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Some explicit identities associated with positive self-similar Markov processes

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  • Chaumont, L.
  • Kyprianou, A.E.
  • Pardo, J.C.

Abstract

We consider some special classes of Lévy processes with no gaussian component whose Lévy measure is of the type , where [nu] is the density of the stable Lévy measure and [gamma] is a positive parameter which depends on its characteristics. These processes were introduced in [M. E. Caballero, L. Chaumont, Conditioned stable Lévy processes and the Lamperti representation, J. Appl. Probab. 43 (2006) 967-983] as the underlying Lévy processes in the Lamperti representation of conditioned stable Lévy processes. In this paper, we compute explicitly the law of these Lévy processes at their first exit time from a finite or semi-finite interval, the law of their exponential functional and the first hitting time probability of a pair of points.

Suggested Citation

  • Chaumont, L. & Kyprianou, A.E. & Pardo, J.C., 2009. "Some explicit identities associated with positive self-similar Markov processes," Stochastic Processes and their Applications, Elsevier, vol. 119(3), pages 980-1000, March.
    • Handle: RePEc:eee:spapps:v:119:y:2009:i:3:p:980-1000
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    References listed on IDEAS

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    1. Chaumont, L., 1996. "Conditionings and path decompositions for Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 64(1), pages 39-54, November.
    2. Peter Carr & Helyette Geman, 2002. "The Fine Structure of Asset Returns: An Empirical Investigation," The Journal of Business, University of Chicago Press, vol. 75(2), pages 305-332, April.
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    Cited by:

    1. Li, Bo & Palmowski, Zbigniew, 2018. "Fluctuations of Omega-killed spectrally negative Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 128(10), pages 3273-3299.
    2. Kuznetsov, A., 2012. "On the distribution of exponential functionals for Lévy processes with jumps of rational transform," Stochastic Processes and their Applications, Elsevier, vol. 122(2), pages 654-663.
    3. Biffis, Enrico & Kyprianou, Andreas E., 2010. "A note on scale functions and the time value of ruin for Lévy insurance risk processes," Insurance: Mathematics and Economics, Elsevier, vol. 46(1), pages 85-91, February.
    4. Kuznetsov, A. & Peng, X., 2012. "On the Wiener–Hopf factorization for Lévy processes with bounded positive jumps," Stochastic Processes and their Applications, Elsevier, vol. 122(7), pages 2610-2638.
    5. Fourati, Sonia, 2012. "Explicit solutions of the exit problem for a class of Lévy processes; applications to the pricing of double-barrier options," Stochastic Processes and their Applications, Elsevier, vol. 122(3), pages 1034-1067.

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