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

Roynette, Bernard; Vallois, Pierre; Volpi, Agnès

ESAIM: Probability and Statistics, Tome 12 (2008), pp. 58-93.

Let $(X_t,t\ge 0)$ be a Lévy process started at $0$, with Lévy measure $\nu$. We consider the first passage time $T_x$ of $(X_t,t\ge 0)$ 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 $(T_x,K_x,L_x)$ satisfies some kind of integral equation. Second, assuming that $\nu$ admits exponential moments, we show that $(\widetilde{T_x},K_x,L_x)$ converges in distribution as $x\to \infty$, where $\widetilde{T_x}$ denotes a suitable renormalization of $T_x$.

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

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

[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

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

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

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

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

[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 | Zbl

[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

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

[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

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

[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 | Zbl

[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 | Zbl

[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

[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.