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Time-Inconsistent Stochastic Linear--Quadratic Control

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  • Ying Hu
  • Hanqing Jin
  • Xun Yu Zhou
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    Abstract

    In this paper, we formulate a general time-inconsistent stochastic linear--quadratic (LQ) control problem. The time-inconsistency arises from the presence of a quadratic term of the expected state as well as a state-dependent term in the objective functional. We define an equilibrium, instead of optimal, solution within the class of open-loop controls, and derive a 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 find an explicit equilibrium control. As an application, we then consider 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. Applying the general sufficient condition, we obtain explicit equilibrium strategies when the risk premium is both deterministic and stochastic.

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    File URL: http://arxiv.org/pdf/1111.0818
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    Bibliographic Info

    Paper provided by arXiv.org in its series Papers with number 1111.0818.

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    Date of creation: Nov 2011
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    Handle: RePEc:arx:papers:1111.0818

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    Web page: http://arxiv.org/

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    1. Marie-Amélie Morlais, 2009. "Quadratic BSDEs driven by a continuous martingale and applications to the utility maximization problem," Finance and Stochastics, Springer, vol. 13(1), pages 121-150, January.
    2. Hanqing Jin & Xun Yu Zhou, 2008. "Behavioral Portfolio Selection In Continuous Time," Mathematical Finance, Wiley Blackwell, vol. 18(3), pages 385-426.
    3. Loewenstein, George & Prelec, Drazen, 1992. "Anomalies in Intertemporal Choice: Evidence and an Interpretation," The Quarterly Journal of Economics, MIT Press, vol. 107(2), pages 573-97, May.
    4. Basak, Suleyman & Chabakauri, Georgy, 2009. "Dynamic Mean-Variance Asset Allocation," CEPR Discussion Papers 7256, C.E.P.R. Discussion Papers.
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