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From Ruin To Bankruptcy For Compound Poisson Surplus Processes

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  • Albrecher, Hansjörg
  • Lautscham, Volkmar

Abstract

In classical risk theory, the infinite-time ruin probability of a surplus process Ct is calculated as the probability of the process becoming negative at some point in time. In this paper, we consider a relaxation of the ruin concept to the concept of bankruptcy, according to which one has a positive surplus-dependent probability to continue despite temporary negative surplus. We study the resulting bankruptcy probability for the compound Poisson risk model with exponential claim sizes for different bankruptcy rate functions, deriving analytical results, upper and lower bounds as well as an efficient simulation method. Numerical examples are given and the results are compared with the classical ruin probabilities. Finally, it is illustrated how the analysis can be extended to study the discounted penalty function under this relaxed ruin criterion.

Suggested Citation

  • Albrecher, Hansjörg & Lautscham, Volkmar, 2013. "From Ruin To Bankruptcy For Compound Poisson Surplus Processes," ASTIN Bulletin, Cambridge University Press, vol. 43(2), pages 213-243, May.
    • Handle: RePEc:cup:astinb:v:43:y:2013:i:02:p:213-243_00
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    Citations

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    Cited by:

    1. Kathrin Glau, 2015. "Feynman-Kac formula for L\'evy processes with discontinuous killing rate," Papers 1502.07531, arXiv.org, revised Nov 2015.
    2. Brinker, Leonie Violetta & Eisenberg, Julia, 2021. "Dividend optimisation: A behaviouristic approach," Insurance: Mathematics and Economics, Elsevier, vol. 101(PB), pages 202-224.
    3. Jos'e Miguel Flores-Contr'o & Kira Henshaw & Sooie-Hoe Loke & S'everine Arnold & Corina Constantinescu, 2021. "Subsidising Inclusive Insurance to Reduce Poverty," Papers 2103.17255, arXiv.org, revised Feb 2024.
    4. Hansjörg Albrecher & Jevgenijs Ivanovs, 2013. "A Risk Model with an Observer in a Markov Environment," Risks, MDPI, vol. 1(3), pages 1-14, November.
    5. Guérin, Hélène & Renaud, Jean-François, 2017. "On the distribution of cumulative Parisian ruin," Insurance: Mathematics and Economics, Elsevier, vol. 73(C), pages 116-123.
    6. Kathrin Glau, 2016. "A Feynman–Kac-type formula for Lévy processes with discontinuous killing rates," Finance and Stochastics, Springer, vol. 20(4), pages 1021-1059, October.
    7. Zhongqin Gao & Jingmin He & Zhifeng Zhao & Bingbing Wang, 2022. "Omega Model for a Jump-Diffusion Process with a Two-Step Premium Rate and a Threshold Dividend Strategy," Methodology and Computing in Applied Probability, Springer, vol. 24(1), pages 233-258, March.
    8. Hansjoerg Albrecher & Jevgenijs Ivanovs, 2013. "Power identities for L\'evy risk models under taxation and capital injections," Papers 1310.3052, arXiv.org, revised Mar 2014.
    9. Choi, Michael C.H. & Cheung, Eric C.K., 2014. "On the expected discounted dividends in the Cramér–Lundberg risk model with more frequent ruin monitoring than dividend decisions," Insurance: Mathematics and Economics, Elsevier, vol. 59(C), pages 121-132.
    10. Li, Xin & Liu, Haibo & Tang, Qihe & Zhu, Jinxia, 2020. "Liquidation risk in insurance under contemporary regulatory frameworks," Insurance: Mathematics and Economics, Elsevier, vol. 93(C), pages 36-49.
    11. Vierkötter, Matthias & Schmidli, Hanspeter, 2017. "On optimal dividends with exponential and linear penalty payments," Insurance: Mathematics and Economics, Elsevier, vol. 72(C), pages 265-270.
    12. Jos'e Miguel Flores-Contr'o & S'everine Arnold, 2023. "The Role of Direct Capital Cash Transfers Towards Poverty and Extreme Poverty Alleviation -- An Omega Risk Process," Papers 2401.06141, arXiv.org, revised Feb 2024.
    13. Wong, Jeff T.Y. & Cheung, Eric C.K., 2015. "On the time value of Parisian ruin in (dual) renewal risk processes with exponential jumps," Insurance: Mathematics and Economics, Elsevier, vol. 65(C), pages 280-290.
    14. Landriault, David & Li, Bin & Wong, Jeff T.Y. & Xu, Di, 2018. "Poissonian potential measures for Lévy risk models," Insurance: Mathematics and Economics, Elsevier, vol. 82(C), pages 152-166.
    15. Boxma, Onno & Kella, Offer & Mandjes, Michel, 2023. "On fluctuation-theoretic decompositions via Lindley-type recursions," Stochastic Processes and their Applications, Elsevier, vol. 165(C), pages 316-336.

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