Maximal inequalities and some applications

Abstract

A maximal inequality is an inequality which involves the (absolute) supremum sups⩽t|Xs| or the running maximum sups⩽tXs of a stochastic process (Xt)t⩾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 also discuss some relations between maximal estimates in probability and the Hardy–Littlewood maximal functions from analysis.