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Equilibria of Time-inconsistent Stopping for One-dimensional Diffusion Processes

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  • Erhan Bayraktar
  • Zhenhua Wang
  • Zhou Zhou

Abstract

We consider three equilibrium concepts proposed in the literature for time-inconsistent stopping problems, including mild equilibria, weak equilibria and strong equilibria. The discount function is assumed to be log sub-additive and the underlying process is one-dimensional diffusion. We first provide necessary and sufficient conditions for the characterization of weak equilibria. The smooth-fit condition is obtained as a by-product. Next, based on the characterization of weak equilibria, we show that an optimal mild equilibrium is also weak. Then we provide conditions under which a weak equilibrium is strong. We further show that an optimal mild equilibrium is also strong under a certain condition. Finally, we provide several examples including one shows a weak equilibrium may not be strong, and another one shows a strong equilibrium may not be optimal mild.

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  • Erhan Bayraktar & Zhenhua Wang & Zhou Zhou, 2022. "Equilibria of Time-inconsistent Stopping for One-dimensional Diffusion Processes," Papers 2201.07659, arXiv.org, revised Nov 2022.
    • Handle: RePEc:arx:papers:2201.07659
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    References listed on IDEAS

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    1. 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.
    2. 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.
    3. Yu-Jui Huang & Adrien Nguyen-Huu, 2018. "Time-consistent stopping under decreasing impatience," Finance and Stochastics, Springer, vol. 22(1), pages 69-95, January.
    4. 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.
    5. 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.
    6. Ebert, Sebastian & Wei, Wei & Zhou, Xun Yu, 2020. "Weighted discounting—On group diversity, time-inconsistency, and consequences for investment," Journal of Economic Theory, Elsevier, vol. 189(C).
    7. Yu‐Jui Huang & Zhou Zhou, 2020. "Optimal equilibria for time‐inconsistent stopping problems in continuous time," Mathematical Finance, Wiley Blackwell, vol. 30(3), pages 1103-1134, July.
    8. Yu-Jui Huang & Zhenhua Wang, 2020. "Optimal Equilibria for Multi-dimensional Time-inconsistent Stopping Problems," Papers 2006.00754, arXiv.org, revised Jan 2021.
    9. Yu-Jui Huang & Xiang Yu, 2019. "Optimal Stopping under Model Ambiguity: a Time-Consistent Equilibrium Approach," Papers 1906.01232, arXiv.org, revised Mar 2021.
    10. 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.
    11. 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.
    12. Yu-Jui Huang & Zhou Zhou, 2021. "Strong and Weak Equilibria for Time-Inconsistent Stochastic Control in Continuous Time," Mathematics of Operations Research, INFORMS, vol. 46(2), pages 428-451, May.
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    Cited by:

    1. 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.

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