IDEAS home Printed from https://ideas.repec.org/a/eee/insuma/v79y2018icp57-68.html
   My bibliography  Save this article

A time of ruin constrained optimal dividend problem for spectrally one-sided Lévy processes

Author

Listed:
  • Hernández, Camilo
  • Junca, Mauricio
  • Moreno-Franco, Harold

Abstract

We introduce a longevity feature to the classical optimal dividend problem by adding a constraint on the time of ruin of the firm. We consider De Finetti’s problem for one-sided Lévy risk models in both scenarios with and without fix transaction costs. To characterize the solution to the aforementioned models we introduce the dual problem and show that the complementary slackness conditions are satisfied and therefore there is no duality gap. As a consequence the optimal value function can be obtained as the pointwise infimum of auxiliary value functions. Finally, we illustrate our findings with a series of numerical examples.

Suggested Citation

  • Hernández, Camilo & Junca, Mauricio & Moreno-Franco, Harold, 2018. "A time of ruin constrained optimal dividend problem for spectrally one-sided Lévy processes," Insurance: Mathematics and Economics, Elsevier, vol. 79(C), pages 57-68.
    • Handle: RePEc:eee:insuma:v:79:y:2018:i:c:p:57-68
    DOI: 10.1016/j.insmatheco.2017.12.011
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0167668717303712
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.insmatheco.2017.12.011?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Pablo Azcue & Nora Muler, 2005. "Optimal Reinsurance And Dividend Distribution Policies In The Cramér‐Lundberg Model," Mathematical Finance, Wiley Blackwell, vol. 15(2), pages 261-308, April.
    2. 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.
    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. Hans U. Gerber, 1972. "Games of Economic Survival with Discrete- and Continuous-Income Processes," Operations Research, INFORMS, vol. 20(1), pages 37-45, February.
    5. Loeffen, R.L., 2009. "An optimal dividends problem with transaction costs for spectrally negative Lévy processes," Insurance: Mathematics and Economics, Elsevier, vol. 45(1), pages 41-48, August.
    6. Michael I. Taksar, 2000. "Optimal risk and dividend distribution control models for an insurance company," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 51(1), pages 1-42, February.
    7. Hernández, Camilo & Junca, Mauricio, 2015. "Optimal dividend payments under a time of ruin constraint: Exponential claims," Insurance: Mathematics and Economics, Elsevier, vol. 65(C), pages 136-142.
    8. Thonhauser, Stefan & Albrecher, Hansjorg, 2007. "Dividend maximization under consideration of the time value of ruin," Insurance: Mathematics and Economics, Elsevier, vol. 41(1), pages 163-184, July.
    9. 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.
    10. Peter Grandits, 2015. "An optimal consumption problem in finite time with a constraint on the ruin probability," Finance and Stochastics, Springer, vol. 19(4), pages 791-847, October.
    11. 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.
    12. Loeffen, Ronnie L. & Renaud, Jean-François, 2010. "De Finetti's optimal dividends problem with an affine penalty function at ruin," Insurance: Mathematics and Economics, Elsevier, vol. 46(1), pages 98-108, February.
    13. Chuancun Yin & Yuzhen Wen, 2013. "Optimal dividends problem with a terminal value for spectrally positive Levy processes," Papers 1302.6011, arXiv.org.
    14. F. Avram & Z. Palmowski & M. R. Pistorius, 2011. "On Gerber-Shiu functions and optimal dividend distribution for a L\'{e}vy risk process in the presence of a penalty function," Papers 1110.4965, arXiv.org, revised Jun 2015.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Yangmin Zhong & Huaping Huang, 2023. "Cash Flow Optimization on Insurance: An Application of Fixed-Point Theory," Mathematics, MDPI, vol. 11(4), pages 1-12, February.
    2. Christian Hipp, 2018. "Company Value with Ruin Constraint in Lundberg Models," Risks, MDPI, vol. 6(3), pages 1-15, July.
    3. 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.
    4. Soren Christensen & Kristoffer Lindensjo, 2019. "Moment constrained optimal dividends: precommitment \& consistent planning," Papers 1909.10749, arXiv.org.
    5. Mauricio Junca & Harold A. Moreno-Franco & José Luis Pérez, 2019. "Optimal Bail-Out Dividend Problem with Transaction Cost and Capital Injection Constraint," Risks, MDPI, vol. 7(1), pages 1-24, January.
    6. 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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Camilo Hernandez & Mauricio Junca & Harold Moreno-Franco, 2016. "A time of ruin constrained optimal dividend problem for spectrally one-sided L\'evy processes," Papers 1608.02550, arXiv.org, revised May 2017.
    2. 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.
    3. Ran Xu & Wenyuan Wang & Jose Garrido, 2022. "Optimal Dividend Strategy Under Parisian Ruin with Affine Penalty," Methodology and Computing in Applied Probability, Springer, vol. 24(3), pages 1385-1409, September.
    4. Ewa Marciniak & Zbigniew Palmowski, 2018. "On the Optimal Dividend Problem in the Dual Model with Surplus-Dependent Premiums," Journal of Optimization Theory and Applications, Springer, vol. 179(2), pages 533-552, November.
    5. 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.
    6. Ewa Marciniak & Zbigniew Palmowski, 2016. "On the Optimal Dividend Problem in the Dual Model with Surplus-Dependent Premiums," Papers 1605.04584, arXiv.org.
    7. Liang, Zhibin & Young, Virginia R., 2012. "Dividends and reinsurance under a penalty for ruin," Insurance: Mathematics and Economics, Elsevier, vol. 50(3), pages 437-445.
    8. 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.
    9. 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.
    10. Masahiko Egami & Kazutoshi Yamazaki, 2010. "Solving Optimal Dividend Problems via Phase-Type Fitting Approximation of Scale Functions," Discussion papers e-10-011, Graduate School of Economics Project Center, Kyoto University.
    11. Xiaoqing Liang & Zbigniew Palmowski, 2016. "A note on optimal expected utility of dividend payments with proportional reinsurance," Papers 1605.06849, arXiv.org, revised May 2017.
    12. Benjamin Avanzi & Debbie Kusch Falden & Mogens Steffensen, 2022. "Stable Dividends under Linear-Quadratic Optimization," Papers 2210.03494, arXiv.org.
    13. Kazutoshi Yamazaki, 2017. "Inventory Control for Spectrally Positive Lévy Demand Processes," Mathematics of Operations Research, INFORMS, vol. 42(1), pages 212-237, January.
    14. Gajek, Lesław & Kuciński, Łukasz, 2017. "Complete discounted cash flow valuation," Insurance: Mathematics and Economics, Elsevier, vol. 73(C), pages 1-19.
    15. Ying Shen & Chuancun Yin & Kam Chuen Yuen, 2011. "Alternative approach to the optimality of the threshold strategy for spectrally negative Levy processes," Papers 1101.0446, arXiv.org, revised Feb 2014.
    16. Pelsser, Antoon A.J. & Laeven, Roger J.A., 2013. "Optimal dividends and ALM under unhedgeable risk," Insurance: Mathematics and Economics, Elsevier, vol. 53(3), pages 515-523.
    17. Benjamin Avanzi & Hayden Lau & Bernard Wong, 2020. "Optimal periodic dividend strategies for spectrally negative L\'evy processes with fixed transaction costs," Papers 2004.01838, arXiv.org, revised Dec 2020.
    18. 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.
    19. Ewa Marciniak & Zbigniew Palmowski, 2016. "On the Optimal Dividend Problem for Insurance Risk Models with Surplus-Dependent Premiums," Journal of Optimization Theory and Applications, Springer, vol. 168(2), pages 723-742, February.
    20. Zhuo Jin & Zuo Quan Xu & Bin Zou, 2020. "A Perturbation Approach to Optimal Investment, Liability Ratio, and Dividend Strategies," Papers 2012.06703, arXiv.org, revised May 2021.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:insuma:v:79:y:2018:i:c:p:57-68. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/locate/inca/505554 .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.