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On Two Multistable Extensions of Stable Lévy Motion and Their Semi-martingale Representations

Author

Listed:
  • Ronan Le Guével

    (Université de Rennes 2-Haute Bretagne)

  • Jacques Lévy Véhel

    (Ecole Centrale Paris)

  • Lining Liu

    (Ecole Centrale Paris)

Abstract

We study two versions of multistable Lévy motion. Such processes are extensions of classical Lévy motion where the stability index is allowed to vary in time, a useful property for modeling non-increment stationary phenomena. We show that the two multistable Lévy motions have distinct properties: in particular, one is a pure jump Markov process, while the other one satisfies neither of these properties. We prove that both are semi-martingales and provide semi-martingale decompositions.

Suggested Citation

  • Ronan Le Guével & Jacques Lévy Véhel & Lining Liu, 2015. "On Two Multistable Extensions of Stable Lévy Motion and Their Semi-martingale Representations," Journal of Theoretical Probability, Springer, vol. 28(3), pages 1125-1144, September.
    • Handle: RePEc:spr:jotpro:v:28:y:2015:i:3:d:10.1007_s10959-013-0528-6
    DOI: 10.1007/s10959-013-0528-6
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    References listed on IDEAS

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    1. Ayache, Antoine, 2013. "Sharp estimates on the tail behavior of a multistable distribution," Statistics & Probability Letters, Elsevier, vol. 83(3), pages 680-688.
    2. Kenneth J. Falconer, 2002. "Tangent Fields and the Local Structure of Random Fields," Journal of Theoretical Probability, Springer, vol. 15(3), pages 731-750, July.
    3. K. J. Falconer & J. Lévy Véhel, 2009. "Multifractional, Multistable, and Other Processes with Prescribed Local Form," Journal of Theoretical Probability, Springer, vol. 22(2), pages 375-401, June.
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    Cited by:

    1. Olivier Le Courtois, 2018. "Some Further Results on the Tempered Multistable Approach," Post-Print hal-02312142, HAL.
    2. K. J. Falconer & J. Lévy Véhel, 2020. "Self-Stabilizing Processes Based on Random Signs," Journal of Theoretical Probability, Springer, vol. 33(1), pages 134-152, March.

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