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On a multi-threshold compound Poisson process perturbed by diffusion

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  • Mitric, Ilie-Radu
  • Sendova, Kristina P.
  • Tsai, Cary Chi-Liang

Abstract

We consider a multi-layer compound Poisson surplus process perturbed by diffusion and examine the behaviour of the Gerber-Shiu discounted penalty function. We derive the general solution to a certain second order integro-differential equation. This permits us to provide explicit expressions for the Gerber-Shiu function depending on the current surplus level. The advantage of our proposed approach is that if the diffusion term converges to zero, the above-mentioned explicit expressions converge to those under the classical compound Poisson model, provided that the same initial conditions apply. This is subsequently illustrated by an extended example related to the probability of ultimate ruin.

Suggested Citation

  • Mitric, Ilie-Radu & Sendova, Kristina P. & Tsai, Cary Chi-Liang, 2010. "On a multi-threshold compound Poisson process perturbed by diffusion," Statistics & Probability Letters, Elsevier, vol. 80(5-6), pages 366-375, March.
    • Handle: RePEc:eee:stapro:v:80:y:2010:i:5-6:p:366-375
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    References listed on IDEAS

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    1. Lin, X. Sheldon & Willmot, Gordon E., 1999. "Analysis of a defective renewal equation arising in ruin theory," Insurance: Mathematics and Economics, Elsevier, vol. 25(1), pages 63-84, September.
    2. Hans Gerber & Elias Shiu, 1998. "On the Time Value of Ruin," North American Actuarial Journal, Taylor & Francis Journals, vol. 2(1), pages 48-72.
    3. 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.
    4. 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.
    5. Jun Cai & Chengming Xu, 2006. "On The Decomposition Of The Ruin Probability For A Jump-Diffusion Surplus Process Compounded By A Geometric Brownian Motion," North American Actuarial Journal, Taylor & Francis Journals, vol. 10(2), pages 120-129.
    6. 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.
    7. 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.
    8. Tsai, Cary Chi-Liang & Willmot, Gordon E., 2002. "A generalized defective renewal equation for the surplus process perturbed by diffusion," Insurance: Mathematics and Economics, Elsevier, vol. 30(1), pages 51-66, February.
    9. 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.
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