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.
KeywordsNash Equilibrium Option Price Nonlinear Differential Equation Operator Versus Stochastic Game
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- 9.G. Carmona, Nash equilibria of games with a continuum of players, preprint.Google Scholar
- 10.W. H. Fleming and H. M. Soner, On controlled Markov Processes and Viscosity Solutions, Springer, Berlin, 1993.Google Scholar
- 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.R. Frey and A. Stremme, Portfolio insurance and volatility, Department of Economics, Univ. of Bonn, discussion paper B−256.Google Scholar
- 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.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.J.-M. Lasry and P.-L. Lions, Towards a self-consistent theory of volatility, J. Math. Pures Appl., 2006-2007, to appear.Google Scholar
- 23.P.-L. Lions, Mathematical Topics in Fluid Mechanics, Oxford Sci. Publ., Oxford Univ. Press, 1 (1996); 2 (1998).Google Scholar