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Optimal Dividends and Capital Injections in the Dual Model with Diffusion

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  • Avanzi, Benjamin
  • Shen, Jonathan
  • Wong, Bernard

Abstract

The dual model with diffusion is appropriate for companies with continuous expenses that are offset by stochastic and irregular gains. Examples include research-based or commission-based companies. In this context, Avanzi and Gerber (2008) showed how to determine the expected present value of dividends, if a barrier strategy is followed. In this paper, we further include capital injections and allow for (proportional) transaction costs both on dividends and capital injections. We determine the optimal dividend and (unconstrained) capital injection strategy (among all possible strategies) when jumps are hyperexponential. This strategy happens to be either a dividend barrier strategy without capital injections, or another dividend barrier strategy with forced injections when the surplus is null to prevent ruin. The latter is also shown to be the optimal dividend and capital injection strategy, if ruin is not allowed to occur. Both the choice to inject capital or not and the level of the optimal barrier depend on the parameters of the model. In all cases, we determine the optimal dividend barrier and show its existence and uniqueness. We also provide closed form representations of the value functions when the optimal strategy is applied. Results are illustrated.

Suggested Citation

  • Avanzi, Benjamin & Shen, Jonathan & Wong, Bernard, 2011. "Optimal Dividends and Capital Injections in the Dual Model with Diffusion," ASTIN Bulletin, Cambridge University Press, vol. 41(2), pages 611-644, November.
    • Handle: RePEc:cup:astinb:v:41:y:2011:i:02:p:611-644_00
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    Cited by:

    1. Peng, Xiaofan & Chen, Mi & Guo, Junyi, 2012. "Optimal dividend and equity issuance problem with proportional and fixed transaction costs," Insurance: Mathematics and Economics, Elsevier, vol. 51(3), pages 576-585.
    2. Jean-Franc{c}ois Renaud & Alexandre Roch & Clarence Simard, 2023. "An optimization dichotomy for capital injections and absolutely continuous dividend strategies," Papers 2311.10191, arXiv.org.
    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. Budhi Surya & Wenyuan Wang & Xianghua Zhao & Xiaowen Zhou, 2020. "Parisian excursion with capital injection for draw-down reflected Levy insurance risk process," Papers 2005.09214, arXiv.org.
    5. Kristoffer Lindensjo & Filip Lindskog, 2019. "Optimal dividends and capital injection under dividend restrictions," Papers 1902.06294, arXiv.org.
    6. Benjamin Avanzi & Ping Chen & Lars Frederik Brandt Henriksen & Bernard Wong, 2022. "On the surplus management of funds with assets and liabilities in presence of solvency requirements," Papers 2203.05139, arXiv.org, revised Aug 2022.
    7. Avanzi, Benjamin & Tu, Vincent & Wong, Bernard, 2014. "On optimal periodic dividend strategies in the dual model with diffusion," Insurance: Mathematics and Economics, Elsevier, vol. 55(C), pages 210-224.
    8. Chuancun Yin & Yuzhen Wen, 2013. "Optimal dividends problem with a terminal value for spectrally positive Levy processes," Papers 1302.6011, arXiv.org.
    9. 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.
    10. Pérez, José-Luis & Yamazaki, Kazutoshi, 2017. "On the optimality of periodic barrier strategies for a spectrally positive Lévy process," Insurance: Mathematics and Economics, Elsevier, vol. 77(C), pages 1-13.
    11. 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.
    12. Bayraktar, Erhan & Kyprianou, Andreas E. & Yamazaki, Kazutoshi, 2013. "On Optimal Dividends In The Dual Model," ASTIN Bulletin, Cambridge University Press, vol. 43(3), pages 359-372, September.
    13. Chen, Shumin & Zeng, Yan & Hao, Zhifeng, 2017. "Optimal dividend strategies with time-inconsistent preferences and transaction costs in the Cramér–Lundberg model," Insurance: Mathematics and Economics, Elsevier, vol. 74(C), pages 31-45.
    14. Yongxia Zhao & Rongming Wang & Dingjun Yao & Ping Chen, 2015. "Optimal Dividends and Capital Injections in the Dual Model with a Random Time Horizon," Journal of Optimization Theory and Applications, Springer, vol. 167(1), pages 272-295, October.
    15. Avanzi, Benjamin & Cheung, Eric C.K. & Wong, Bernard & Woo, Jae-Kyung, 2013. "On a periodic dividend barrier strategy in the dual model with continuous monitoring of solvency," Insurance: Mathematics and Economics, Elsevier, vol. 52(1), pages 98-113.
    16. Zhao, Yongxia & Chen, Ping & Yang, Hailiang, 2017. "Optimal periodic dividend and capital injection problem for spectrally positive Lévy processes," Insurance: Mathematics and Economics, Elsevier, vol. 74(C), pages 135-146.
    17. Xu, Ran & Woo, Jae-Kyung, 2020. "Optimal dividend and capital injection strategy with a penalty payment at ruin: Restricted dividend payments," Insurance: Mathematics and Economics, Elsevier, vol. 92(C), pages 1-16.
    18. Dong, Hua & Zhou, Xiaowen, 2019. "On a spectrally negative Lévy risk process with periodic dividends and capital injections," Statistics & Probability Letters, Elsevier, vol. 155(C), pages 1-1.
    19. Kristoffer Lindensjö & Filip Lindskog, 2020. "Optimal dividends and capital injection under dividend restrictions," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 92(3), pages 461-487, December.
    20. Yin, Chuancun & Wen, Yuzhen, 2013. "Optimal dividend problem with a terminal value for spectrally positive Lévy processes," Insurance: Mathematics and Economics, Elsevier, vol. 53(3), pages 769-773.
    21. Florin Avram & Dan Goreac & Juan Li & Xiaochi Wu, 2021. "Equity Cost Induced Dichotomy for Optimal Dividends with Capital Injections in the Cramér-Lundberg Model," Mathematics, MDPI, vol. 9(9), pages 1-27, April.
    22. Benjamin Avanzi & Jos'e-Luis P'erez & Bernard Wong & Kazutoshi Yamazaki, 2016. "On optimal joint reflective and refractive dividend strategies in spectrally positive L\'evy models," Papers 1607.01902, arXiv.org, revised Nov 2016.
    23. Chuancun Yin & Kam Chuen Yuen, 2014. "Optimal dividend problems for a jump-diffusion model with capital injections and proportional transaction costs," Papers 1409.0407, arXiv.org.
    24. 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.
    25. 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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