IDEAS home Printed from https://ideas.repec.org/p/hal/journl/hal-01139343.html
   My bibliography  Save this paper

Time-Inconsistent Stochastic Linear--Quadratic Control: Characterization and Uniqueness of Equilibrium

Author

Listed:
  • Ying Hu

    (IRMAR - Institut de Recherche Mathématique de Rennes - UR - Université de Rennes - INSA Rennes - Institut National des Sciences Appliquées - Rennes - INSA - Institut National des Sciences Appliquées - ENS Rennes - École normale supérieure - Rennes - UR2 - Université de Rennes 2 - CNRS - Centre National de la Recherche Scientifique - INSTITUT AGRO Agrocampus Ouest - Institut Agro - Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement)

  • Hanqing Jin

    (University of Oxford)

  • Xun Yu Zhou

    (University of Oxford)

Abstract

In this paper, we continue our study on a general time-inconsistent stochastic linear--quadratic (LQ) control problem originally formulated in [6]. We derive a necessary and sufficient condition for equilibrium controls via a flow of forward--backward stochastic differential equations. When the state is one dimensional and the coefficients in the problem are all deterministic, we prove that the explicit equilibrium control constructed in \cite{HJZ} is indeed unique. Our proof is based on the derived equivalent condition for equilibria as well as a stochastic version of the Lebesgue differentiation theorem. Finally, we show that the equilibrium strategy is unique for a mean--variance portfolio selection model in a complete financial market where the risk-free rate is a deterministic function of time but all the other market parameters are possibly stochastic processes.

Suggested Citation

  • Ying Hu & Hanqing Jin & Xun Yu Zhou, 2017. "Time-Inconsistent Stochastic Linear--Quadratic Control: Characterization and Uniqueness of Equilibrium," Post-Print hal-01139343, HAL.
  • Handle: RePEc:hal:journl:hal-01139343
    DOI: 10.1137/15M1019040
    Note: View the original document on HAL open archive server: https://hal.science/hal-01139343v2
    as

    Download full text from publisher

    File URL: https://hal.science/hal-01139343v2/document
    Download Restriction: no

    File URL: https://libkey.io/10.1137/15M1019040?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
    ---><---

    References listed on IDEAS

    as
    1. Ying Hu & Hanqing Jin & Xun Yu Zhou, 2012. "Time-Inconsistent Stochastic Linear--Quadratic Control," Post-Print hal-00691816, HAL.
    2. 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.
    3. Grenadier, Steven R. & Wang, Neng, 2007. "Investment under uncertainty and time-inconsistent preferences," Journal of Financial Economics, Elsevier, vol. 84(1), pages 2-39, April.
    4. 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.
    Full references (including those not matched with items on IDEAS)

    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. Yu-Jui Huang & Adrien Nguyen-Huu, 2018. "Time-consistent stopping under decreasing impatience," Finance and Stochastics, Springer, vol. 22(1), pages 69-95, January.
    2. Xue Dong He & Xun Yu Zhou, 2021. "Who Are I: Time Inconsistency and Intrapersonal Conflict and Reconciliation," Papers 2105.01829, arXiv.org.
    3. Yu-Jui Huang & Zhou Zhou, 2017. "Optimal Equilibria for Time-Inconsistent Stopping Problems in Continuous Time," Papers 1712.07806, arXiv.org, revised Oct 2018.
    4. Liyuan Wang & Zhiping Chen, 2019. "Stochastic Game Theoretic Formulation for a Multi-Period DC Pension Plan with State-Dependent Risk Aversion," Mathematics, MDPI, vol. 7(1), pages 1-16, January.
    5. 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.
    6. Marcel Nutz & Yuchong Zhang, 2019. "Conditional Optimal Stopping: A Time-Inconsistent Optimization," Papers 1901.05802, arXiv.org, revised Oct 2019.
    7. Ying Hu & Hanqing Jin & Xun Yu Zhou, 2020. "Consistent Investment of Sophisticated Rank-Dependent Utility Agents in Continuous Time," Working Papers hal-02624308, HAL.
    8. Zhao, Qian & Shen, Yang & Wei, Jiaqin, 2014. "Consumption–investment strategies with non-exponential discounting and logarithmic utility," European Journal of Operational Research, Elsevier, vol. 238(3), pages 824-835.
    9. Yu-Jui Huang & Zhenhua Wang, 2020. "Optimal Equilibria for Multi-dimensional Time-inconsistent Stopping Problems," Papers 2006.00754, arXiv.org, revised Jan 2021.
    10. 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.
    11. 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).
    12. Felix Fie{ss}inger & Mitja Stadje, 2023. "Time-Consistent Asset Allocation for Risk Measures in a L\'evy Market," Papers 2305.09471, arXiv.org, revised Oct 2024.
    13. Ying Hu & Hanqing Jin & Xun Yu Zhou, 2020. "Consistent Investment of Sophisticated Rank-Dependent Utility Agents in Continuous Time," Papers 2006.01979, arXiv.org.
    14. Zongxia Liang & Sheng Wang & Jianming Xia & Fengyi Yuan, 2024. "Dynamic portfolio selection under generalized disappointment aversion," Papers 2401.08323, arXiv.org, revised Mar 2024.
    15. Jin Ma & Ting-Kam Leonard Wong & Jianfeng Zhang, 2018. "Time-consistent conditional expectation under probability distortion," Papers 1809.08262, arXiv.org, revised Jun 2020.
    16. Haiyang Wang & Ruimin Xu, 2023. "Time-Inconsistent LQ Games for Large-Population Systems and Applications," Journal of Optimization Theory and Applications, Springer, vol. 197(3), pages 1249-1268, June.
    17. 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.
    18. 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.
    19. Yan, Tingjin & Wong, Hoi Ying, 2020. "Open-loop equilibrium reinsurance-investment strategy under mean–variance criterion with stochastic volatility," Insurance: Mathematics and Economics, Elsevier, vol. 90(C), pages 105-119.
    20. Zongxia Liang & Jianming Xia & Fengyi Yuan, 2023. "Dynamic portfolio selection for nonlinear law-dependent preferences," Papers 2311.06745, arXiv.org, revised Nov 2023.

    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:hal:journl:hal-01139343. 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: CCSD (email available below). General contact details of provider: https://hal.archives-ouvertes.fr/ .

    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.