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On the optimality of the refraction–reflection strategies for Lévy processes

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  • Noba, Kei

Abstract

In this paper, we study de Finetti’s optimal dividend problem with capital injection under the assumption that the dividend strategies are absolutely continuous. In many previous studies, the process before being controlled was assumed to be a spectrally one-sided Lévy process, however in this paper we use a Lévy process that may have both positive and negative jumps. In the main theorem, we show that a refraction–reflection strategy is an optimal strategy. We also mention the existence and uniqueness of solutions of the stochastic differential equations that define refracted Lévy processes.

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  • 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.
    • Handle: RePEc:eee:spapps:v:160:y:2023:i:c:p:174-217
    DOI: 10.1016/j.spa.2023.02.006
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    References listed on IDEAS

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    1. 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.
    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. 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.
    4. Yin, Chuancun & Wen, Yuzhen & Zhao, Yongxia, 2014. "On The Optimal Dividend Problem For A Spectrally Positive Lévy Process," ASTIN Bulletin, Cambridge University Press, vol. 44(3), pages 635-651, September.
    5. 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.
    6. Asmussen, Soren & Taksar, Michael, 1997. "Controlled diffusion models for optimal dividend pay-out," Insurance: Mathematics and Economics, Elsevier, vol. 20(1), pages 1-15, June.
    7. Chuancun Yin & Yuzhen Wen & Yongxia Zhao, 2013. "On the optimal dividend problem for a spectrally positive Levy process," Papers 1302.2231, arXiv.org, revised Mar 2014.
    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. Hans Gerber & Elias Shiu, 2006. "On Optimal Dividend Strategies In The Compound Poisson Model," North American Actuarial Journal, Taylor & Francis Journals, vol. 10(2), pages 76-93.
    10. 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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