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Asymptotic behaviour of first passage time distributions for Lévy processes

Le : 09/07/2012 11h00
Par : Víctor Rivero (CIMAT Guanajuato)
Lieu : I 103
Lien web :
Résumé : Let X be a real valued Lévy process that is in the domain of attraction of a stable law without centering with norming function c. As an analogue of the random walk results obtained by Vatutin and Watchel and Doney we study the local behaviour of the distribution of the lifetime Z under the characteristic measure N of excursions away from zero of the process X reflected in its past infimum, and of the first passage time of $X$ below zero T_{0}=\inf \{t>0:X_{t}<0\}, when X starts from x, in two different regimes for x, viz. x=o(c(t)) and x>D c(t), for some D>0. We sharpen our estimates by distinguishing between two types of path behaviour, viz. continuous passage at T_{0} and discontinuous passage. In order to prove our main results we establish some sharp local estimates for the entrance law of the excursion process associated to X reflected in its past infimum.