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Kac–Lévy Processes

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  • Nikita Ratanov

    (Universidad del Rosario)

Abstract

Markov-modulated Lévy processes with two different regimes of restarting are studied. These regimes correspond to the completely renewed process and to the process of Markov modulation, accompanied by jumps. We give explicit expressions for the Lévy–Khintchine exponent in the case of a two-state underlying Markov chain. For the renewal case, the limit distributions (as $$t\rightarrow \infty $$t→∞) are obtained. In the case of processes with jumps, we present some results for the exponential functional.

Suggested Citation

  • Nikita Ratanov, 2020. "Kac–Lévy Processes," Journal of Theoretical Probability, Springer, vol. 33(1), pages 239-267, March.
    • Handle: RePEc:spr:jotpro:v:33:y:2020:i:1:d:10.1007_s10959-018-0873-6
    DOI: 10.1007/s10959-018-0873-6
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    References listed on IDEAS

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    1. Bogachev, Leonid & Ratanov, Nikita, 2011. "Occupation time distributions for the telegraph process," Stochastic Processes and their Applications, Elsevier, vol. 121(8), pages 1816-1844, August.
    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. Thomas W Epps, 2007. "Pricing Derivative Securities," World Scientific Books, World Scientific Publishing Co. Pte. Ltd., number 6243, February.
    4. Robert J. Elliott & Leunglung Chan & Tak Kuen Siu, 2005. "Option pricing and Esscher transform under regime switching," Annals of Finance, Springer, vol. 1(4), pages 423-432, October.
    5. Ratanov, Nikita, 2017. "Piecewise linear process with renewal starting points," Statistics & Probability Letters, Elsevier, vol. 131(C), pages 78-86.
    6. Ning Cai & Steven Kou, 2012. "Pricing Asian Options Under a Hyper-Exponential Jump Diffusion Model," Operations Research, INFORMS, vol. 60(1), pages 64-77, February.
    7. Chevallier Julien & Goutte Stéphane, 2017. "On the estimation of regime-switching Lévy models," Studies in Nonlinear Dynamics & Econometrics, De Gruyter, vol. 21(1), pages 3-29, February.
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