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Optimizing Dividends and Capital Injections Limited by Bankruptcy, and Practical Approximations for the Cramér-Lundberg Process

Author

Listed:
  • Florin Avram

    (Université de Pau)

  • Dan Goreac

    (Shandong University
    UPEM, Univ Paris Est Creteil, CNRS)

  • Rim Adenane

    (Université Ibn-Tofail)

  • Ulyses Solon

    (Université de Pau
    University of the Philippines Diliman)

Abstract

The recent papers Gajek and Kucinsky (Insur Math Econ 73:1–19, 2017) and Avram et al. (Mathematics 9(9):931, 2021) cost induced dichotomy for optimal dividends in the cramr-lundberg model. Avram et al. (Mathematics 9(9):931, 2021) investigated the control problem of optimizing dividends when limiting capital injections stopped upon bankruptcy. The first paper works under the spectrally negative Lévy model; the second works under the Cramér-Lundberg model with exponential jumps, where the results are considerably more explicit. The current paper has three purposes. First, it illustrates the fact that quite reasonable approximations of the general problem may be obtained using the particular exponential case studied in Avram et al. cost induced dichotomy for optimal dividends in the Cramér-Lundberg model (Avram et al. in Mathematics 9(9):931, 2021). Secondly, it extends the results to the case when a final penalty P is taken into consideration as well besides a proportional cost $$k>1$$ k > 1 for capital injections. This requires amending the “scale and Gerber-Shiu functions” already introduced in Gajek and Kucinsky (Insur Math Econ 73:1–19, 2017). Thirdly, in the exponential case, the results will be made even more explicit by employing the Lambert-W function. This tool has particular importance in computational aspects and can be employed in theoretical aspects such as asymptotics.

Suggested Citation

  • Florin Avram & Dan Goreac & Rim Adenane & Ulyses Solon, 2022. "Optimizing Dividends and Capital Injections Limited by Bankruptcy, and Practical Approximations for the Cramér-Lundberg Process," Methodology and Computing in Applied Probability, Springer, vol. 24(4), pages 2339-2371, December.
    • Handle: RePEc:spr:metcap:v:24:y:2022:i:4:d:10.1007_s11009-021-09916-z
    DOI: 10.1007/s11009-021-09916-z
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    References listed on IDEAS

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    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. 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. Løkka, Arne & Zervos, Mihail, 2008. "Optimal dividend and issuance of equity policies in the presence of proportional costs," Insurance: Mathematics and Economics, Elsevier, vol. 42(3), pages 954-961, June.
    4. 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.
    5. Kulenko, Natalie & Schmidli, Hanspeter, 2008. "Optimal dividend strategies in a Cramér-Lundberg model with capital injections," Insurance: Mathematics and Economics, Elsevier, vol. 43(2), pages 270-278, October.
    6. Kristoffer Lindensjo & Filip Lindskog, 2019. "Optimal dividends and capital injection under dividend restrictions," Papers 1902.06294, arXiv.org.
    7. Gerber, Hans U. & Shiu, Elias S.W. & Smith, Nathaniel, 2008. "Methods for estimating the optimal dividend barrier and the probability of ruin," Insurance: Mathematics and Economics, Elsevier, vol. 42(1), pages 243-254, February.
    8. Avram, F. & Pistorius, M., 2014. "On matrix exponential approximations of ruin probabilities for the classic and Brownian perturbed Cramér–Lundberg processes," Insurance: Mathematics and Economics, Elsevier, vol. 59(C), pages 57-64.
    9. Gajek, Lesław & Kuciński, Łukasz, 2017. "Complete discounted cash flow valuation," Insurance: Mathematics and Economics, Elsevier, vol. 73(C), pages 1-19.
    10. Florin Avram & Andras Horváth & Serge Provost & Ulyses Solon, 2019. "On the Padé and Laguerre–Tricomi–Weeks Moments Based Approximations of the Scale Function W and of the Optimal Dividends Barrier for Spectrally Negative Lévy Risk Processes," Risks, MDPI, vol. 7(4), pages 1-24, December.
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