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A Drawdown Reflected Spectrally Negative Lévy Process

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  • Wenyuan Wang

    (Xiamen University)

  • Xiaowen Zhou

    (Concordia University)

Abstract

In this paper, we study a spectrally negative Lévy process that is reflected at its drawdown level whenever a drawdown time from the running supremum arrives. Using an excursion-theoretical approach, for such a reflected process we find the Laplace transform of the upper exiting time and an expression of the associated potential measure. When the reflected process is identified as a risk process with capital injections, the expected total amount of discounted capital injections prior to the exiting time and the Laplace transform of the accumulated capital injections until the exiting time are also obtained. The results are expressed in terms of scale functions for the spectrally negative Lévy process.

Suggested Citation

  • Wenyuan Wang & Xiaowen Zhou, 2021. "A Drawdown Reflected Spectrally Negative Lévy Process," Journal of Theoretical Probability, Springer, vol. 34(1), pages 283-306, March.
    • Handle: RePEc:spr:jotpro:v:34:y:2021:i:1:d:10.1007_s10959-019-00971-4
    DOI: 10.1007/s10959-019-00971-4
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    References listed on IDEAS

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    1. M. R. Pistorius, 2004. "On Exit and Ergodicity of the Spectrally One-Sided Lévy Process Reflected at Its Infimum," Journal of Theoretical Probability, Springer, vol. 17(1), pages 183-220, January.
    2. Florin Avram & Zbigniew Palmowski & Martijn R. Pistorius, 2007. "On the optimal dividend problem for a spectrally negative L\'{e}vy process," Papers math/0702893, arXiv.org.
    3. Dickson, David C.M. & Waters, Howard R., 2004. "Some Optimal Dividends Problems," ASTIN Bulletin, Cambridge University Press, vol. 34(1), pages 49-74, May.
    4. Gerber, Hans U., 1990. "When does the surplus reach a given target?," Insurance: Mathematics and Economics, Elsevier, vol. 9(2-3), pages 115-119, September.
    5. Avram, Florin & Vu, Nhat Linh & Zhou, Xiaowen, 2017. "On taxed spectrally negative Lévy processes with draw-down stopping," Insurance: Mathematics and Economics, Elsevier, vol. 76(C), pages 69-74.
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    Cited by:

    1. Wenyuan Wang & Yuebao Wang & Ping Chen & Xueyuan Wu, 2022. "Dividend and Capital Injection Optimization with Transaction Cost for Lévy Risk Processes," Journal of Optimization Theory and Applications, Springer, vol. 194(3), pages 924-965, September.
    2. Xuan Huang & Jieming Zhou, 2022. "General Draw-Down Times for Refracted Spectrally Negative Lévy Processes," Methodology and Computing in Applied Probability, Springer, vol. 24(2), pages 875-891, June.

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