Small and large time stability of the time taken for a Lévy process to cross curved boundaries
Annales de l'I.H.P. Probabilités et statistiques, Tome 49 (2013) no. 1, pp. 208-235.

Ce papier traite du comportement en temps court d’un processus de Lévy X. En particulier, nous étudions la stabilité des temps T ¯ b (r) et T b * (r) auxquels X, partant de X 0 =0, quitte pour la première fois les domaines {(t,y) 2 :yrt b ,t0} (sortie unilatérale), ou {(t,y) 2 :|y|rt b ,t0} (sortie bilatérale), 0b<1, quand r0. Nous déterminons si ces temps de passage se comportent ou non comme des fonctions déterministes selon différents modes de convergence : en probabilité, presque sûrement et dans L p . Dans de nombreux cas, ceci est équivalent à la stabilité du processus X. Le problème analogue à temps grand est aussi discuté.

This paper is concerned with the small time behaviour of a Lévy process X. In particular, we investigate the stabilities of the times, T ¯ b (r) and T b * (r), at which X, started with X 0 =0, first leaves the space-time regions {(t,y) 2 :yrt b ,t0} (one-sided exit), or {(t,y) 2 :|y|rt b ,t0} (two-sided exit), 0b<1, as r0. Thus essentially we determine whether or not these passage times behave like deterministic functions in the sense of different modes of convergence; specifically convergence in probability, almost surely and in L p . In many instances these are seen to be equivalent to relative stability of the process X itself. The analogous large time problem is also discussed.

DOI : 10.1214/11-AIHP449
Classification : 60G51, 60F15, 60F25, 60K05
Mots clés : Lévy process, passage times across power law boundaries, relative stability, overshoot, random walks
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     title = {Small and large time stability of the time taken for a {L\'evy} process to cross curved boundaries},
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Griffin, Philip S.; Maller, Ross A. Small and large time stability of the time taken for a Lévy process to cross curved boundaries. Annales de l'I.H.P. Probabilités et statistiques, Tome 49 (2013) no. 1, pp. 208-235. doi : 10.1214/11-AIHP449. http://www.numdam.org/articles/10.1214/11-AIHP449/

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