Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

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

Philip S. Griffin and Ross A. Maller

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This paper is concerned with the small time behaviour of a Lévy process $X$. In particular, we investigate the stabilities of the times, $\overline{T} _{b}(r)$ and $T^{*}_{b}(r)$, at which $X$, started with $X_{0}=0$, first leaves the space-time regions $\{(t,y)\in \mathbb{R} ^{2}\colon\ y\le rt^{b},t\ge0\}$ (one-sided exit), or $\{(t,y)\in \mathbb{R} ^{2}\colon\ |y|\le rt^{b},t\ge0\}$ (two-sided exit), $0\le b<1$, as $r\downarrow 0$. 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 $\overline{T} _{b}(r)$ et $T^{*}_{b}(r)$ auxquels $X$, partant de $X_{0}=0$, quitte pour la première fois les domaines $\{(t,y)\in \mathbb{R} ^{2}\colon\ y\le rt^{b},t\ge0\}$ (sortie unilatérale), ou $\{(t,y)\in \mathbb{R} ^{2}\colon\ |y|\le rt^{b},t\ge0\}$ (sortie bilatérale), $0\le b<1$, quand $r\downarrow 0$. 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.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 49, Number 1 (2013), 208-235.

First available in Project Euclid: 29 January 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G51: Processes with independent increments; Lévy processes 60F15: Strong theorems 60F25: $L^p$-limit theorems 60K05: Renewal theory

Lévy process Passage times across power law boundaries Relative stability Overshoot Random walks


Griffin, Philip S.; Maller, Ross A. Small and large time stability of the time taken for a Lévy process to cross curved boundaries. Ann. Inst. H. Poincaré Probab. Statist. 49 (2013), no. 1, 208--235. doi:10.1214/11-AIHP449.

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