Maximal Inequalities and Some Applications
Abstract
A maximal inequality is an inequality which involves the (absolute) supremum s≤ t|Xs| or the running maximum s≤ 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, 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
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