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Recursive methods for a multi-dimensional risk process with common shocks

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  • Gong, Lan
  • Badescu, Andrei L.
  • Cheung, Eric C.K.

Abstract

In this paper, a multi-dimensional risk model with common shocks is studied. Using a simple probabilistic approach via observing the risk processes at claim instants, recursive integral formulas are developed for the survival probabilities as well as for a class of Gerber–Shiu expected discounted penalty functions that include the surplus levels at ruin. Under the assumption of exponential or mixed Erlang claims, the recursive integrals can be simplified to give recursive sums which are computationally more tractable. Numerical examples including an optimal capital allocation problem are also given towards the end.

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  • 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.
    • Handle: RePEc:eee:insuma:v:50:y:2012:i:1:p:109-120
    DOI: 10.1016/j.insmatheco.2011.10.007
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    References listed on IDEAS

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    3. G. A. Delsing & M. R. H. Mandjes & P. J. C. Spreij & E. M. M. Winands, 2020. "Asymptotics and Approximations of Ruin Probabilities for Multivariate Risk Processes in a Markovian Environment," Methodology and Computing in Applied Probability, Springer, vol. 22(3), pages 927-948, September.
    4. Badila, E.S. & Boxma, O.J. & Resing, J.A.C., 2015. "Two parallel insurance lines with simultaneous arrivals and risks correlated with inter-arrival times," Insurance: Mathematics and Economics, Elsevier, vol. 61(C), pages 48-61.
    5. 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.
    6. 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.
    7. Ivanovs, Jevgenijs & Boxma, Onno, 2015. "A bivariate risk model with mutual deficit coverage," Insurance: Mathematics and Economics, Elsevier, vol. 64(C), pages 126-134.
    8. Nicole Bauerle & Gregor Leimcke, 2021. "Bayesian optimal investment and reinsurance with dependent financial and insurance risks," Papers 2103.05777, arXiv.org.
    9. Yan Zhang & Peibiao Zhao & Rufei Ma, 2022. "Robust Optimal Excess-of-Loss Reinsurance and Investment Problem with more General Dependent Claim Risks and Defaultable Risk," Methodology and Computing in Applied Probability, Springer, vol. 24(4), pages 2743-2777, December.
    10. Albrecher, Hansjörg & Cheung, Eric C.K. & Liu, Haibo & Woo, Jae-Kyung, 2022. "A bivariate Laguerre expansions approach for joint ruin probabilities in a two-dimensional insurance risk process," Insurance: Mathematics and Economics, Elsevier, vol. 103(C), pages 96-118.
    11. Xiang Hu & Lianzeng Zhang, 2016. "Ruin Probability in a Correlated Aggregate Claims Model with Common Poisson Shocks: Application to Reinsurance," Methodology and Computing in Applied Probability, Springer, vol. 18(3), pages 675-689, September.
    12. Florin Avram & Romain Biard & Christophe Dutang & Stéphane Loisel & Landy Rabehasaina, 2014. "A survey of some recent results on Risk Theory," Post-Print hal-01616178, HAL.
    13. Junna Bi & Zhibin Liang & Kam Chuen Yuen, 2019. "Optimal mean–variance investment/reinsurance with common shock in a regime-switching market," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 90(1), pages 109-135, August.
    14. Bi, Junna & Liang, Zhibin & Xu, Fangjun, 2016. "Optimal mean–variance investment and reinsurance problems for the risk model with common shock dependence," Insurance: Mathematics and Economics, Elsevier, vol. 70(C), pages 245-258.

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