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Continuity inequalities for multidimensional renewal risk models

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  • Gordienko, E.
  • Vázquez-Ortega, P.

Abstract

In this paper we study the continuity properties of the surplus process in multidimensional renewal risk models. Under certain conditions on the distributions of claim sizes and inter-claim times we prove continuity (stability) inequalities expressed in terms of the total variation distance between the processes. The usage of the uniform metric is also discussed.

Suggested Citation

  • Gordienko, E. & Vázquez-Ortega, P., 2018. "Continuity inequalities for multidimensional renewal risk models," Insurance: Mathematics and Economics, Elsevier, vol. 82(C), pages 48-54.
    • Handle: RePEc:eee:insuma:v:82:y:2018:i:c:p:48-54
    DOI: 10.1016/j.insmatheco.2018.06.005
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    References listed on IDEAS

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    1. Ewa Marciniak & Zbigniew Palmowski, 2016. "On the Optimal Dividend Problem for Insurance Risk Models with Surplus-Dependent Premiums," Papers 1604.06892, arXiv.org.
    2. Evgueni Gordienko & Juan Ruiz de Chávez, 2002. "New continuity estimates of geometric sums," International Journal of Stochastic Analysis, Hindawi, vol. 15, pages 1-15, January.
    3. Cai, Jun & Li, Haijun, 2007. "Dependence properties and bounds for ruin probabilities in multivariate compound risk models," Journal of Multivariate Analysis, Elsevier, vol. 98(4), pages 757-773, April.
    4. Yin, Chuancun & Wen, Yuzhen & Zhao, Yongxia, 2014. "On The Optimal Dividend Problem For A Spectrally Positive Lévy Process," ASTIN Bulletin, Cambridge University Press, vol. 44(3), pages 635-651, September.
    5. Ewa Marciniak & Zbigniew Palmowski, 2016. "On the Optimal Dividend Problem for Insurance Risk Models with Surplus-Dependent Premiums," Journal of Optimization Theory and Applications, Springer, vol. 168(2), pages 723-742, February.
    6. Yiqing Chen & Kam C. Yuen & Kai W. Ng, 2011. "Asymptotics for the ruin probabilities of a two‐dimensional renewal risk model with heavy‐tailed claims," Applied Stochastic Models in Business and Industry, John Wiley & Sons, vol. 27(3), pages 290-300, May.
    7. Chuancun Yin & Yuzhen Wen & Yongxia Zhao, 2013. "On the optimal dividend problem for a spectrally positive Levy process," Papers 1302.2231, arXiv.org, revised Mar 2014.
    8. Chan, Wai-Sum & Yang, Hailiang & Zhang, Lianzeng, 2003. "Some results on ruin probabilities in a two-dimensional risk model," Insurance: Mathematics and Economics, Elsevier, vol. 32(3), pages 345-358, July.
    9. Cai, Jun & Li, Haijun, 2005. "Multivariate risk model of phase type," Insurance: Mathematics and Economics, Elsevier, vol. 36(2), pages 137-152, April.
    10. Albrecher Hansjörg & Bäuerle Nicole & Thonhauser Stefan, 2011. "Optimal dividend-payout in random discrete time," Statistics & Risk Modeling, De Gruyter, vol. 28(3), pages 251-276, September.
    11. Ivanovs, Jevgenijs & Boxma, Onno, 2015. "A bivariate risk model with mutual deficit coverage," Insurance: Mathematics and Economics, Elsevier, vol. 64(C), pages 126-134.
    12. Beirlant, J. & Rachev, S. T., 1987. "The problem of stability in insurance mathematics," Insurance: Mathematics and Economics, Elsevier, vol. 6(3), pages 179-188, July.
    13. 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.
    14. Li, Junhai & Liu, Zaiming & Tang, Qihe, 2007. "On the ruin probabilities of a bidimensional perturbed risk model," Insurance: Mathematics and Economics, Elsevier, vol. 41(1), pages 185-195, July.
    15. Gordienko, Evgueni & Vázquez-Ortega, Patricia, 2016. "Simple Continuity Inequalities For Ruin Probability In The Classical Risk Model," ASTIN Bulletin, Cambridge University Press, vol. 46(3), pages 801-814, September.
    16. Wei Huang & Chengguo Weng & Yi Zhang, 2014. "Multivariate risk models under heavy‐tailed risks," Applied Stochastic Models in Business and Industry, John Wiley & Sons, vol. 30(3), pages 341-360, May.
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    More about this item

    Keywords

    Multidimensional renewal risk model; Continuity inequalities for surplus process; Probability metrics; Total variation distance; Approximating risk model;
    All these keywords.

    JEL classification:

    • C3 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables
    • G22 - Financial Economics - - Financial Institutions and Services - - - Insurance; Insurance Companies; Actuarial Studies

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