Dynamic Games and Applications

, Volume 3, Issue 4, pp 537–552 | Cite as

Linear–Quadratic Time-Inconsistent Mean Field Games

  • A. BensoussanEmail author
  • K. C. J. Sung
  • S. C. P. Yam


In this paper, we study a class of time-inconsistent analogs (in the sense of Hu et al. (Time-inconsistent stochastic linear–quadratic control. Preprint, 2012) which is originated from the mean-variance portfolio selection problem with state-dependent risk aversion in the context of financial economics) of the standard Linear–Quadratic Mean Field Games considered in Huang et al. (Commun. Inf. Syst. 6(3):221–252, 2006) and Bensoussan et al. (Linear–quadratic mean field games., submitted, 2012). For the one-dimensional case, we first establish the unique time-consistent optimal strategy under an arbitrary guiding path, with which we further obtain the unique time-consistent mean-field equilibrium strategy under a mild convexity condition. Second, for the dimension greater than one, by applying the adjoint equation approach, we formulate a sufficient condition under which the unique existence of both, time-consistent optimal strategy under a given guiding path and time-consistent equilibrium strategy, can be guaranteed.


Mean field games Time-inconsistent stochastic control problems Adjoint equations Linear–quadratic type Banach fixed point theorem 



The first author Alain Bensoussan acknowledges the financial support from WCU (World Class University) program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (R31-20007) and The Hong Kong RGC GRF 500111. The third author Phillip Yam acknowledges the financial support from The Hong Kong RGC GRF 404012 with the project title: Advanced Topics In Multivariate Risk Management In Finance And Insurance, The Chinese University of Hong Kong Direct Grant 2010/2011 Project ID: 2060422, and The Chinese University of Hong Kong Direct Grant 2011/2012 Project ID: 2060444. Phillip Yam also expresses his sincere gratitude to the hospitality of both Hausdorff Center for Mathematics of the University of Bonn and Mathematisches Forschungsinstitut Oberwolfach (MFO) in the German Black Forest during the preparation of the present work.

Thanks are expressed to Xun Yu Zhou for raising the question on the possible connection of time-consistent stochastic control with mean-field games to the first author in a conference talk in Ajou, Korea.


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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • A. Bensoussan
    • 1
    • 2
    • 3
    Email author
  • K. C. J. Sung
    • 4
  • S. C. P. Yam
    • 5
  1. 1.International Center for Decision and Risk Analysis, School of ManagementThe University of Texas at DallasRichardsonUSA
  2. 2.Department of Systems Engineering and Engineering Management, College of Science and EngineeringCity University of Hong KongHong KongChina
  3. 3.Graduate Department of Financial EngineeringAjou UniversitySuwonSouth Korea
  4. 4.Department of Statistics and Actuarial ScienceThe University of Hong KongHong KongChina
  5. 5.Department of StatisticsThe Chinese University of Hong KongHong KongChina

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