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

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  • Ying Hu

    (IRMAR)

  • Hanqing Jin
  • Xun Yu Zhou

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, 2015. "Time-Inconsistent Stochastic Linear--Quadratic Control: Characterization and Uniqueness of Equilibrium," Papers 1504.01152, arXiv.org, revised May 2015.
    • Handle: RePEc:arx:papers:1504.01152
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    References listed on IDEAS

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    1. 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.
    2. Suleyman Basak & Georgy Chabakauri, 2010. "Dynamic Mean-Variance Asset Allocation," The Review of Financial Studies, Society for Financial Studies, vol. 23(8), pages 2970-3016, August.
    3. 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.
    4. Nicolas Vieille & Jörgen Weibull, 2009. "Multiple solutions under quasi-exponential discounting," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 39(3), pages 513-526, June.
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    Cited by:

    1. Qinglong Zhou & Gaofeng Zong, 2016. "Time-Inconsistent Stochastic Linear-quadratic Differential Game," Papers 1607.00638, arXiv.org.
    2. Li, Bin & Li, Danping & Xiong, Dewen, 2016. "Alpha-robust mean-variance reinsurance-investment strategy," Journal of Economic Dynamics and Control, Elsevier, vol. 70(C), pages 101-123.
    3. 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.
    4. Wei, Jiaqin & Wang, Tianxiao, 2017. "Time-consistent mean–variance asset–liability management with random coefficients," Insurance: Mathematics and Economics, Elsevier, vol. 77(C), pages 84-96.

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