IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v184y2020i3d10.1007_s10957-019-01595-8.html
   My bibliography  Save this article

Minimizing the Probability of Lifetime Exponential Parisian Ruin

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10957-019-01595-8
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10957-019-01595-8?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Li, Shu & Zhou, Xiaowen, 2022. "The Parisian and ultimate drawdowns of Lévy insurance models," Insurance: Mathematics and Economics, Elsevier, vol. 107(C), pages 140-160.
    2. Lkabous, Mohamed Amine & Wang, Zijia, 2023. "On the area in the red of Lévy risk processes and related quantities," Insurance: Mathematics and Economics, Elsevier, vol. 111(C), pages 257-278.
    3. 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.
    4. Erhan Bayraktar & Asaf Cohen, 2015. "Risk Sensitive Control of the Lifetime Ruin Problem," Papers 1503.05769, arXiv.org, revised Jul 2016.
    5. Cohen, Asaf & Young, Virginia R., 2016. "Minimizing lifetime poverty with a penalty for bankruptcy," Insurance: Mathematics and Economics, Elsevier, vol. 69(C), pages 156-167.
    6. Landriault, David & Li, Bin & Lkabous, Mohamed Amine, 2020. "On occupation times in the red of Lévy risk models," Insurance: Mathematics and Economics, Elsevier, vol. 92(C), pages 17-26.
    7. Erhan Bayraktar & Yuchong Zhang, 2014. "Minimizing the Probability of Lifetime Ruin Under Ambiguity Aversion," Papers 1402.1809, arXiv.org, revised Nov 2014.
    8. Lkabous, Mohamed Amine & Czarna, Irmina & Renaud, Jean-François, 2017. "Parisian ruin for a refracted Lévy process," Insurance: Mathematics and Economics, Elsevier, vol. 74(C), pages 153-163.
    9. Cheung, Eric C.K. & Wong, Jeff T.Y., 2017. "On the dual risk model with Parisian implementation delays in dividend payments," European Journal of Operational Research, Elsevier, vol. 257(1), pages 159-173.
    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. Ran Xu & Wenyuan Wang & Jose Garrido, 2022. "Optimal Dividend Strategy Under Parisian Ruin with Affine Penalty," Methodology and Computing in Applied Probability, Springer, vol. 24(3), pages 1385-1409, September.
    12. Brinker, Leonie Violetta & Eisenberg, Julia, 2021. "Dividend optimisation: A behaviouristic approach," Insurance: Mathematics and Economics, Elsevier, vol. 101(PB), pages 202-224.
    13. 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.
    14. Landriault, David & Li, Bin & Lkabous, Mohamed Amine, 2021. "On the analysis of deep drawdowns for the Lévy insurance risk model," Insurance: Mathematics and Economics, Elsevier, vol. 100(C), pages 147-155.
    15. Hansjörg Albrecher & Jevgenijs Ivanovs, 2013. "A Risk Model with an Observer in a Markov Environment," Risks, MDPI, vol. 1(3), pages 1-14, November.
    16. 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.
    17. Landriault, David & Li, Bin & Lkabous, Mohamed Amine & Wang, Zijia, 2023. "Bridging the first and last passage times for Lévy models," Stochastic Processes and their Applications, Elsevier, vol. 157(C), pages 308-334.
    18. Wang, Ting & Young, Virginia R., 2012. "Optimal commutable annuities to minimize the probability of lifetime ruin," Insurance: Mathematics and Economics, Elsevier, vol. 50(1), pages 200-216.
    19. Erhan Bayraktar & Virginia Young, 2011. "Proving regularity of the minimal probability of ruin via a game of stopping and control," Finance and Stochastics, Springer, vol. 15(4), pages 785-818, December.
    20. Chen, Xinfu & Landriault, David & Li, Bin & Li, Dongchen, 2015. "On minimizing drawdown risks of lifetime investments," Insurance: Mathematics and Economics, Elsevier, vol. 65(C), pages 46-54.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:joptap:v:184:y:2020:i:3:d:10.1007_s10957-019-01595-8. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.