ESAIM: Probability and Statistics

Table of Contents - 2008 - Volume 12  

Research Article

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

Roynette, Bernarda1, Vallois, Pierrea1 and Volpi, Agnèsa1a2

a1 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

a2 ESSTIN, 2 rue Jean Lamour, Parc Robert Bentz, 54500 Vandœuvre-lès-Nancy, France; volpi@esstin.uhp-nancy.fr

Abstract

Let (Xt, t ≥ 0) be a Lévy process started at 0, with Lévy measure ν. We consider the first passage time T x of (Xt, t ≥ 0) to level x > 0, and Kx := XTx - x the overshoot and Lx := x- XTx- the undershoot. We first prove that the Laplace transform of the random triple (Tx,Kx,Lx ) satisfies some kind of integral equation. Second, assuming that ν admits exponential moments, we show that $(\widetilde{T_x},K_x,L_x)$ converges in distribution as x → ∞, where $\widetilde{T_x}$ denotes a suitable renormalization of T x .


(Received February 16 2007)

(Online publication November 13 2007)

Key Words:

  • Lévy processes;
  • ruin problem;
  • hitting time;
  • overshoot;
  • undershoot;
  • asymptotic estimates;
  • functional equation.

Mathematics Subject Classification:

  • 60E10;
  • 60F05;
  • 60G17;
  • 60G40;
  • 60G51;
  • 60J65;
  • 60J75;
  • 60J80;
  • 60K05
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