This paper is concerned with the small time behaviour of a Lévy process . In particular, we investigate the stabilities of the times, and , at which , started with , first leaves the space-time regions (one-sided exit), or (two-sided exit), , as . 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 . In many instances these are seen to be equivalent to relative stability of the process itself. The analogous large time problem is also discussed.
Ce papier traite du comportement en temps court d’un processus de Lévy . En particulier, nous étudions la stabilité des temps et auxquels , partant de , quitte pour la première fois les domaines (sortie unilatérale), ou (sortie bilatérale), , quand . 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 . Dans de nombreux cas, ceci est équivalent à la stabilité du processus . Le problème analogue à temps grand est aussi discuté.
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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