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Strong and Weak Equilibria for Time-Inconsistent Stochastic Control in Continuous Time

Author

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  • Yu-Jui Huang

    (Department of Applied Mathematics, University of Colorado, Boulder, Boulder, Colorado 80309)

  • Zhou Zhou

    (School of Mathematics and Statistics, University of Sydney, New South Wales 2006, Australia)

Abstract

A new definition of continuous-time equilibrium controls is introduced. As opposed to the standard definition, which involves a derivative-type operation, the new definition parallels how a discrete-time equilibrium is defined and allows for unambiguous economic interpretation. The terms “strong equilibria” and “weak equilibria” are coined for controls under the new and standard definitions, respectively. When the state process is a time-homogeneous continuous-time Markov chain, a careful asymptotic analysis gives complete characterizations of weak and strong equilibria. Thanks to the Kakutani–Fan fixed-point theorem, the general existence of weak and strong equilibria is also established under an additional compactness assumption. Our theoretic results are applied to a two-state model under nonexponential discounting. In particular, we demonstrate explicitly that there can be incentive to deviate from a weak equilibrium, which justifies the need for strong equilibria. Our analysis also provides new results for the existence and characterization of discrete-time equilibria under infinite horizon.

Suggested Citation

  • 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.
    • Handle: RePEc:inm:ormoor:v:46:y:2021:i:2:p:428-451
    DOI: 10.1287/moor.2020.1066
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    References listed on IDEAS

    as
    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. Ying Hu & Hanqing Jin & Xun Yu Zhou, 2012. "Time-Inconsistent Stochastic Linear--Quadratic Control," Post-Print hal-00691816, HAL.
    3. Pirvu, Traian A. & Zhang, Huayue, 2014. "Investment–consumption with regime-switching discount rates," Mathematical Social Sciences, Elsevier, vol. 71(C), pages 142-150.
    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. Yu-Jui Huang & Adrien Nguyen-Huu, 2018. "Time-consistent stopping under decreasing impatience," Finance and Stochastics, Springer, vol. 22(1), pages 69-95, January.
    6. repec:dau:papers:123456789/11473 is not listed on IDEAS
    7. 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. Oumar Mbodji & Traian A. Pirvu, 2023. "Portfolio Time Consistency and Utility Weighted Discount Rates," Papers 2402.05113, arXiv.org.
    2. 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.
    3. Mariana Khapko, 2023. "Asset pricing with dynamically inconsistent agents," Finance and Stochastics, Springer, vol. 27(4), pages 1017-1046, October.
    4. Alain Bensoussan & Guiyuan Ma & Chi Chung Siu & Sheung Chi Phillip Yam, 2022. "Dynamic mean–variance problem with frictions," Finance and Stochastics, Springer, vol. 26(2), pages 267-300, April.
    5. Yunfei Peng & Wei Wei, 2023. "Solutions to Equilibrium HJB Equations for Time-Inconsistent Deterministic Linear Quadratic Control: Characterization and Uniqueness," Papers 2308.13850, arXiv.org.
    6. Zongxia Liang & Fengyi Yuan, 2023. "Weak equilibria for time‐inconsistent control: With applications to investment‐withdrawal decisions," Mathematical Finance, Wiley Blackwell, vol. 33(3), pages 891-945, July.
    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. Pengyu Wei & Wei Wei, 2024. "Irreversible investment under weighted discounting: effects of decreasing impatience," Papers 2409.01478, arXiv.org.

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