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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
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    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.
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    Cited by:

    1. Goreac, Dan & Li, Juan & Wang, Pangbo & Xu, Boxiang, 2024. "Linearisation techniques and the dual algorithm for a class of mixed singular/continuous control problems in reinsurance. Part II: Numerical aspects," Applied Mathematics and Computation, Elsevier, vol. 473(C).
    2. 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).

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