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Linear–Quadratic Time-Inconsistent Mean Field Games

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Abstract

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. http://www.sta.cuhk.edu.hk/scpy, 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.

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References

  1. Bardi M (2012) Explicit solutions of some linear–quadratic mean field games. Netw Heterog Media 7(2):243–261

    Article  MathSciNet  MATH  Google Scholar 

  2. Bensoussan A, Sung KCJ, Yam SCP, Yung SP (2012) Linear–quadratic mean field games. For a preprint, see http://www.sta.cuhk.edu.hk/scpy (submitted)

  3. Bensoussan A, Sung KCJ, Yam SCP (2012) On notion of time-inconsistent ϵ-Nash equilibrium. Working paper

  4. Björk T, Murgoci A (2010) A general theory of Markovian time inconsistent stochastic control problem. Preprint

  5. Björk T, Murgoci A, Zhou XY (2012) Mean variance portfolio optimization with state dependent risk aversion. Math Finance. doi:10.1111/j.1467-9965.2011.00515.x

    Google Scholar 

  6. Cardaliaguet P (2010) Notes on mean field games

  7. Ekeland I, Pirvu TA (2008) Investment and consumption without commitment. Math Financ Econ 2:57–86

    Article  MathSciNet  MATH  Google Scholar 

  8. Ekeland I, Mbodji O, Pirvu TA (2012) Time-consistent portfolio management. SIAM J Financ Math 3:1–32

    Article  MathSciNet  MATH  Google Scholar 

  9. Freiling G (2002) A survey of nonsymmetric Riccati equations. Linear Algebra Appl 351:243–270

    Article  MathSciNet  Google Scholar 

  10. Guéant O (2009) Mean field games and applications to economics. PhD Thesis, Université Paris-Dauphine

  11. Guéant O (2012) Mean field games equation with quadratic Hamiltonian: a specific approach. Math Models Methods Appl Sci 22:1250022 [37 pages]. doi:10.1142/S0218202512500224

    Article  MathSciNet  Google Scholar 

  12. Guéant O, Lasry JM, Lions PL (2011) Mean field games and applications. In: Carmona AR et al. (eds) Paris–Princeton lectures on mathematical finance 2010. Lecture notes in mathematics, vol 2003, pp 205–266

    Chapter  Google Scholar 

  13. Hu Y, Jin H, Zhou XY (2012) Time-inconsistent stochastic linear–quadratic control. SIAM J Control Optim 50(3):1548–1572 (25 pages)

    Article  MathSciNet  MATH  Google Scholar 

  14. Huang M, Caines PE, Malhamé RP (2003) Individual and mass behaviour in large population stochastic wireless power control problems: centralized and Nash equilibrium solutions. In: Proceedings of the 42nd IEEE conference on decision and control. Maui, Hawaii, December, pp 98–103

    Google Scholar 

  15. Huang M, Malhamé RP, Caines PE (2006) Large population stochastic dynamic games: closed-loop McKean–Vlasov systems and the Nash certainty equivalence principle. Commun Inf Syst 6(3):221–252

    MathSciNet  MATH  Google Scholar 

  16. Huang M, Caines PE, Malhamé RP (2007) Large-population cost-coupled LQG problems with nonuniform agents: individual-mass behavior and decentralized ϵ-Nash equilibria. IEEE Trans Autom Control 52(9):1560–1571

    Article  Google Scholar 

  17. Huang M, Caines PE, Malhamé RP (2007) An invariance principle in large population stochastic dynamic games. J Syst Sci Complex 20(2):162–172

    Article  MathSciNet  Google Scholar 

  18. Lasry J-M, Lions P-L (2006) Jeux à champ moyen I—le cas stationnaire. C R Acad Sci, Sér 1 Math 343:619–625

    Article  MathSciNet  MATH  Google Scholar 

  19. Lasry J-M, Lions P-L (2006) Jeux à champ moyen II. Horizon fini et contrôle optimal. C R Acad Sci, Sér 1 Math 343:679–684

    Article  MathSciNet  MATH  Google Scholar 

  20. Lasry JM, Lions PL (2007) Mean field games. Jpn J Math 2(1):229–260

    Article  MathSciNet  MATH  Google Scholar 

  21. Ma J, Yong J (1999) Forward–backward stochastic differential equations and their applications. Lecture notes in mathematics, vol 1702. Springer, Berlin

    MATH  Google Scholar 

  22. Peleg B, Yaari ME (1973) On the existence of a consistent course of action when tastes are changing. Rev Econ Stud 40:391–401

    Article  MATH  Google Scholar 

  23. Strotz RH (1955) Myopia and inconsistency in dynamic utility maximization. Rev Econ Stud 23:165–180

    Article  Google Scholar 

  24. Tembine H, Zhu Q, Başar T (2011) Risk-sensitive mean-field stochastic differential games. In: Proceedings of the 18th IFAC world congress, Milan, August 2011, pp 3222–3227

    Google Scholar 

  25. Wei JQ, Wong KC, Yam SCP, Yung SP (2013) Markowitz’s mean-variance asset-liability management with regime switching: a time-consistent approach. Insur Math Econ 53(1):281–291

    Article  MathSciNet  Google Scholar 

  26. Yong J (2011) A deterministic linear quadratic time-inconsistent optimal control problem. Math Control Relat Fields 1:83–118

    Article  MathSciNet  MATH  Google Scholar 

  27. Yong J (2012) Time-inconsistent optimal control problems and the equilibrium HJB equation. Math Control Relat Fields 2:271–329

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

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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Authors and Affiliations

  1. International Center for Decision and Risk Analysis, School of Management, The University of Texas at Dallas, Richardson, TX, USA

    A. Bensoussan

  2. Department of Systems Engineering and Engineering Management, College of Science and Engineering, City University of Hong Kong, Hong Kong, China

    A. Bensoussan

  3. Graduate Department of Financial Engineering, Ajou University, Suwon, South Korea

    A. Bensoussan

  4. Department of Statistics and Actuarial Science, The University of Hong Kong, Hong Kong, China

    K. C. J. Sung

  5. Department of Statistics, The Chinese University of Hong Kong, Hong Kong, China

    S. C. P. Yam

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  1. A. Bensoussan
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  2. K. C. J. Sung
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  3. S. C. P. Yam
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Correspondence to A. Bensoussan.

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Bensoussan, A., Sung, K.C.J. & Yam, S.C.P. Linear–Quadratic Time-Inconsistent Mean Field Games. Dyn Games Appl 3, 537–552 (2013). https://doi.org/10.1007/s13235-013-0090-y

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  • DOI: https://doi.org/10.1007/s13235-013-0090-y

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