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A two-dimensional ruin problem on the positive quadrant


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  • Avram, Florin
  • Palmowski, Zbigniew
  • Pistorius, Martijn
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    In this paper we study the joint ruin problem for two insurance companies that divide between them both claims and premia in some specified proportions (modeling two branches of the same insurance company or an insurance and re-insurance company). Modeling the risk processes of the insurance companies by Cramér-Lundberg processes we obtain the Laplace transform in space of the probability that either of the insurance companies is ruined in finite time. Subsequently, for exponentially distributed claims, we derive an explicit analytical expression for this joint ruin probability by explicitly inverting this Laplace transform. We also provide a characterization of the Laplace transform of the joint ruin time.

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    Bibliographic Info

    Article provided by Elsevier in its journal Insurance: Mathematics and Economics.

    Volume (Year): 42 (2008)
    Issue (Month): 1 (February)
    Pages: 227-234

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    Handle: RePEc:eee:insuma:v:42:y:2008:i:1:p:227-234

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    Cited by:
    1. Macci, Claudio, 2011. "Large deviations for estimators of unknown probabilities, with applications in risk theory," Statistics & Probability Letters, Elsevier, vol. 81(1), pages 16-24, January.
    2. Liu, Jingchen & Woo, Jae-Kyung, 2014. "Asymptotic analysis of risk quantities conditional on ruin for multidimensional heavy-tailed random walks," Insurance: Mathematics and Economics, Elsevier, vol. 55(C), pages 1-9.
    3. Bäuerle, Nicole & Blatter, Anja, 2011. "Optimal control and dependence modeling of insurance portfolios with Lévy dynamics," Insurance: Mathematics and Economics, Elsevier, vol. 48(3), pages 398-405, May.
    4. Zhang, Yuanyuan & Wang, Wensheng, 2012. "Ruin probabilities of a bidimensional risk model with investment," Statistics & Probability Letters, Elsevier, vol. 82(1), pages 130-138.
    5. Irmina Czarna & Zbigniew Palmowski, 2009. "De Finetti's dividend problem and impulse control for a two-dimensional insurance risk process," Papers 0906.2100,, revised Feb 2011.
    6. Debicki, K. & Kosinski, K.M. & Mandjes, M. & Rolski, T., 2010. "Extremes of multidimensional Gaussian processes," Stochastic Processes and their Applications, Elsevier, vol. 120(12), pages 2289-2301, December.
    7. Castañer, A. & Claramunt, M.M. & Lefèvre, C., 2013. "Survival probabilities in bivariate risk models, with application to reinsurance," Insurance: Mathematics and Economics, Elsevier, vol. 53(3), pages 632-642.
    8. Gong, Lan & Badescu, Andrei L. & Cheung, Eric C.K., 2012. "Recursive methods for a multi-dimensional risk process with common shocks," Insurance: Mathematics and Economics, Elsevier, vol. 50(1), pages 109-120.


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