Maximal Inequalities and Some Applications

Abstract

A maximal inequality is an inequality which involves the (absolute) supremum ${\sup }_{s\le t}|X_s|$ or the running maximum ${\sup }_{s\le t}{X}_s$ of a stochastic process $(X_t{)}_{t\ge 0}$. We discuss maximal inequalities for several classes of stochastic processes with values in an Euclidean space: Martingales, Lévy processes, Lévy-type - including Feller processes, (compound) pseudo Poisson processes, stable-like processes and solutions to SDEs driven by a Lévy 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