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Asymptotic behavior of the hitting time, overshootandundershoot for some Lévy processes

Published online by Cambridge University Press:  13 November 2007

Bernard Roynette ,
Pierre Vallois  and
Agnès Volpi
Bernard Roynette
Affiliation:
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
Pierre Vallois
Affiliation:
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
Agnès Volpi
Affiliation:
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 ESSTIN, 2 rue Jean Lamour, Parc Robert Bentz, 54500 Vandœuvre-lès-Nancy, France; volpi@esstin.uhp-nancy.fr
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Abstract

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


Keywords

Lévy processesruin problemhitting timeovershootundershootasymptotic estimatesfunctional equation.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 12 , 2008 , pp. 58 - 93
Copyright
© EDP Sciences, SMAI, 2008

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