IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v130y2020i5p2886-2917.html
   My bibliography  Save this article

On time-inconsistent stopping problems and mixed strategy stopping times

Author

Listed:
  • Christensen, Sören
  • Lindensjö, Kristoffer

Abstract

A game-theoretic framework for time-inconsistent stopping problems where the time-inconsistency is due to the consideration of a non-linear function of an expected reward is developed. A class of mixed strategy stopping times that allows the agents in the game to jointly choose the intensity function of a Cox process is introduced and motivated. A subgame perfect Nash equilibrium is defined. The equilibrium is characterized and other necessary and sufficient equilibrium conditions including a smooth fit result are proved. Existence and uniqueness are investigated. A mean–variance and a variance problem are studied. The state process is a general one-dimensional Itô diffusion.

Suggested Citation

  • Christensen, Sören & Lindensjö, Kristoffer, 2020. "On time-inconsistent stopping problems and mixed strategy stopping times," Stochastic Processes and their Applications, Elsevier, vol. 130(5), pages 2886-2917.
    • Handle: RePEc:eee:spapps:v:130:y:2020:i:5:p:2886-2917
    DOI: 10.1016/j.spa.2019.08.010
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304414918301273
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.spa.2019.08.010?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. Erhan Bayraktar & Jingjie Zhang & Zhou Zhou, 2018. "Time Consistent Stopping For The Mean-Standard Deviation Problem --- The Discrete Time Case," Papers 1802.08358, arXiv.org, revised Apr 2019.
    2. Yu-Jui Huang & Zhou Zhou, 2017. "Optimal Equilibria for Time-Inconsistent Stopping Problems in Continuous Time," Papers 1712.07806, arXiv.org, revised Oct 2018.
    3. Yu-Jui Huang & Zhou Zhou, 2017. "The Optimal Equilibrium for Time-Inconsistent Stopping Problems -- the Discrete-Time Case," Papers 1707.04981, arXiv.org, revised Dec 2018.
    4. Harry Markowitz, 1952. "Portfolio Selection," Journal of Finance, American Finance Association, vol. 7(1), pages 77-91, March.
    5. Tomas Björk & Mariana Khapko & Agatha Murgoci, 2017. "On time-inconsistent stochastic control in continuous time," Finance and Stochastics, Springer, vol. 21(2), pages 331-360, April.
    6. Li, Yongwu & Li, Zhongfei, 2013. "Optimal time-consistent investment and reinsurance strategies for mean–variance insurers with state dependent risk aversion," Insurance: Mathematics and Economics, Elsevier, vol. 53(1), pages 86-97.
    7. Yu-Jui Huang & Adrien Nguyen-Huu, 2018. "Time-consistent stopping under decreasing impatience," Finance and Stochastics, Springer, vol. 22(1), pages 69-95, January.
    8. Tomas Björk & Agatha Murgoci & Xun Yu Zhou, 2014. "Mean–Variance Portfolio Optimization With State-Dependent Risk Aversion," Mathematical Finance, Wiley Blackwell, vol. 24(1), pages 1-24, January.
    9. He, Lin & Liang, Zongxia, 2013. "Optimal investment strategy for the DC plan with the return of premiums clauses in a mean–variance framework," Insurance: Mathematics and Economics, Elsevier, vol. 53(3), pages 643-649.
    10. R. H. Strotz, 1955. "Myopia and Inconsistency in Dynamic Utility Maximization," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 23(3), pages 165-180.
    11. Tomas Björk & Agatha Murgoci, 2014. "A theory of Markovian time-inconsistent stochastic control in discrete time," Finance and Stochastics, Springer, vol. 18(3), pages 545-592, July.
    12. Touzi, N. & Vieille, N., 1999. "Continuous-Time Dynkin Games with Mixed Strategies," Papiers d'Economie Mathématique et Applications 1999.112, Université Panthéon-Sorbonne (Paris 1).
    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. Fang Chen & Xianping Guo & Zhong-Wei Liao, 2022. "Optimal Stopping Time on Semi-Markov Processes with Finite Horizon," Journal of Optimization Theory and Applications, Springer, vol. 194(2), pages 408-439, August.
    2. Erhan Bayraktar & Zhenhua Wang & Zhou Zhou, 2023. "Equilibria of time‐inconsistent stopping for one‐dimensional diffusion processes," Mathematical Finance, Wiley Blackwell, vol. 33(3), pages 797-841, July.
    3. Yu-Jui Huang & Zhou Zhou, 2022. "A time-inconsistent Dynkin game: from intra-personal to inter-personal equilibria," Finance and Stochastics, Springer, vol. 26(2), pages 301-334, April.
    4. Kristoffer Lindensjö & Filip Lindskog, 2020. "Optimal dividends and capital injection under dividend restrictions," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 92(3), pages 461-487, December.
    5. Yu‐Jui Huang & Xiang Yu, 2021. "Optimal stopping under model ambiguity: A time‐consistent equilibrium approach," Mathematical Finance, Wiley Blackwell, vol. 31(3), pages 979-1012, July.
    6. Zongxia Liang & Fengyi Yuan, 2021. "Weak equilibria for time-inconsistent control: with applications to investment-withdrawal decisions," Papers 2105.06607, arXiv.org, revised Jun 2023.

    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. Soren Christensen & Kristoffer Lindensjo, 2019. "Time-inconsistent stopping, myopic adjustment & equilibrium stability: with a mean-variance application," Papers 1909.11921, arXiv.org, revised Jan 2020.
    2. Bian, Lihua & Li, Zhongfei & Yao, Haixiang, 2018. "Pre-commitment and equilibrium investment strategies for the DC pension plan with regime switching and a return of premiums clause," Insurance: Mathematics and Economics, Elsevier, vol. 81(C), pages 78-94.
    3. Xue Dong He & Xun Yu Zhou, 2021. "Who Are I: Time Inconsistency and Intrapersonal Conflict and Reconciliation," Papers 2105.01829, arXiv.org.
    4. Yu-Jui Huang & Zhou Zhou, 2021. "A Time-Inconsistent Dynkin Game: from Intra-personal to Inter-personal Equilibria," Papers 2101.00343, arXiv.org, revised Dec 2021.
    5. Yu-Jui Huang & Zhou Zhou, 2017. "Optimal Equilibria for Time-Inconsistent Stopping Problems in Continuous Time," Papers 1712.07806, arXiv.org, revised Oct 2018.
    6. Marcel Nutz & Yuchong Zhang, 2019. "Conditional Optimal Stopping: A Time-Inconsistent Optimization," Papers 1901.05802, arXiv.org, revised Oct 2019.
    7. Zhou, Zhou & Jin, Zhuo, 2020. "Optimal equilibrium barrier strategies for time-inconsistent dividend problems in discrete time," Insurance: Mathematics and Economics, Elsevier, vol. 94(C), pages 100-108.
    8. Yu-Jui Huang & Zhou Zhou, 2022. "A time-inconsistent Dynkin game: from intra-personal to inter-personal equilibria," Finance and Stochastics, Springer, vol. 26(2), pages 301-334, April.
    9. Chi Kin Lam & Yuhong Xu & Guosheng Yin, 2016. "Dynamic portfolio selection without risk-free assets," Papers 1602.04975, arXiv.org.
    10. Yu-Jui Huang & Adrien Nguyen-Huu, 2018. "Time-consistent stopping under decreasing impatience," Finance and Stochastics, Springer, vol. 22(1), pages 69-95, January.
    11. Junna Bi & Jun Cai & Yan Zeng, 2021. "Equilibrium reinsurance-investment strategies with partial information and common shock dependence," Annals of Operations Research, Springer, vol. 307(1), pages 1-24, December.
    12. Yu-Jui Huang & Zhou Zhou, 2018. "Strong and Weak Equilibria for Time-Inconsistent Stochastic Control in Continuous Time," Papers 1809.09243, arXiv.org, revised Aug 2019.
    13. Li, Yongwu & Qiao, Han & Wang, Shouyang & Zhang, Ling, 2015. "Time-consistent investment strategy under partial information," Insurance: Mathematics and Economics, Elsevier, vol. 65(C), pages 187-197.
    14. Gad, Kamille Sofie Tågholt & Matomäki, Pekka, 2020. "Optimal variance stopping with linear diffusions," Stochastic Processes and their Applications, Elsevier, vol. 130(4), pages 2349-2383.
    15. Bingyan Han & Chi Seng Pun & Hoi Ying Wong, 2023. "Robust Time-inconsistent Linear-Quadratic Stochastic Controls: A Stochastic Differential Game Approach," Papers 2306.16982, arXiv.org, revised Sep 2024.
    16. Erhan Bayraktar & Jingjie Zhang & Zhou Zhou, 2021. "Equilibrium concepts for time‐inconsistent stopping problems in continuous time," Mathematical Finance, Wiley Blackwell, vol. 31(1), pages 508-530, January.
    17. Alia, Ishak & Chighoub, Farid & Sohail, Ayesha, 2016. "A characterization of equilibrium strategies in continuous-time mean–variance problems for insurers," Insurance: Mathematics and Economics, Elsevier, vol. 68(C), pages 212-223.
    18. Yu-Jui Huang & Zhenhua Wang, 2020. "Optimal Equilibria for Multi-dimensional Time-inconsistent Stopping Problems," Papers 2006.00754, arXiv.org, revised Jan 2021.
    19. Soren Christensen & Kristoffer Lindensjo, 2019. "Moment constrained optimal dividends: precommitment \& consistent planning," Papers 1909.10749, arXiv.org.
    20. Liang, Zongxia & Song, Min, 2015. "Time-consistent reinsurance and investment strategies for mean–variance insurer under partial information," Insurance: Mathematics and Economics, Elsevier, vol. 65(C), pages 66-76.

    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:eee:spapps:v:130:y:2020:i:5:p:2886-2917. 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: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/505572/description#description .

    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.