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

Griffin, Philip S.; Maller, Ross A.

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 $\{(t,y)\in {ℝ}^2:y\le r{t}^b,t\ge 0\}$ (sortie unilatérale), ou $\{(t,y)\in {ℝ}^2:|y|\le r{t}^b,t\ge 0\}$ (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.

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 $\{(t,y)\in {ℝ}^2:y\le r{t}^b,t\ge 0\}$ (one-sided exit), or $\{(t,y)\in {ℝ}^2:|y|\le r{t}^b,t\ge 0\}$ (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.

Classification : 60G51, 60F15, 60F25, 60K05
Mots clés : Lévy process, passage times across power law boundaries, relative stability, overshoot, random walks

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