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Refraction–Reflection Strategies In The Dual Model

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  • Pérez, José-Luis
  • Yamazaki, Kazutoshi

Abstract

We study the dual model with capital injection under the additional condition that the dividend strategy is absolutely continuous. We consider a refraction–reflection strategy that pays dividends at the maximal rate whenever the surplus is above a certain threshold, while capital is injected so that it stays non-negative. The resulting controlled surplus process becomes the spectrally positive version of the refracted–reflected process recently studied by Pérez and Yamazaki (2015). We study various fluctuation identities of this process and prove the optimality of the refraction–reflection strategy. Numerical results on the optimal dividend problem are also given.

Suggested Citation

  • 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.
    • Handle: RePEc:cup:astinb:v:47:y:2017:i:01:p:199-238_00
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    Citations

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    Cited by:

    1. 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.
    2. Jos'e-Luis P'erez & Kazutoshi Yamazaki & Xiang Yu, 2017. "On the Bail-Out Optimal Dividend Problem," Papers 1709.06348, arXiv.org, revised Jun 2018.
    3. Kazutoshi Yamazaki, 2017. "Phase-type Approximation of the Gerber-Shiu Function," Papers 1701.02798, arXiv.org.
    4. Zhuo Jin & Huafu Liao & Yue Yang & Xiang Yu, 2019. "Optimal Dividend Strategy for an Insurance Group with Contagious Default Risk," Papers 1909.09511, arXiv.org, revised Oct 2020.
    5. Czarna, Irmina & Pérez, José-Luis & Rolski, Tomasz & Yamazaki, Kazutoshi, 2019. "Fluctuation theory for level-dependent Lévy risk processes," Stochastic Processes and their Applications, Elsevier, vol. 129(12), pages 5406-5449.
    6. Czarna, Irmina & Pérez, José-Luis & Yamazaki, Kazutoshi, 2018. "Optimality of multi-refraction control strategies in the dual model," Insurance: Mathematics and Economics, Elsevier, vol. 83(C), pages 148-160.
    7. 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.
    8. 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.
    9. Avram, Florin & Pérez, José-Luis & Yamazaki, Kazutoshi, 2018. "Spectrally negative Lévy processes with Parisian reflection below and classical reflection above," Stochastic Processes and their Applications, Elsevier, vol. 128(1), pages 255-290.
    10. 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.
    11. Avanzi, Benjamin & Pérez, José-Luis & Wong, Bernard & Yamazaki, Kazutoshi, 2017. "On optimal joint reflective and refractive dividend strategies in spectrally positive Lévy models," Insurance: Mathematics and Economics, Elsevier, vol. 72(C), pages 148-162.
    12. Noba, Kei, 2021. "On the optimality of double barrier strategies for Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 131(C), pages 73-102.

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