood maximal functions from analysis. This paper has been accepted for publication in Probability Surveys