Mean field games
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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.
- Aumann, R. (1964) Markets with a continuum of traders. Econometrica 32: pp. 39-50 CrossRef
- Avellaneda, M., Lipkin, M.D. (2003) A market induced mechanism for stock pinning. Quant. Finance 3: pp. 417-425 CrossRef
- Back, K. (1992) Insider trading in continuous time. Review of Financial Studies 5: pp. 387-409 CrossRef
- Back, K., Cao, C.-H., Willard, G. (2000) Imperfect competition among informed traders. J. Finance 55: pp. 2117-2155 CrossRef
- Bardi, M., Capuzzo-Dolcetta, I. (1997) Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman equations. Birkhäuser, Boston
- Bensoussan, A., Frehse, J. (1984) Nonlinear elliptic systems in stochastic game theory. J. Reine Angew. Math. 350: pp. 23-67
- Bensoussan, A., Frehse, J. (1995) Ergodic Bellman systems for stochastic games in arbitrary dimension. Proc. Roy. Soc. London Ser. A. 449: pp. 65-77 CrossRef
- Black, F., Scholes, M. (1973) The pricing of options and corporate liabilities. J. Polit. Econ. 81: pp. 637-654 CrossRef
- G. Carmona, Nash equilibria of games with a continuum of players, preprint.
- Fleming, W.H., Soner, H.M. (1993) On controlled Markov Processes and Viscosity Solutions. Springer, Berlin
- Föllmer, H. Stock price fluctuations as a diffusion in a random environment. In: Howison, S. D., Kelly, F. P., Wilmott, P. eds. (1995) Mathematical Models in Finance. Chapman & Hall, London, pp. 21-33
- R. Frey and A. Stremme, Portfolio insurance and volatility, Department of Economics, Univ. of Bonn, discussion paper B−256.
- Guionnet, A. (2004) First order asymptotics of matrix integrals; a rigorous approach towards the understanding of matrix models. Comm. Math. Phys. 244: pp. 527-569 CrossRef
- Guionnet, A., Zeitouni, O. (2002) Large deviation asymptotics for spherical integrals. J. Funct. Anal. 188: pp. 461-515 CrossRef
- Kyle, A.S. (1985) Continuous auctions and insider trading. Econometrica 53: pp. 1315-1335 CrossRef
- Lasry, J.-M., Lions, P.-L. (2000) Une classe nouvelle de problémes singuliers de contrôle stochastique. C. R. Acad. Sci. Paris Ser. I Math. 331: pp. 879-889
- 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.
- J.-M. Lasry and P.-L. Lions, Large investor trading impacts on volatility, Ann. Inst. H. Poincaré Anal. Non Linéaire, 2006-2007, to appear.
- J.-M. Lasry and P.-L. Lions, Towards a self-consistent theory of volatility, J. Math. Pures Appl., 2006-2007, to appear.
- Lasry, J.-M., Lions, P.-L. (2006) Jeux á champ moyen. I. Le cas stationnaire. C. R. Acad. Sci. Paris 343: pp. 619-625
- Lasry, J.-M., Lions, P.-L. (2006) Jeux á champ moyen. II. Horizon fini et contrôle optimal. C. R. Acad. Sci. Paris 343: pp. 679-684
- Lasserre, G. (2004) Asymmetric information and imperfect competition in a continuous time multivariate security model. Finance Stoch. 8: pp. 285-309 CrossRef
- P.-L. Lions, Mathematical Topics in Fluid Mechanics, Oxford Sci. Publ., Oxford Univ. Press, 1 (1996); 2 (1998).
- Merton, R. (1973) Theory of rational option pricing. Bell J. Econom. Manag. Sci. 4: pp. 141-183 CrossRef
- Mean field games
Japanese Journal of Mathematics
Volume 2, Issue 1 , pp 229-260
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- 1. Institut de Finance, Université Paris-Dauphine, Place du Maréchal de Lattre de Tassigny, 75775, Paris Cedex 16, France
- 2. Collège de France, 3 rue d’Ulm, 75005, Paris, France
- 3. Ceremade-UMR CNRS 7549, Université Paris-Dauphine, Place du Maréchal de Lattre de Tassigny, 75775, Paris Cedex 16, France