Volume 12, April 2008
|Page(s)||58 - 93|
|Published online||13 November 2007|
Asymptotic behavior of the hitting time, overshoot and undershoot for some Lévy processes
Département de mathématiques, Institut Élie
Cartan,Université Henri Poincaré, BP 239, 54506 Vandœ
uvre-lès-Nancy cedex, France; email@example.com; firstname.lastname@example.org
2 ESSTIN, 2 rue Jean Lamour, Parc Robert Bentz, 54500 Vandœuvre-lès-Nancy, France; email@example.com
Let (Xt, t ≥ 0) be a Lévy process started at 0, with Lévy measure ν. We consider the first passage time Tx 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 converges in distribution as x → ∞, where denotes a suitable renormalization of Tx.
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
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