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On the decomposition of the absolute ruin probability in a perturbed compound Poisson surplus process with debit interest

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  • Jun Cai
  • Hailiang Yang

Abstract

We consider a compound Poisson surplus process perturbed by diffusion with debit interest. When the surplus is below zero or the company is on deficit, the company is allowed to borrow money at a debit interest rate to continue its business as long as its debt is at a reasonable level. When the surplus of a company is below a certain critical level, the company is no longer profitable, we say that absolute ruin occurs at this situation. In this risk model, absolute ruin may be caused by a claim or by oscillation. Thus, the absolute ruin probability in the model is decomposed as the sum of two absolute ruin probabilities, where one is the probability that absolute ruin is caused by a claim and the other is the probability that absolute ruin is caused by oscillation. In this paper, we first give the integro-differential equations satisfied by the absolute ruin probabilities and then derive the defective renewal equations for the absolute ruin probabilities. Using these defective renewal equations, we derive the asymptotical forms of the absolute ruin probabilities when the distributions of claim sizes are heavy-tailed and light-tailed. Finally, we derive explicit expressions for the absolute ruin probabilities when claim sizes are exponentially distributed. Copyright Springer Science+Business Media, LLC 2014

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  • Jun Cai & Hailiang Yang, 2014. "On the decomposition of the absolute ruin probability in a perturbed compound Poisson surplus process with debit interest," Annals of Operations Research, Springer, vol. 212(1), pages 61-77, January.
    • Handle: RePEc:spr:annopr:v:212:y:2014:i:1:p:61-77:10.1007/s10479-011-1032-y
    DOI: 10.1007/s10479-011-1032-y
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    References listed on IDEAS

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    1. Wang, Guojing, 2001. "A decomposition of the ruin probability for the risk process perturbed by diffusion," Insurance: Mathematics and Economics, Elsevier, vol. 28(1), pages 49-59, February.
    2. Yang, Hu & Zhang, Zhimin & Lan, Chunmei, 2008. "On the time value of absolute ruin for a multi-layer compound Poisson model under interest force," Statistics & Probability Letters, Elsevier, vol. 78(13), pages 1835-1845, September.
    3. Hans Gerber & Hailiang Yang, 2007. "Absolute Ruin Probabilities in a Jump Diffusion Risk Model with Investment," North American Actuarial Journal, Taylor & Francis Journals, vol. 11(3), pages 159-169.
    4. Chunwei Wang & Chuancun Yin, 2009. "Dividend payments in the classical risk model under absolute ruin with debit interest," Applied Stochastic Models in Business and Industry, John Wiley & Sons, vol. 25(3), pages 247-262, May.
    5. Chuancun Yin & Chunwei Wang, 2010. "The Perturbed Compound Poisson Risk Process with Investment and Debit Interest," Methodology and Computing in Applied Probability, Springer, vol. 12(3), pages 391-413, September.
    6. 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.
    7. Yuan, Haili & Hu, Yijun, 2008. "Absolute ruin in the compound Poisson risk model with constant dividend barrier," Statistics & Probability Letters, Elsevier, vol. 78(14), pages 2086-2094, October.
    8. Veraverbeke, Noel, 1993. "Asymptotic estimates for the probability of ruin in a Poisson model with diffusion," Insurance: Mathematics and Economics, Elsevier, vol. 13(1), pages 57-62, September.
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    Cited by:

    1. Mehmet Akif Yazici & Nail Akar, 2017. "The finite/infinite horizon ruin problem with multi-threshold premiums: a Markov fluid queue approach," Annals of Operations Research, Springer, vol. 252(1), pages 85-99, May.

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