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Distribution Of The Time To Ruin In Some Sparre Andersen Risk Models

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  • Shi, Tianxiang
  • Landriault, David

Abstract

The finite-time ruin problem, which implicitly involves the inversion of the Laplace transform of the time to ruin, has been a long-standing research problem in risk theory. Existing results in the Sparre Andersen risk models are mainly based on an exponential assumption either on the interclaim times or on the claim sizes. In this paper, we utilize the multivariate version of Lagrange expansion theorem to obtain a series expansion for the density of the time to ruin under a more general distribution assumption, namely the combination of n exponentials. A remark is further made to emphasize that this technique can also be applied to other areas of applied probability. For instance, the proposed methodology can be used to obtain the distribution of some first passage times for particular stochastic processes. As an illustration, the duration of a busy period in a queueing risk model will be examined.

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  • Shi, Tianxiang & Landriault, David, 2013. "Distribution Of The Time To Ruin In Some Sparre Andersen Risk Models," ASTIN Bulletin, Cambridge University Press, vol. 43(1), pages 39-59, January.
    • Handle: RePEc:cup:astinb:v:43:y:2013:i:01:p:39-59_00
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    Cited by:

    1. Goffard, Pierre-Olivier & Lefèvre, Claude, 2018. "Duality in ruin problems for ordered risk models," Insurance: Mathematics and Economics, Elsevier, vol. 78(C), pages 44-52.
    2. Jin, Can & Li, Shuanming & Wu, Xueyuan, 2016. "On the occupation times in a delayed Sparre Andersen risk model with exponential claims," Insurance: Mathematics and Economics, Elsevier, vol. 71(C), pages 304-316.
    3. Landriault, David & Li, Bin & Shi, Tianxiang & Xu, Di, 2019. "On the distribution of classic and some exotic ruin times," Insurance: Mathematics and Economics, Elsevier, vol. 89(C), pages 38-45.
    4. Pierre-Olivier Goffard & Claude Lefèvre, 2018. "Duality in ruin problems for ordered risk models," Post-Print hal-01398910, HAL.
    5. Lee, Wing Yan & Li, Xiaolong & Liu, Fangda & Shi, Yifan & Yam, Sheung Chi Phillip, 2021. "A Fourier-cosine method for finite-time ruin probabilities," Insurance: Mathematics and Economics, Elsevier, vol. 99(C), pages 256-267.

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