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 ${\overline{T}}_{b}\left(r\right)$ et ${T}_{b}^{*}\left(r\right)$ auxquels $X$, partant de ${X}_{0}=0$, quitte pour la première fois les domaines $\left\{\left(t,y\right)\in {ℝ}^{2}:\phantom{\rule{4pt}{0ex}}y\le r{t}^{b},t\ge 0\right\}$ (sortie unilatérale), ou $\left\{\left(t,y\right)\in {ℝ}^{2}:\phantom{\rule{4pt}{0ex}}|y|\le r{t}^{b},t\ge 0\right\}$ (sortie bilatérale), $0\le b<1$, 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.

This paper is concerned with the small time behaviour of a Lévy process $X$. In particular, we investigate the stabilities of the times, ${\overline{T}}_{b}\left(r\right)$ and ${T}_{b}^{*}\left(r\right)$, at which $X$, started with ${X}_{0}=0$, first leaves the space-time regions $\left\{\left(t,y\right)\in {ℝ}^{2}:\phantom{\rule{4pt}{0ex}}y\le r{t}^{b},t\ge 0\right\}$ (one-sided exit), or $\left\{\left(t,y\right)\in {ℝ}^{2}:\phantom{\rule{4pt}{0ex}}|y|\le r{t}^{b},t\ge 0\right\}$ (two-sided exit), $0\le b<1$, 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.

DOI : https://doi.org/10.1214/11-AIHP449
Classification : 60G51,  60F15,  60F25,  60K05
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},
zbl = {1267.60053},
mrnumber = {3060154},
language = {en},
url = {http://www.numdam.org/articles/10.1214/11-AIHP449/}
}
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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