Small and large time stability of the time taken for a Lévy process to cross curved boundaries

Griffin, Philip S.; Maller, Ross A.

Griffin, Philip S.; Maller, Ross A.

Annales de l'I.H.P. Probabilités et statistiques, Volume 49 (2013) no. 1, p. 208-235

Abstract
Résumé

This paper is concerned with the small time behaviour of a Lévy process $X$ . In particular, we investigate the stabilities of the times, ${\bar{T}}_{b} (r)$ and $T_{b}^{*} (r)$ , at which $X$ , started with $X_{0} = 0$ , first leaves the space-time regions ${(t, y) \in ℝ^{2} : y \leq r t^{b}, t \geq 0}$ (one-sided exit), or ${(t, y) \in ℝ^{2} : | y | \leq r t^{b}, t \geq 0}$ (two-sided exit), $0 \leq b < 1$ , as $r ↓ 0$ . 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 ${\bar{T}}_{b} (r)$ et $T_{b}^{*} (r)$ auxquels $X$ , partant de $X_{0} = 0$ , quitte pour la première fois les domaines ${(t, y) \in ℝ^{2} : y \leq r t^{b}, t \geq 0}$ (sortie unilatérale), ou ${(t, y) \in ℝ^{2} : | y | \leq r t^{b}, t \geq 0}$ (sortie bilatérale), $0 \leq b < 1$ , quand $r ↓ 0$ . 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.

MR 3060154 | Zbl 1267.60053

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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