IDEAS home Printed from https://ideas.repec.org/a/gam/jrisks/v7y2019i4p120-d296153.html
   My bibliography  Save this article

The Løkka–Zervos Alternative for a Cramér–Lundberg Process with Exponential Jumps

Author

Listed:
  • Florin Avram

    (Laboratoire de Mathématiques Appliquées, Université de Pau, 64012 Pau, France)

  • Dan Goreac

    (School of Mathematics and Statistics, Shandong University, Weihai 264209, China
    LAMA, Univ Gustave Eiffel, UPEM, Univ Paris Est Creteil, CNRS, F-77447 Marne-la-Vallée, France)

  • Jean-François Renaud

    (Département de Mathématiques, Université du Québec à Montréal (UQAM), Montréal, QC H2X 3Y7, Canada)

Abstract

In this paper, we study a stochastic control problem faced by an insurance company allowed to pay out dividends and make capital injections. As in (Løkka and Zervos (2008); Lindensjö and Lindskog (2019)), for a Brownian motion risk process, and in Zhu and Yang (2016), for diffusion processes, we will show that the so-called Løkka–Zervos alternative also holds true in the case of a Cramér–Lundberg risk process with exponential claims. More specifically, we show that: if the cost of capital injections is low , then according to a double-barrier strategy, it is optimal to pay dividends and inject capital, meaning ruin never occurs; and if the cost of capital injections is high , then according to a single-barrier strategy, it is optimal to pay dividends and never inject capital, meaning ruin occurs at the first passage below zero.

Suggested Citation

  • Florin Avram & Dan Goreac & Jean-François Renaud, 2019. "The Løkka–Zervos Alternative for a Cramér–Lundberg Process with Exponential Jumps," Risks, MDPI, vol. 7(4), pages 1-9, December.
    • Handle: RePEc:gam:jrisks:v:7:y:2019:i:4:p:120-:d:296153
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-9091/7/4/120/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-9091/7/4/120/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Zhu, Jinxia & Yang, Hailiang, 2016. "Optimal capital injection and dividend distribution for growth restricted diffusion models with bankruptcy," Insurance: Mathematics and Economics, Elsevier, vol. 70(C), pages 259-271.
    2. 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.
    3. 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.
    4. 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.
    5. Gerber, Hans U. & Shiu, Elias S.W. & Smith, Nathaniel, 2006. "Maximizing Dividends without Bankruptcy," ASTIN Bulletin, Cambridge University Press, vol. 36(1), pages 5-23, May.
    6. Kristoffer Lindensjo & Filip Lindskog, 2019. "Optimal dividends and capital injection under dividend restrictions," Papers 1902.06294, arXiv.org.
    7. Merton H. Miller & Franco Modigliani, 1961. "Dividend Policy, Growth, and the Valuation of Shares," The Journal of Business, University of Chicago Press, vol. 34, pages 411-411.
    8. Dufresne, Francois & Gerber, Hans U., 1991. "Risk theory for the compound Poisson process that is perturbed by diffusion," Insurance: Mathematics and Economics, Elsevier, vol. 10(1), pages 51-59, March.
    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. GOREAC, Dan & LI, Juan & XU, Boxiang, 2022. "Linearisation Techniques and the Dual Algorithm for a Class of Mixed Singular/Continuous Control Problems in Reinsurance. Part I: Theoretical Aspects," Applied Mathematics and Computation, Elsevier, vol. 431(C).

    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. 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.
    2. Ernst, Philip A. & Imerman, Michael B. & Shepp, Larry & Zhou, Quan, 2022. "Fiscal stimulus as an optimal control problem," Stochastic Processes and their Applications, Elsevier, vol. 150(C), pages 1091-1108.
    3. 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.
    4. 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.
    5. 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.
    6. Décamps, Jean-Paul & Villeneuve, Stéphane, 2022. "Learning about profitability and dynamic cash management," Journal of Economic Theory, Elsevier, vol. 205(C).
    7. Gajek, Lesław & Kuciński, Łukasz, 2017. "Complete discounted cash flow valuation," Insurance: Mathematics and Economics, Elsevier, vol. 73(C), pages 1-19.
    8. Alex S. L. Tse, 2020. "Dividend policy and capital structure of a defaultable firm," Mathematical Finance, Wiley Blackwell, vol. 30(3), pages 961-994, July.
    9. 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.
    10. 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.
    11. Kristoffer Lindensjo & Filip Lindskog, 2019. "Optimal dividends and capital injection under dividend restrictions," Papers 1902.06294, arXiv.org.
    12. Ferrari, Giorgio, 2018. "On a Class of Singular Stochastic Control Problems for Reflected Diffusions," Center for Mathematical Economics Working Papers 592, Center for Mathematical Economics, Bielefeld University.
    13. Jean-François Renaud, 2019. "De Finetti’s Control Problem with Parisian Ruin for Spectrally Negative Lévy Processes," Risks, MDPI, vol. 7(3), pages 1-11, July.
    14. Gerber, Hans U. & Shiu, Elias S.W. & Yang, Hailiang, 2013. "Valuing equity-linked death benefits in jump diffusion models," Insurance: Mathematics and Economics, Elsevier, vol. 53(3), pages 615-623.
    15. Giorgio Ferrari & Patrick Schuhmann, 2018. "An Optimal Dividend Problem with Capital Injections over a Finite Horizon," Papers 1804.04870, arXiv.org, revised May 2019.
    16. 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.
    17. 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.
    18. 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.
    19. Décamps, Jean-Paul & Villeneuve, Stéphane, 2019. "Dynamics of cash holdings, learning about profitability, and access to the market," TSE Working Papers 19-1046, Toulouse School of Economics (TSE), revised Sep 2020.
    20. Yao, Dingjun & Yang, Hailiang & Wang, Rongming, 2014. "Optimal risk and dividend control problem with fixed costs and salvage value: Variance premium principle," Economic Modelling, Elsevier, vol. 37(C), pages 53-64.

    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:gam:jrisks:v:7:y:2019:i:4:p:120-:d:296153. 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: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    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.