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On the refracted–reflected spectrally negative Lévy processes

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  • Pérez, José-Luis
  • Yamazaki, Kazutoshi

Abstract

We study a combination of the refracted and reflected Lévy processes. Given a spectrally negative Lévy process and two boundaries, it is reflected at the lower boundary while, whenever it is above the upper boundary, a linear drift at a constant rate is subtracted from the increments of the process. Using the scale functions, we compute the resolvent measure, the Laplace transform of the occupation times as well as other fluctuation identities that will be useful in applied probability including insurance, queues, and inventory management.

Suggested Citation

  • Pérez, José-Luis & Yamazaki, Kazutoshi, 2018. "On the refracted–reflected spectrally negative Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 128(1), pages 306-331.
    • Handle: RePEc:eee:spapps:v:128:y:2018:i:1:p:306-331
    DOI: 10.1016/j.spa.2017.03.024
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    References listed on IDEAS

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    1. Pistorius, M. R., 2003. "On doubly reflected completely asymmetric Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 107(1), pages 131-143, September.
    2. Pérez, José-Luis & Yamazaki, Kazutoshi, 2017. "Refraction–Reflection Strategies In The Dual Model," ASTIN Bulletin, Cambridge University Press, vol. 47(1), pages 199-238, January.
    3. Bayraktar, Erhan & Kyprianou, Andreas E. & Yamazaki, Kazutoshi, 2014. "Optimal dividends in the dual model under transaction costs," Insurance: Mathematics and Economics, Elsevier, vol. 54(C), pages 133-143.
    4. 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.
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    Cited by:

    1. Kei Noba & Jos'e-Luis P'erez & Xiang Yu, 2019. "On the bail-out dividend problem for spectrally negative Markov additive models," Papers 1901.03021, arXiv.org, revised Feb 2020.
    2. José-Luis Pérez & Kazutoshi Yamazaki & Xiang Yu, 2018. "On the Bail-Out Optimal Dividend Problem," Journal of Optimization Theory and Applications, Springer, vol. 179(2), pages 553-568, November.
    3. Noba, Kei, 2023. "On the optimality of the refraction–reflection strategies for Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 160(C), pages 174-217.

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