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Optimal Equilibria for Multi-dimensional Time-inconsistent Stopping Problems

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  • Yu-Jui Huang
  • Zhenhua Wang

Abstract

We study an optimal stopping problem under non-exponential discounting, where the state process is a multi-dimensional continuous strong Markov process. The discount function is taken to be log sub-additive, capturing decreasing impatience in behavioral economics. On strength of probabilistic potential theory, we establish the existence of an optimal equilibrium among a sufficiently large collection of equilibria, consisting of finely closed equilibria satisfying a boundary condition. This generalizes the existence of optimal equilibria for one-dimensional stopping problems in prior literature.

Suggested Citation

  • Yu-Jui Huang & Zhenhua Wang, 2020. "Optimal Equilibria for Multi-dimensional Time-inconsistent Stopping Problems," Papers 2006.00754, arXiv.org, revised Jan 2021.
    • Handle: RePEc:arx:papers:2006.00754
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    References listed on IDEAS

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    1. 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.
    2. Yu‐Jui Huang & Adrien Nguyen‐Huu & Xun Yu Zhou, 2020. "General stopping behaviors of naïve and noncommitted sophisticated agents, with application to probability distortion," Mathematical Finance, Wiley Blackwell, vol. 30(1), pages 310-340, January.
    3. Ying Hu & Hanqing Jin & Xun Yu Zhou, 2012. "Time-Inconsistent Stochastic Linear--Quadratic Control," Post-Print hal-00691816, HAL.
    4. 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.
    5. 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.
    6. Yu-Jui Huang & Adrien Nguyen-Huu, 2018. "Time-consistent stopping under decreasing impatience," Finance and Stochastics, Springer, vol. 22(1), pages 69-95, January.
    7. Yu-Jui Huang & Adrien Nguyen-Huu & Xun Yu Zhou, 2018. "General stopping behaviors of naïve and non-committed sophisticated agents, with application to probability distortion," CEE-M Working Papers 18-16, CEE-M, Universitiy of Montpellier, CNRS, INRA, Montpellier SupAgro.
    8. 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.
    9. repec:dau:papers:123456789/11473 is not listed on IDEAS
    10. George Loewenstein & Drazen Prelec, 1992. "Anomalies in Intertemporal Choice: Evidence and an Interpretation," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 107(2), pages 573-597.
    11. Thaler, Richard, 1981. "Some empirical evidence on dynamic inconsistency," Economics Letters, Elsevier, vol. 8(3), pages 201-207.
    12. Yu-Jui Huang & Xiang Yu, 2019. "Optimal Stopping under Model Ambiguity: a Time-Consistent Equilibrium Approach," Papers 1906.01232, arXiv.org, revised Mar 2021.
    13. 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.
    14. Loewenstein, George & Thaler, Richard H, 1989. "Intertemporal Choice," Journal of Economic Perspectives, American Economic Association, vol. 3(4), pages 181-193, Fall.
    15. 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.
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    Citations

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    Cited by:

    1. 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.
    2. 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.
    3. Oumar Mbodji & Traian A. Pirvu, 2023. "Portfolio Time Consistency and Utility Weighted Discount Rates," Papers 2402.05113, arXiv.org.
    4. Luca De Gennaro Aquino & Sascha Desmettre & Yevhen Havrylenko & Mogens Steffensen, 2024. "Equilibrium control theory for Kihlstrom-Mirman preferences in continuous time," Papers 2407.16525, arXiv.org, revised Oct 2024.
    5. Shuoqing Deng & Xiang Yu & Jiacheng Zhang, 2023. "On time-consistent equilibrium stopping under aggregation of diverse discount rates," Papers 2302.07470, arXiv.org, revised Dec 2023.
    6. Jiaqin Wei & Jianming Xia & Qian Zhao, 2024. "Time-Consistent Portfolio Selection for Rank-Dependent Utilities in an Incomplete Market," Papers 2409.19259, arXiv.org.
    7. 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.
    8. Yu-Jui Huang & Li-Hsien Sun, 2023. "Partial Information in a Mean-Variance Portfolio Selection Game," Papers 2312.04045, arXiv.org, revised Dec 2024.

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