IDEAS home Printed from https://ideas.repec.org/a/eee/insuma/v44y2009i2p296-303.html
   My bibliography  Save this article

The Markovian regime-switching risk model with a threshold dividend strategy

Author

Listed:
  • Lu, Yi
  • Li, Shuanming

Abstract

In this paper, we study a regime-switching risk model with a threshold dividend strategy, in which the rate for the Poisson claim arrivals and the distribution of the claim amounts are driven by an underlying (external) Markov jump process. The purpose of this paper is to study the unified Gerber-Shiu discounted penalty function and the moments of the total dividend payments until ruin. We adopt an approach which is akin to the one used in [Lin, X.S., Pavlova, K.P., 2006. The compound Poisson risk model with a threshold dividend strategy. Insu.: Math. and Econ. 38, 57-80] to extend the results for the classical risk model with a threshold dividend strategy to our model. The matrix form of systems of integro-differential equations is presented and the analytical solutions to these systems are derived. Finally, numerical illustrations with exponential claim amounts are also given.

Suggested Citation

  • Lu, Yi & Li, Shuanming, 2009. "The Markovian regime-switching risk model with a threshold dividend strategy," Insurance: Mathematics and Economics, Elsevier, vol. 44(2), pages 296-303, April.
    • Handle: RePEc:eee:insuma:v:44:y:2009:i:2:p:296-303
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0167-6687(08)00054-1
    Download Restriction: Full text for ScienceDirect subscribers only
    ---><---

    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. Hansjörg Albrecher & Jürgen Hartinger, 2007. "A Risk Model with Multilayer Dividend Strategy," North American Actuarial Journal, Taylor & Francis Journals, vol. 11(2), pages 43-64.
    2. Lin, X. Sheldon & Sendova, Kristina P., 2008. "The compound Poisson risk model with multiple thresholds," Insurance: Mathematics and Economics, Elsevier, vol. 42(2), pages 617-627, April.
    3. Hans Gerber & Elias Shiu, 1998. "On the Time Value of Ruin," North American Actuarial Journal, Taylor & Francis Journals, vol. 2(1), pages 48-72.
    4. Ng, Andrew C.Y. & Yang, Hailiang, 2006. "On the joint distribution of surplus before and after ruin under a Markovian regime switching model," Stochastic Processes and their Applications, Elsevier, vol. 116(2), pages 244-266, February.
    5. Zhu, Jinxia & Yang, Hailiang, 2008. "Ruin theory for a Markov regime-switching model under a threshold dividend strategy," Insurance: Mathematics and Economics, Elsevier, vol. 42(1), pages 311-318, February.
    6. Dickson, David C. M. & Hipp, Christian, 2001. "On the time to ruin for Erlang(2) risk processes," Insurance: Mathematics and Economics, Elsevier, vol. 29(3), pages 333-344, December.
    7. Hans Gerber & Elias Shiu, 2005. "The Time Value of Ruin in a Sparre Andersen Model," North American Actuarial Journal, Taylor & Francis Journals, vol. 9(2), pages 49-69.
    8. Lin, X.Sheldon & Pavlova, Kristina P., 2006. "The compound Poisson risk model with a threshold dividend strategy," Insurance: Mathematics and Economics, Elsevier, vol. 38(1), pages 57-80, February.
    9. Li, Shuanming & Garrido, Jose, 2004. "On a class of renewal risk models with a constant dividend barrier," Insurance: Mathematics and Economics, Elsevier, vol. 35(3), pages 691-701, December.
    10. Yang, Hu & Zhang, Zhimin, 2008. "Gerber-Shiu discounted penalty function in a Sparre Andersen model with multi-layer dividend strategy," Insurance: Mathematics and Economics, Elsevier, vol. 42(3), pages 984-991, June.
    11. Yi Lu & Cary Tsai, 2007. "The Expected Discounted Penalty at Ruin for a Markov-Modulated Risk Process Perturbed by Diffusion," North American Actuarial Journal, Taylor & Francis Journals, vol. 11(2), pages 136-149.
    12. Shuanming Li & Yi Lu, 2007. "Moments of the Dividend Payments and Related Problems in a Markov-Modulated Risk Model," North American Actuarial Journal, Taylor & Francis Journals, vol. 11(2), pages 65-76.
    13. Reinhard, Jean-Marie, 1984. "On a Class of Semi-Markov Risk Models Obtained as Classical Risk Models in a Markovian Environment," ASTIN Bulletin, Cambridge University Press, vol. 14(1), pages 23-43, April.
    14. Albrecher, Hansjorg & Boxma, Onno J., 2005. "On the discounted penalty function in a Markov-dependent risk model," Insurance: Mathematics and Economics, Elsevier, vol. 37(3), pages 650-672, December.
    15. Hans Gerber & Elias Shiu, 2006. "On Optimal Dividend Strategies In The Compound Poisson Model," North American Actuarial Journal, Taylor & Francis Journals, vol. 10(2), pages 76-93.
    16. Li, Shuanming & Lu, Yi, 2008. "The Decompositions of the Discounted Penalty Functions and Dividends-Penalty Identity in a Markov-Modulated Risk Model," ASTIN Bulletin, Cambridge University Press, vol. 38(1), pages 53-71, May.
    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. Brinker, Leonie Violetta & Eisenberg, Julia, 2021. "Dividend optimisation: A behaviouristic approach," Insurance: Mathematics and Economics, Elsevier, vol. 101(PB), pages 202-224.
    2. Qinjing Qiu & Reiichiro Kawai, 2023. "Iterative Weak Approximation and Hard Bounds for Switching Diffusion," Journal of Theoretical Probability, Springer, vol. 36(2), pages 1003-1036, June.
    3. Li, Shuanming & Lu, Yi, 2009. "The distribution of total dividend payments in a Sparre Andersen model," Statistics & Probability Letters, Elsevier, vol. 79(9), pages 1246-1251, May.
    4. Yue He & Reiichiro Kawai & Yasutaka Shimizu & Kazutoshi Yamazaki, 2022. "The Gerber-Shiu discounted penalty function: A review from practical perspectives," Papers 2203.10680, arXiv.org, revised Dec 2022.
    5. He, Yue & Kawai, Reiichiro & Shimizu, Yasutaka & Yamazaki, Kazutoshi, 2023. "The Gerber-Shiu discounted penalty function: A review from practical perspectives," Insurance: Mathematics and Economics, Elsevier, vol. 109(C), pages 1-28.
    6. Wuyuan Jiang & Zhaojun Yang, 2014. "The expected discounted penalty function for two classes of risk processes perturbed by diffusion with multiple thresholds," Indian Journal of Pure and Applied Mathematics, Springer, vol. 45(4), pages 479-495, August.
    7. Jiang, Wuyuan & Yang, Zhaojun & Li, Xinping, 2012. "The discounted penalty function with multi-layer dividend strategy in the phase-type risk model," Statistics & Probability Letters, Elsevier, vol. 82(7), pages 1358-1366.
    8. Chen, Mi & Yuen, Kam Chuen & Guo, Junyi, 2014. "Survival probabilities in a discrete semi-Markov risk model," Applied Mathematics and Computation, Elsevier, vol. 232(C), pages 205-215.
    9. Sotomayor, Luz R. & Cadenillas, Abel, 2011. "Classical and singular stochastic control for the optimal dividend policy when there is regime switching," Insurance: Mathematics and Economics, Elsevier, vol. 48(3), pages 344-354, May.
    10. Brill, Percy H. & Yu, Kaiqi, 2011. "Analysis of risk models using a level crossing technique," Insurance: Mathematics and Economics, Elsevier, vol. 49(3), pages 298-309.
    11. Chadjiconstantinidis, Stathis & Papaioannou, Apostolos D., 2009. "Analysis of the Gerber-Shiu function and dividend barrier problems for a risk process with two classes of claims," Insurance: Mathematics and Economics, Elsevier, vol. 45(3), pages 470-484, December.

    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. He, Yue & Kawai, Reiichiro & Shimizu, Yasutaka & Yamazaki, Kazutoshi, 2023. "The Gerber-Shiu discounted penalty function: A review from practical perspectives," Insurance: Mathematics and Economics, Elsevier, vol. 109(C), pages 1-28.
    2. Yue He & Reiichiro Kawai & Yasutaka Shimizu & Kazutoshi Yamazaki, 2022. "The Gerber-Shiu discounted penalty function: A review from practical perspectives," Papers 2203.10680, arXiv.org, revised Dec 2022.
    3. Cheung, Eric C.K., 2011. "A generalized penalty function in Sparre Andersen risk models with surplus-dependent premium," Insurance: Mathematics and Economics, Elsevier, vol. 48(3), pages 384-397, May.
    4. Jiang, Wuyuan & Yang, Zhaojun & Li, Xinping, 2012. "The discounted penalty function with multi-layer dividend strategy in the phase-type risk model," Statistics & Probability Letters, Elsevier, vol. 82(7), pages 1358-1366.
    5. Yang, Hu & Zhang, Zhimin, 2008. "Gerber-Shiu discounted penalty function in a Sparre Andersen model with multi-layer dividend strategy," Insurance: Mathematics and Economics, Elsevier, vol. 42(3), pages 984-991, June.
    6. Zhou, Zhongbao & Xiao, Helu & Deng, Yingchun, 2015. "Markov-dependent risk model with multi-layer dividend strategy," Applied Mathematics and Computation, Elsevier, vol. 252(C), pages 273-286.
    7. Feng, Runhuan, 2009. "On the total operating costs up to default in a renewal risk model," Insurance: Mathematics and Economics, Elsevier, vol. 45(2), pages 305-314, October.
    8. Olena Ragulina & Jonas Šiaulys, 2020. "Upper Bounds and Explicit Formulas for the Ruin Probability in the Risk Model with Stochastic Premiums and a Multi-Layer Dividend Strategy," Mathematics, MDPI, vol. 8(11), pages 1-35, October.
    9. Cheung, Eric C.K. & Feng, Runhuan, 2013. "A unified analysis of claim costs up to ruin in a Markovian arrival risk model," Insurance: Mathematics and Economics, Elsevier, vol. 53(1), pages 98-109.
    10. Yang, Hu & Zhang, Zhimin, 2009. "The perturbed compound Poisson risk model with multi-layer dividend strategy," Statistics & Probability Letters, Elsevier, vol. 79(1), pages 70-78, January.
    11. Lin, X. Sheldon & Sendova, Kristina P., 2008. "The compound Poisson risk model with multiple thresholds," Insurance: Mathematics and Economics, Elsevier, vol. 42(2), pages 617-627, April.
    12. Landriault, David, 2008. "Constant dividend barrier in a risk model with interclaim-dependent claim sizes," Insurance: Mathematics and Economics, Elsevier, vol. 42(1), pages 31-38, February.
    13. Runhuan Feng & Yasutaka Shimizu, 2013. "On a Generalization from Ruin to Default in a Lévy Insurance Risk Model," Methodology and Computing in Applied Probability, Springer, vol. 15(4), pages 773-802, December.
    14. Albrecher, Hansjörg & Constantinescu, Corina & Pirsic, Gottlieb & Regensburger, Georg & Rosenkranz, Markus, 2010. "An algebraic operator approach to the analysis of Gerber-Shiu functions," Insurance: Mathematics and Economics, Elsevier, vol. 46(1), pages 42-51, February.
    15. Li, Shu & Landriault, David & Lemieux, Christiane, 2015. "A risk model with varying premiums: Its risk management implications," Insurance: Mathematics and Economics, Elsevier, vol. 60(C), pages 38-46.
    16. Ehyter Matías Martín-González & Antonio Murillo-Salas & Henry Pantí, 2022. "Gerber-Shiu Function for a Class of Markov-Modulated Lévy Risk Processes with Two-Sided Jumps," Methodology and Computing in Applied Probability, Springer, vol. 24(4), pages 2779-2800, December.
    17. Li, Shuanming & Lu, Yi, 2009. "The distribution of total dividend payments in a Sparre Andersen model," Statistics & Probability Letters, Elsevier, vol. 79(9), pages 1246-1251, May.
    18. Liu, Xiangdong & Xiong, Jie & Zhang, Shuaiqi, 2015. "The Gerber–Shiu discounted penalty function in the classical risk model with impulsive dividend policy," Statistics & Probability Letters, Elsevier, vol. 107(C), pages 183-190.
    19. Xin Zhang, 2008. "On the Ruin Problem in a Markov-Modulated Risk Model," Methodology and Computing in Applied Probability, Springer, vol. 10(2), pages 225-238, June.
    20. Cheung, Eric C.K. & Landriault, David, 2010. "A generalized penalty function with the maximum surplus prior to ruin in a MAP risk model," Insurance: Mathematics and Economics, Elsevier, vol. 46(1), pages 127-134, February.

    More about this item

    Keywords

    C02 Gerber-Shiu function Integro-differential equation Present value of dividend payments Regime-switching model Threshold dividend strategy;

    JEL classification:

    • C02 - Mathematical and Quantitative Methods - - General - - - Mathematical Economics

    Statistics

    Access and download statistics

    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:44:y:2009:i:2:p:296-303. 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.