Asymptotic behavior of the hitting time, overshoot and undershoot for some Lévy processes
ESAIM: Probability and Statistics, Tome 12 (2008), pp. 58-93.

Let $\left({X}_{t},\phantom{\rule{0.277778em}{0ex}}t\ge 0\right)$ be a Lévy process started at $0$, with Lévy measure $\nu$. We consider the first passage time ${T}_{x}$ of $\left({X}_{t},\phantom{\rule{0.277778em}{0ex}}t\ge 0\right)$ to level $x>0$, and ${K}_{x}:={X}_{{T}_{x}}-𝑥$ the overshoot and ${L}_{x}:=x-{X}_{{T}_{{𝑥}^{-}}}$ the undershoot. We first prove that the Laplace transform of the random triple $\left({T}_{x},{K}_{x},{L}_{x}\right)$ satisfies some kind of integral equation. Second, assuming that $\nu$ admits exponential moments, we show that $\left(\stackrel{˜}{{T}_{x}},{K}_{x},{L}_{x}\right)$ converges in distribution as $x\to \infty$, where $\stackrel{˜}{{T}_{x}}$ denotes a suitable renormalization of ${T}_{x}$.

DOI : https://doi.org/10.1051/ps:2007034
Classification : 60E10,  60F05,  60G17,  60G40,  60G51,  60J65,  60J75,  60J80,  60K05
Mots clés : Lévy processes, ruin problem, hitting time, overshoot, undershoot, asymptotic estimates, functional equation
@article{PS_2008__12__58_0,
author = {Roynette, Bernard and Vallois, Pierre and Volpi, Agn\es},
title = {Asymptotic behavior of the hitting time, overshoot and undershoot for some {L\'evy} processes},
journal = {ESAIM: Probability and Statistics},
pages = {58--93},
publisher = {EDP-Sciences},
volume = {12},
year = {2008},
doi = {10.1051/ps:2007034},
mrnumber = {2367994},
language = {en},
url = {http://www.numdam.org/articles/10.1051/ps:2007034/}
}
TY  - JOUR
AU  - Roynette, Bernard
AU  - Vallois, Pierre
AU  - Volpi, Agnès
TI  - Asymptotic behavior of the hitting time, overshoot and undershoot for some Lévy processes
JO  - ESAIM: Probability and Statistics
PY  - 2008
DA  - 2008///
SP  - 58
EP  - 93
VL  - 12
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/ps:2007034/
UR  - https://www.ams.org/mathscinet-getitem?mr=2367994
UR  - https://doi.org/10.1051/ps:2007034
DO  - 10.1051/ps:2007034
LA  - en
ID  - PS_2008__12__58_0
ER  - 
Roynette, Bernard; Vallois, Pierre; Volpi, Agnès. Asymptotic behavior of the hitting time, overshoot and undershoot for some Lévy processes. ESAIM: Probability and Statistics, Tome 12 (2008), pp. 58-93. doi : 10.1051/ps:2007034. http://www.numdam.org/articles/10.1051/ps:2007034/`

[1] J. Bertoin, Lévy processes, Cambridge Tracts in Mathematics, vol. 121. Cambridge University Press, Cambridge (1996). | MR 1406564 | Zbl 0861.60003

[2] J. Bertoin and R.A. Doney, Cramér's estimate for Lévy processes. Statist. Probab. Lett. 21 (1994) 363-365. | MR 1325211 | Zbl 0809.60085

[3] 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. | MR 90177

[4] H. Cramér, On the mathematical Theory of Risk. Skandia Jubilee Volume, Stockholm (1930). | JFM 56.1100.03

[5] R.A. Doney, Hitting probabilities for spectrally positive Lévy processes. J. London Math. Soc. 44 (1991) 566-576. | MR 1149016 | Zbl 0699.60061

[6] R.A. Doney and A.E. Kyprianou, Overshoots and undershoots of Lévy processes. Ann. Appl. Probab. 16 (2006) 91-106. | MR 2209337 | Zbl 1101.60029

[7] R.A. Doney and R.A. Maller. Stability of the overshoot for Lévy processes. Ann. Probab. 30 (2002) 188-212. | MR 1894105 | Zbl 1016.60052

[8] F. Dufresne and H.U. Gerber, Risk theory for the compound Poisson process that is perturbed by diffusion. Insurance Math. Econom. 10 (1991) 51-59. | MR 1114429 | Zbl 0723.62065

[9] 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. | Zbl 0521.33001

[10] P.S. Griffin and R.A. Maller, On the rate of growth of the overshoot and the maximum partial sum. Adv. in Appl. Probab. 30 (1998) 181-196. | MR 1618833 | Zbl 0905.60064

[11] 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. | MR 916870 | Zbl 0634.60061

[12] I. Karatzas and S.E. Shreve. Brownian motion and stochastic calculus, Graduate Texts in Mathematics, vol.113. Springer-Verlag, New York, second edition (1991). | MR 1121940 | Zbl 0734.60060

[13] A.E. Kyprianou, Introductory lectures on fluctuations of Lévy processes with applications. Universitext. Springer-Verlag, Berlin (2006). | MR 2250061 | Zbl 1104.60001

[14] 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. | Zbl 0271.33001

[15] M. Loève, Probability theory. II. Springer-Verlag, New York, fourth edition (1978). Graduate Texts in Mathematics, Vol. 46. | MR 651018 | Zbl 0385.60001

[16] F. Lundberg, I- Approximerad Framställning av Sannolikhetsfunktionen. II- Aterförsäkering av Kollectivrisker. Almqvist and Wiksell, Uppsala (1903).

[17] 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). | MR 1680267 | Zbl 0940.60005

[18] 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. | MR 1739520 | Zbl 0973.60001

[19] 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]. | Zbl 0531.30002

[20] 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.

Cité par Sources :