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Asymptotic behavior of the hitting time, overshoot andundershoot 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 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 $(\widetilde{T_x},K_x,L_x)$ converges in distribution as x → ∞, where $\widetilde{T_x}$ denotes a suitable renormalization of Tx.


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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References

J. Bertoin, Lévy processes, Cambridge Tracts in Mathematics, vol. 121. Cambridge University Press, Cambridge (1996).
Bertoin, J. and Doney, R.A., Cramér's estimate for Lévy processes. Statist. Probab. Lett. 21 (1994) 363–365. CrossRef
H. Cramér, Collective risk theory: A survey of the theory from the point of view of the theory of stochastic processes. Skandia Insurance Company, Stockholm, (1955). Reprinted from the Jubilee Volume of Försäkringsaktiebolaget Skandia.
H. Cramér, On the mathematical Theory of Risk. Skandia Jubilee Volume, Stockholm (1930).
Doney, R.A., Hitting probabilities for spectrally positive Lévy processes. J. London Math. Soc. 44 (1991) 566–576. CrossRef
Doney, R.A. and Kyprianou, A.E., Overshoots and undershoots of Lévy processes. Ann. Appl. Probab. 16 (2006) 91–106. CrossRef
R.A. Doney and R.A. Maller. Stability of the overshoot for Lévy processes. Ann. Probab. 30 (2002) 188–212.
Dufresne, F. and Gerber, H.U., Risk theory for the compound Poisson process that is perturbed by diffusion. Insurance Math. Econom. 10 (1991) 51–59. CrossRef
I.S. Gradshteyn and I.M. Ryzhik, Table of integrals, series, and products. Academic Press [Harcourt Brace Jovanovich Publishers], New York (1980). Corrected and enlarged edition edited by Alan Jeffrey, Incorporating the fourth edition edited by Yu. V. Geronimus [Yu. V. Geronimus] and M. Yu. Tseytlin [M. Yu. Tseĭtlin], Translated from Russian.
Griffin, P.S. and Maller, R.A., On the rate of growth of the overshoot and the maximum partial sum. Adv. in Appl. Probab. 30 (1998) 181–196. CrossRef
A. Gut, Stopped random walks, Applied Probability, vol. 5, A Series of the Applied Probability Trust. Springer-Verlag, New York, (1988). Limit theorems and applications.
I. Karatzas and S.E. Shreve. Brownian motion and stochastic calculus, Graduate Texts in Mathematics, vol.113. Springer-Verlag, New York, second edition (1991).
A.E. Kyprianou, Introductory lectures on fluctuations of Lévy processes with applications. Universitext. Springer-Verlag, Berlin (2006).
N.N. Lebedev, Special functions and their applications. Dover Publications Inc., New York (1972). Revised edition, translated from the Russian and edited by Richard A. Silverman, Unabridged and corrected republication.
M. Loève, Probability theory. II. Springer-Verlag, New York, fourth edition (1978). Graduate Texts in Mathematics, Vol. 46.
F. Lundberg, I- Approximerad Framställning av Sannolikhetsfunktionen. II- Aterförsäkering av Kollectivrisker. Almqvist and Wiksell, Uppsala (1903).
T. Rolski, H. Schmidli, V. Schmidt and J. Teugels, Stochastic processes for insurance and finance. Wiley Series in Probability and Statistics. John Wiley & Sons Ltd., Chichester (1999).
K. Sato, Lévy processes and infinitely divisible distributions, volume 68 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, (1999). Translated from the 1990 Japanese original, Revised by the author.
A.G. Sveshnikov and A.N. Tikhonov, The theory of functions of a complex variable. “Mir”, Moscow (1982). Translated from the Russian by George Yankovsky [G. Yankovskiĭ].
A. Volpi, Processus associés à l'équation de diffusion rapide; Étude asymptotique du temps de ruine et de l'overshoot. Univ. Henri Poincaré, Nancy I, Vandoeuvre les Nancy (2003). Thèse.