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Exit Problems for Spectrally Negative Lévy Processes Reflected at Either the Supremum or the Infimum

Part of: Stochastic processes Probability theory on algebraic and topological structures

Published online by Cambridge University Press:  14 July 2016

Xiaowen Zhou
Xiaowen Zhou*
Affiliation:
Concordia University
*
∗Postal address: Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve Blvd. W., Montreal, Quebec, H3G 1M8, Canada. Email address: zhou@alcor.concordia.ca
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Abstract

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For a spectrally negative Lévy process X on the real line, let S denote its supremum process and let I denote its infimum process. For a > 0, let τ(a) and κ(a) denote the times when the reflected processes Ŷ := S − X and Y := X − I first exit level a, respectively; let τ−(a) and κ−(a) denote the times when X first reaches Sτ(a) and Iκ(a), respectively. The main results of this paper concern the distributions of (τ(a), Sτ(a), τ−(a), Ŷτ(a)) and of (κ(a), Iκ(a), κ−(a)). They generalize some recent results on spectrally negative Lévy processes. Our approach relies on results concerning the solution to the two-sided exit problem for X. Such an approach is also adapted to study the excursions for the reflected processes. More explicit expressions are obtained when X is either a Brownian motion with drift or a completely asymmetric stable process.

Keywords

Spectrally negative Lévy processreflected Lévy processfluctuation theoryexit problemexcursionrisk modelruin time

MSC classification

Secondary: 60G51: Processes with independent increments; L'evy processes 60B15: Probability measures on groups or semigroups, Fourier transforms, factorization
Type
Research Papers
Information
Journal of Applied Probability , Volume 44 , Issue 4 , December 2007 , pp. 1012 - 1030
Copyright
Copyright © Applied Probability Trust 2007 

Footnotes

Supported by an NSERC operating grant.

References

[1] Avram, F., Kyprianou, A. E. and Pistorius, M. R. (2004). Exit problems for spectrally negative Lévy processes and applications to (Canadized) Russian options. Ann. Appl. Prob. 14, 215–238.CrossRefGoogle Scholar
[2] Bertoin, J. (1992). An extension of Pitman's theorem for spectrally positive Lévy processes. Ann. Prob. 20, 1464–1483.CrossRefGoogle Scholar
[3] Bertoin, J. (1996a). Lévy Processes. Campbridge University Press.Google Scholar
[4] Bertoin, J. (1996b). On the first exit-time of a completely asymmetric stable process from a finite interval. Bull. London Math. Soc. 5, 514–520.CrossRefGoogle Scholar
[5] Bertoin, J. (1997). Exponential decay and ergodicity of completely asymmetric Lévy processes in a finite interval. Ann. Appl. Prob. 7, 156–169.CrossRefGoogle Scholar
[6] Bingham, N. H. (1975). Fluctuation theory in continuous time. Adv. Appl. Prob. 7, 705–766.CrossRefGoogle Scholar
[7] Doney, R. A. (2004). Some excursion calculations for spectrally one-sided Lévy processes. In Séminaire de Probabilités XXXVIII (Lecture Notes Math. 1857), Springer, Berlin, pp. 5–15.Google Scholar
[8] Doney, R. A. (2007). Fluctuation Theory for Lévy Processes (Lecture Notes Math. 1897). Springer, Berlin.Google Scholar
[9] Gerber, H. U. and Shiu, E. S. W. (2004). Optimal dividends: analysis with Brownian motion. N. Amer. Actuarial J. 8, 1–20.CrossRefGoogle Scholar
[10] Greenwood, P. and Pitman, J. W. (1980). Fluctuation identities for Lévy processes and splitting at the maximum. Adv. Appl. Prob. 12, 893–902.CrossRefGoogle Scholar
[11] Harrison, J. M. (1985). Brownian Motion and Stochastic Flow Systems. John Wiley, New York.Google Scholar
[12] Kyprianou, A. E. (2006). First passage of reflected strictly stable processes. ALEA Latin Amer. J. Prob. Math. Statist. 2, 119–123.Google Scholar
[13] Kyprianou, A. E. (2006). Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin.Google Scholar
[14] Kyprianou, A. E. and Palmowski, Z. (2007). Distributional study of De Finetti's dividend problem for a general Lévy insurance risk process. J. Appl. Prob. 44, 428–443.CrossRefGoogle Scholar
[15] Lambert, A. (2000). Completely asymmetric Lévy processes confined in a finite interval. Ann. Inst. H. Poincaré Prob. Statist. 36, 251–274.CrossRefGoogle Scholar
[16] Le Gall, J. and Le Jan, Y. (1998). Branching processes in Lévy processes: the exploration process. Ann. Prob. 26, 213–252.Google Scholar
[17] Nguyen-Ngoc, L. and Yor, M. (2004). Some martingales associated to reflected Lévy processes. In Séminaire de Probabilités XXXVIII (Lecture Notes Math. 1857), Springer, Berlin, pp. 42–69.Google Scholar
[18] Pistorius, M. R. (2004). On exit and ergodicity of the completely asymmetric Lévy process reflected at its infimum. J. Theoret. Prob. 17, 183–220.CrossRefGoogle Scholar
[19] Pistorius, M. R. (2007). An excursion theoretical approach to some boundary crossing problems and the Skorokhod embedding for reflected Lévy processes. In Séminaire de Probabilités XL, eds Donati-Martin, C. et al., Springer, Berlin, pp. 287–308.CrossRefGoogle Scholar
[20] Renaud, J. and Zhou, X. (2007). Distribution of the present value dividend payments in a Lévy risk model. J. Appl. Prob. 44, 420–428.CrossRefGoogle Scholar
[21] Revuz, D. and Yor, M. (1991). Continuous Martingales and Brownian Motion. Springer, New York.CrossRefGoogle Scholar
[22] Rogers, L. C. G. (1984). A new identity for real Lévy processes. Ann. Inst. H. Poincaré Prob. Statist. 20, 12–34.Google Scholar
[23] Taylor, H. M. (1975). A stopped Brownian motion formula. Ann. Prob. 3, 234–246.CrossRefGoogle Scholar
[24] Whitt, W. (2000). An overview of Brownian and non-Brownian FCLTs for the single-server queue. Queueing Systems Theory Appl. 36, 39–70.CrossRefGoogle Scholar
[25] Williams, R. J. (1992). Asymptotic variance parameters for the boundary local times of reflected Brownian motion on a compact interval. J. Appl. Prob. 29, 996–1002.CrossRefGoogle Scholar