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Minimizing the Probability of Lifetime Exponential Parisian Ruin

Author

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  • Xiaoqing Liang

    (Hebei University of Technology)

  • Virginia R. Young

    (University of Michigan)

Abstract

We find the optimal investment strategy in a Black–Scholes market to minimize the probability of so-called lifetime exponential Parisian ruin, that is, the probability that wealth exhibits an excursion below zero of an exponentially distributed time before the individual dies. We find that leveraging the risky asset is worse for negative wealth when minimizing the probability of lifetime exponential Parisian ruin than when minimizing the probability of lifetime ruin. Moreover, when wealth is negative, the optimal amount invested in the risky asset increases as the hazard rate of the exponential “excursion clock” increases. In view of the heavy leveraging when wealth is negative, we also compute the minimum probability of lifetime exponential Parisian ruin under a constraint on investment. Finally, we derive an asymptotic expansion of the minimum probability of lifetime exponential Parisian ruin for small values of the hazard rate of the excursion clock. It is interesting to find that for small values of this hazard rate, the minimum probability of lifetime exponential Parisian ruin is proportional to the minimum occupation time studied in Bayraktar and Young, and the proportion equals the hazard rate. To the best of our knowledge, our work is the first to control the probability of Parisian ruin.

Suggested Citation

  • Xiaoqing Liang & Virginia R. Young, 2020. "Minimizing the Probability of Lifetime Exponential Parisian Ruin," Journal of Optimization Theory and Applications, Springer, vol. 184(3), pages 1036-1064, March.
    • Handle: RePEc:spr:joptap:v:184:y:2020:i:3:d:10.1007_s10957-019-01595-8
    DOI: 10.1007/s10957-019-01595-8
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    References listed on IDEAS

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    1. Bayraktar, Erhan & Hu, Xueying & Young, Virginia R., 2011. "Minimizing the probability of lifetime ruin under stochastic volatility," Insurance: Mathematics and Economics, Elsevier, vol. 49(2), pages 194-206, September.
    2. Kristen Moore & Virginia Young, 2006. "Optimal and Simple, Nearly Optimal Rules for Minimizing the Probability Of Financial Ruin in Retirement," North American Actuarial Journal, Taylor & Francis Journals, vol. 10(4), pages 145-161.
    3. Bayraktar, Erhan & Young, Virginia R., 2007. "Minimizing the probability of lifetime ruin under borrowing constraints," Insurance: Mathematics and Economics, Elsevier, vol. 41(1), pages 196-221, July.
    4. David Landriault & Jean-François Renaud & Xiaowen Zhou, 2014. "An Insurance Risk Model with Parisian Implementation Delays," Methodology and Computing in Applied Probability, Springer, vol. 16(3), pages 583-607, September.
    5. Virginia Young, 2004. "Optimal Investment Strategy to Minimize the Probability of Lifetime Ruin," North American Actuarial Journal, Taylor & Francis Journals, vol. 8(4), pages 106-126.
    6. Erhan Bayraktar & Virginia Young, 2010. "Optimal investment strategy to minimize occupation time," Annals of Operations Research, Springer, vol. 176(1), pages 389-408, April.
    7. Moshe Milevsky & Chris Robinson, 2000. "Self-Annuitization and Ruin in Retirement," North American Actuarial Journal, Taylor & Francis Journals, vol. 4(4), pages 112-124.
    8. Liang, Xiaoqing & Young, Virginia R., 2018. "Minimizing the probability of ruin: Two riskless assets with transaction costs and proportional reinsurance," Statistics & Probability Letters, Elsevier, vol. 140(C), pages 167-175.
    9. Landriault, David & Renaud, Jean-François & Zhou, Xiaowen, 2011. "Occupation times of spectrally negative Lévy processes with applications," Stochastic Processes and their Applications, Elsevier, vol. 121(11), pages 2629-2641, November.
    10. Guérin, Hélène & Renaud, Jean-François, 2017. "On the distribution of cumulative Parisian ruin," Insurance: Mathematics and Economics, Elsevier, vol. 73(C), pages 116-123.
    11. Ronnie Loeffen & Irmina Czarna & Zbigniew Palmowski, 2011. "Parisian ruin probability for spectrally negative L\'{e}vy processes," Papers 1102.4055, arXiv.org, revised Mar 2013.
    12. Dassios, Angelos & Wu, Shanle, 2008. "Parisian ruin with exponential claims," LSE Research Online Documents on Economics 32033, London School of Economics and Political Science, LSE Library.
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    Cited by:

    1. Liang, Xiaoqing & Young, Virginia R., 2023. "Annuitizing at a bounded, absolutely continuous rate to minimize the probability of lifetime ruin," Insurance: Mathematics and Economics, Elsevier, vol. 112(C), pages 80-96.
    2. Osatakul, Dhiti & Li, Shuanming & Wu, Xueyuan, 2023. "Discrete-time risk models with surplus-dependent premium corrections," Applied Mathematics and Computation, Elsevier, vol. 437(C).
    3. Zhuo Jin & Zuo Quan Xu & Bin Zou, 2023. "Optimal moral-hazard-free reinsurance under extended distortion premium principles," Papers 2304.08819, arXiv.org.

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