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.
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.
Mots clés : 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}, pages = {208--235}, publisher = {Gauthier-Villars}, volume = {49}, number = {1}, year = {2013}, doi = {10.1214/11-AIHP449}, mrnumber = {3060154}, zbl = {1267.60053}, language = {en}, url = {http://www.numdam.org/articles/10.1214/11-AIHP449/} }
TY - JOUR AU - Griffin, Philip S. AU - Maller, Ross A. TI - Small and large time stability of the time taken for a Lévy process to cross curved boundaries JO - Annales de l'I.H.P. Probabilités et statistiques PY - 2013 SP - 208 EP - 235 VL - 49 IS - 1 PB - Gauthier-Villars UR - http://www.numdam.org/articles/10.1214/11-AIHP449/ DO - 10.1214/11-AIHP449 LA - en ID - AIHPB_2013__49_1_208_0 ER -
%0 Journal Article %A Griffin, Philip S. %A Maller, Ross A. %T Small and large time stability of the time taken for a Lévy process to cross curved boundaries %J Annales de l'I.H.P. Probabilités et statistiques %D 2013 %P 208-235 %V 49 %N 1 %I Gauthier-Villars %U http://www.numdam.org/articles/10.1214/11-AIHP449/ %R 10.1214/11-AIHP449 %G en %F 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, 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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