Maximal inequalities and some applications

A maximal inequality is an which involves the (absolute) supremum sups⩽t|Xs| or running maximum sups⩽tXs of a stochastic process (Xt)t⩾0. We discuss inequalities for several classes processes with values in Euclidean space: Martingales, Lévy processes, Lévy-type – including Feller (compound) pseudo Poisson stable-like and solutions to SDEs driven by –, strong Markov Gaussian processes. Using Burkholder–Davis–Gundy we also some relations between estimates probability Hardy–Littlewood functions from analysis.