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On a Class of Semi-Markov Risk Models Obtained as Classical Risk Models in a Markovian Environment

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  • Reinhard, Jean-Marie

Abstract

We consider a risk model in which the claim inter-arrivals and amounts depend on a markovian environment process. Semi-Markov risk models are so introduced in a quite natural way. We derive some quantities of interest for the risk process and obtain a necessary and sufficient condition for the fairness of the risk (positive asymptotic non-ruin probabilities). These probabilities are explicitly calculated in a particular case (two possible states for the environment, exponential claim amounts distributions).

Suggested Citation

  • 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.
    • Handle: RePEc:cup:astinb:v:14:y:1984:i:01:p:23-43_00
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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. Jasiulewicz, Helena, 2001. "Probability of ruin with variable premium rate in a Markovian environment," Insurance: Mathematics and Economics, Elsevier, vol. 29(2), pages 291-296, October.
    3. Shao, Jia & Papaioannou, Apostolos D. & Pantelous, Athanasios A., 2017. "Pricing and simulating catastrophe risk bonds in a Markov-dependent environment," Applied Mathematics and Computation, Elsevier, vol. 309(C), pages 68-84.
    4. XIAO, Lin, 2022. "Compound binomial risk model in a Markovian environment with capital cost and the calculation algorithm," Applied Mathematics and Computation, Elsevier, vol. 424(C).
    5. Gajek, Lesław & Rudź, Marcin, 2018. "Banach Contraction Principle and ruin probabilities in regime-switching models," Insurance: Mathematics and Economics, Elsevier, vol. 80(C), pages 45-53.
    6. Siegl, Thomas & Tichy, Robert F., 1999. "A process with stochastic claim frequency and a linear dividend barrier," Insurance: Mathematics and Economics, Elsevier, vol. 24(1-2), pages 51-65, March.
    7. 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.
    8. 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.
    9. H. Albrecher, 1998. "Dependent Risks and Ruin Probabilities in Insurance," Working Papers ir98072, International Institute for Applied Systems Analysis.
    10. Yuen, K. C. & Guo, J. Y., 2001. "Ruin probabilities for time-correlated claims in the compound binomial model," Insurance: Mathematics and Economics, Elsevier, vol. 29(1), pages 47-57, August.
    11. Lu, Yi & Li, Shuanming, 2005. "On the probability of ruin in a Markov-modulated risk model," Insurance: Mathematics and Economics, Elsevier, vol. 37(3), pages 522-532, December.
    12. Li, Jingchao & Dickson, David C.M. & Li, Shuanming, 2015. "Some ruin problems for the MAP risk model," Insurance: Mathematics and Economics, Elsevier, vol. 65(C), pages 1-8.
    13. 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.
    14. Bauerle, Nicole, 1996. "Some results about the expected ruin time in Markov-modulated risk models," Insurance: Mathematics and Economics, Elsevier, vol. 18(2), pages 119-127, July.
    15. Cossette, Helene & Landriault, David & Marceau, Etienne, 2004. "Compound binomial risk model in a markovian environment," Insurance: Mathematics and Economics, Elsevier, vol. 35(2), pages 425-443, October.
    16. Gajek, Lesław & Rudź, Marcin, 2017. "A generalization of Gerber’s inequality for ruin probabilities in risk-switching models," Statistics & Probability Letters, Elsevier, vol. 129(C), pages 236-240.

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