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Ruin theory for a Markov regime-switching model under a threshold dividend strategy

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  • Zhu, Jinxia
  • Yang, Hailiang

Abstract

In this paper, we study a Markov regime-switching risk model where dividends are paid out according to a certain threshold strategy depending on the underlying Markovian environment process. We are interested in these quantities: ruin probabilities, deficit at ruin and expected ruin time. To study them, we introduce functions involving the deficit at ruin and the indicator of the event that ruin occurs. We show that the above functions and the expectations of the time to ruin as functions of the initial capital satisfy systems of integro-differential equations. Closed form solutions are derived when the underlying Markovian environment process has only two states and the claim size distributions are exponential.

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  • 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.
    • Handle: RePEc:eee:insuma:v:42:y:2008:i:1:p:311-318
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    References listed on IDEAS

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    1. Hans Gerber & Elias Shiu, 1998. "On the Time Value of Ruin," North American Actuarial Journal, Taylor & Francis Journals, vol. 2(1), pages 48-72.
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    3. 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.
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    7. 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.
    8. Sheldon Lin, X. & E. Willmot, Gordon & Drekic, Steve, 2003. "The classical risk model with a constant dividend barrier: analysis of the Gerber-Shiu discounted penalty function," Insurance: Mathematics and Economics, Elsevier, vol. 33(3), pages 551-566, December.
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    Cited by:

    1. 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.
    2. Zhengjun Jiang & Martijn Pistorius, 2008. "Optimal dividend distribution under Markov-regime switching," Papers 0812.4978, arXiv.org, revised Apr 2011.
    3. 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.
    4. Zhengjun Jiang & Martijn Pistorius, 2012. "Optimal dividend distribution under Markov regime switching," Finance and Stochastics, Springer, vol. 16(3), pages 449-476, July.
    5. Julia Eisenberg & Paul Kruhner, 2016. "The Impact of Negative Interest Rates on Optimal Capital Injections," Papers 1612.06654, arXiv.org.
    6. Zhu, Jinxia & Chen, Feng, 2013. "Dividend optimization for regime-switching general diffusions," Insurance: Mathematics and Economics, Elsevier, vol. 53(2), pages 439-456.
    7. Wang, Guanqing & Wang, Guojing & Yang, Hailiang, 2016. "On a multi-dimensional risk model with regime switching," Insurance: Mathematics and Economics, Elsevier, vol. 68(C), pages 73-83.
    8. Eisenberg, Julia & Krühner, Paul, 2018. "The impact of negative interest rates on optimal capital injections," Insurance: Mathematics and Economics, Elsevier, vol. 82(C), pages 1-10.
    9. 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.
    10. Jiaqin Wei & Hailiang Yang & Rongming Wang, 2010. "Classical and Impulse Control for the Optimization of Dividend and Proportional Reinsurance Policies with Regime Switching," Journal of Optimization Theory and Applications, Springer, vol. 147(2), pages 358-377, November.
    11. 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.
    12. Liang, Xue & Wang, Guojing, 2012. "On a reduced form credit risk model with common shock and regime switching," Insurance: Mathematics and Economics, Elsevier, vol. 51(3), pages 567-575.
    13. Elliott, Robert J. & Chen, Zhiping & Duan, Qihong, 2009. "Insurance claims modulated by a hidden Brownian marked point process," Insurance: Mathematics and Economics, Elsevier, vol. 45(2), pages 163-172, October.

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