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Some fluctuation identities for Lévy processes with jumps of the same sign

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

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, 7141 Sherbrooke Street West, Montreal QC, H4B 1R6, Canada. Email address: [email protected]
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Abstract

We consider a two-sided exit problem for a Lévy process with no positive jumps. The Laplace transform of the time when the process first exits an interval from above is obtained. It is expressed in terms of another Laplace transform for the one-sided exit problem. Applications of this result are discussed. In particular, a new expression for the solution to the two-sided exit problem is obtained. The joint distribution of the minimum and the maximum values of such a Lévy process is also studied.

Keywords

Fluctuation identityLaplace transformLévy process with no positive jumpsone-sided exit problemtwo-sided exit problemcompletely asymmetric stable process

MSC classification

Primary: 60G51: Processes with independent increments; L'evy processes
Secondary: 60B15: Probability measures on groups or semigroups, Fourier transforms, factorization
Type
Short Communications
Information
Journal of Applied Probability , Volume 41 , Issue 4 , December 2004 , pp. 1191 - 1198
Copyright
Copyright © Applied Probability Trust 2004 

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Footnotes

Supported by an NSERC operating grant.

References

Asmussen, S. (2000). Ruin Probabilities (Adv. Ser. Statist. Sci. Appl. Prob. 2). World Scientific, River Edge, NJ.Google Scholar
Bertoin, J. (1996a). Lévy Processes. Cambridge University Press.Google Scholar
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.Google Scholar
Bertoin, J. (1997). Exponential decay and ergodicity of completely asymmetric Lévy processes in a finite interval. Ann. Appl. Prob. 7, 156–169.Google Scholar
Bingham, N. H. (1975). Fluctuation theory in continuous time. Adv. Appl. Prob. 7, 705–766.Google Scholar
Emery, D. J. (1973). Exit problem for a spectrally positive process. Adv. Appl. Prob. 5, 498–520.Google Scholar
Lachal, A. (1993). L'intégrale du mouvement brownien. J. Appl. Prob. 30, 17–27.Google Scholar
Pecherskii, E. A., and Rogozin, B. A. (1969). On joint distributions of random variables associated with fluctuations of a process with independent increments. Theory Prob. Appl. 14, 410–423.Google Scholar
Rogers, L. C. G. (1984). A new identity for real Lévy processes. Ann. Inst. H. Poincaré Prob. Statist. 20, 12–34.Google Scholar
Rogers, L. C. G. (1990). The two-sided exit problem for spectrally positive Lévy processes. Adv. Appl. Prob. 22, 486–487.Google Scholar
Rogers, L. C. G. (2000). Evaluating first-passage probabilities for spectrally one-sided Lévy processes. J. Appl. Prob. 37, 1173–1180.Google Scholar
Rolski, T., Schmidli, H., Schmidt, V., and Teugels, J. (1999). Stochastic Processes for Insurance and Finance. John Wiley, Chichester.Google Scholar
Sato, K. (1999). Lévy Processes and Infinitely Divisible Distributions (Cambridge Stud. Adv. Math 68). Cambridge University Press.Google Scholar