Japanese Journal of Mathematics

, Volume 2, Issue 1, pp 229–260 | Cite as

Mean field games

  • Jean-Michel Lasry
  • Pierre-Louis LionsEmail author
Special Feature: The 1st Takagi Lecture


We survey here some recent studies concerning what we call mean-field models by analogy with Statistical Mechanics and Physics. More precisely, we present three examples of our mean-field approach to modelling in Economics and Finance (or other related subjects...). Roughly speaking, we are concerned with situations that involve a very large number of “rational players” with a limited information (or visibility) on the “game”. Each player chooses his optimal strategy in view of the global (or macroscopic) informations that are available to him and that result from the actions of all players. In the three examples we mention here, we derive a mean-field problem which consists in nonlinear differential equations. These equations are of a new type and our main goal here is to study them and establish their links with various fields of Analysis. We show in particular that these nonlinear problems are essentially well-posed problems i.e., have unique solutions. In addition, we give various limiting cases, examples and possible extensions. And we mention many open problems.


Nash Equilibrium Option Price Nonlinear Differential Equation Operator Versus Stochastic Game 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    R. Aumann, Markets with a continuum of traders, Econometrica, 32 (1964), 39–50.zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    M. Avellaneda and M. D. Lipkin, A market induced mechanism for stock pinning, Quant. Finance, 3 (2003), 417–425.CrossRefMathSciNetGoogle Scholar
  3. 3.
    K. Back, Insider trading in continuous time, Review of Financial Studies, 5 (1992), 387–409.CrossRefGoogle Scholar
  4. 4.
    K. Back, C.-H. Cao and G. Willard, Imperfect competition among informed traders, J. Finance, 55 (2000), 2117–2155.CrossRefGoogle Scholar
  5. 5.
    M. Bardi and I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman equations, Birkhäuser, Boston, 1997.zbMATHGoogle Scholar
  6. 6.
    A. Bensoussan and J. Frehse, Nonlinear elliptic systems in stochastic game theory, J. Reine Angew. Math., 350 (1984), 23–67.zbMATHMathSciNetGoogle Scholar
  7. 7.
    A. Bensoussan and J. Frehse, Ergodic Bellman systems for stochastic games in arbitrary dimension, Proc. Roy. Soc. London Ser. A., 449 (1995), 65–77.zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    F. Black and M. Scholes, The pricing of options and corporate liabilities, J. Polit. Econ., 81 (1973), 637–654.CrossRefGoogle Scholar
  9. 9.
    G. Carmona, Nash equilibria of games with a continuum of players, preprint.Google Scholar
  10. 10.
    W. H. Fleming and H. M. Soner, On controlled Markov Processes and Viscosity Solutions, Springer, Berlin, 1993.Google Scholar
  11. 11.
    H. Föllmer, Stock price fluctuations as a diffusion in a random environment, In: Mathematical Models in Finance, (eds. S. D. Howison, F. P. Kelly and P. Wilmott), Chapman & Hall, London, 1995, pp. 21–33.Google Scholar
  12. 12.
    R. Frey and A. Stremme, Portfolio insurance and volatility, Department of Economics, Univ. of Bonn, discussion paper B256.Google Scholar
  13. 13.
    A. Guionnet, First order asymptotics of matrix integrals; a rigorous approach towards the understanding of matrix models, Comm. Math. Phys., 244 (2004), 527–569.zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    A. Guionnet and O. Zeitouni, Large deviation asymptotics for spherical integrals, J. Funct. Anal., 188 (2002), 461–515.zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    A. S. Kyle, Continuous auctions and insider trading, Econometrica, 53 (1985),1315–1335.zbMATHCrossRefGoogle Scholar
  16. 16.
    J.-M. Lasry and P.-L. Lions, Une classe nouvelle de problémes singuliers de contrôle stochastique, C. R. Acad. Sci. Paris Ser. I Math., 331 (2000), 879–889.zbMATHMathSciNetGoogle Scholar
  17. 17.
    J.-M. Lasry and P.-L. Lions, Instantaneous self-fulfilling of long-term prophecies on the probabilistic distribution of financial asset values, Ann. Inst. H. Poincaré Anal. Non Linéaire, 2006-2007, to appear.Google Scholar
  18. 18.
    J.-M. Lasry and P.-L. Lions, Large investor trading impacts on volatility, Ann. Inst. H. Poincaré Anal. Non Linéaire, 2006-2007, to appear.Google Scholar
  19. 19.
    J.-M. Lasry and P.-L. Lions, Towards a self-consistent theory of volatility, J. Math. Pures Appl., 2006-2007, to appear.Google Scholar
  20. 20.
    J.-M. Lasry and P.-L. Lions, Jeux á champ moyen. I. Le cas stationnaire, C. R. Acad. Sci. Paris, 343 (2006), 619–625.zbMATHMathSciNetGoogle Scholar
  21. 21.
    J.-M. Lasry and P.-L. Lions, Jeux á champ moyen. II. Horizon fini et contrôle optimal, C. R. Acad. Sci. Paris, 343 (2006), 679–684.zbMATHMathSciNetGoogle Scholar
  22. 22.
    G. Lasserre, Asymmetric information and imperfect competition in a continuous time multivariate security model, Finance Stoch., 8 (2004), 285–309.zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    P.-L. Lions, Mathematical Topics in Fluid Mechanics, Oxford Sci. Publ., Oxford Univ. Press, 1 (1996); 2 (1998).Google Scholar
  24. 24.
    R. Merton, Theory of rational option pricing, Bell J. Econom. Manag. Sci., 4 (1973), 141–183.CrossRefMathSciNetGoogle Scholar

Copyright information

© The Mathematical Society of Japan and Springer 2007

Authors and Affiliations

  1. 1.Institut de FinanceUniversité Paris-DauphineParis Cedex 16France
  2. 2.Collège de FranceParisFrance
  3. 3.Ceremade-UMR CNRS 7549Université Paris-DauphineParis Cedex 16France

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