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, Volume 49 (2013) no. 1, p. 208-235

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.

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.

DOI : https://doi.org/10.1214/11-AIHP449
Classification:  60G51,  60F15,  60F25,  60K05
Keywords: Lévy process, passage times across power law boundaries, relative stability, overshoot, random walks
@article{AIHPB_2013__49_1_208_0,
     author = {Griffin, Philip S. and Maller, Ross A.},
     title = {Small and large time stability of the time taken for a L\'evy process to cross curved boundaries},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {49},
     number = {1},
     year = {2013},
     pages = {208-235},
     doi = {10.1214/11-AIHP449},
     zbl = {1267.60053},
     mrnumber = {3060154},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2013__49_1_208_0}
}
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, Volume 49 (2013) no. 1, pp. 208-235. doi : 10.1214/11-AIHP449. http://www.numdam.org/item/AIHPB_2013__49_1_208_0/

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