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

¹ Département de mathématiques, Institut Élie Cartan,Université Henri Poincaré, BP 239, 54506 Vandœ 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 Vandœuvre-lès-Nancy, France; volpi@esstin.uhp-nancy.fr

Received: 16 February 2007

Abstract

Let (X_t, t ≥ 0) be a Lévy process started at 0, with Lévy measure ν. We consider the first passage time T_x of (X_t, t ≥ 0) to level x > 0, and K_x := X_Tx - x the overshoot and L_x := x- X_Tx- 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 ν admits exponential moments, we show that converges in distribution as x → ∞, where denotes a suitable renormalization of T_x.