ESAIM: Probability and Statistics (ESAIM: P&S)

ESAIM: P&S, 2008, Vol. 12, p. 58-93
DOI: 10.1051/ps:2007034

Asymptotic behavior of the hitting time, overshoot and undershoot for some Lévy processes

Bernard Roynette¹, Pierre Vallois¹ and Agnès Volpi^{1, 2}

¹ Département de mathématiques, Institut Élie Cartan,Université Henri Poincaré, BP 239, 54506 Vandoe uvre-lès-Nancy cedex, France; roynette@iecn.u-nancy.fr; vallois@iecn.u-nancy.fr
² ESSTIN, 2 rue Jean Lamour, Parc Robert Bentz, 54500 Vandoeuvre-lès-Nancy, France; volpi@esstin.uhp-nancy.fr

Received February 16, 2007. Published online 13 November 2007

Abstract

Let $(X_t, \; t\ge0)$ be a Lévy process started at 0, with Lévy measure $\nu$ . We consider the first passage time T_x of $(X_t, \; t\ge0)$ to level x > 0, and $K_x:=X_-{\it x}$ the overshoot and $L_x:=x-X_{T_{{\it x}^-}}$ the undershoot. We first prove that the Laplace transform of the random triple (T_x,K_x,L_x) satisfies some kind of integral equation. Second, assuming that $\nu$ admits exponential moments, we show that $(\widetilde,K_x,L_x)$ converges in distribution as $x\rightarrow \infty$ , where $\widetilde$ denotes a suitable renormalization of T_x.

Mathematics Subject Classification. 60E10, 60F05, 60G17, 60G40, 60G51, 60J65, 60J75, 60J80, 60K05

Key words: Lévy processes, ruin problem, hitting time, overshoot, undershoot, asymptotic estimates, functional equation.

© EDP Sciences, SMAI 2008