Annals of Applied Probability

A probabilistic weak formulation of mean field games and applications

René Carmona and Daniel Lacker

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Mean field games are studied by means of the weak formulation of stochastic optimal control. This approach allows the mean field interactions to enter through both state and control processes and take a form which is general enough to include rank and nearest-neighbor effects. Moreover, the data may depend discontinuously on the state variable, and more generally its entire history. Existence and uniqueness results are proven, along with a procedure for identifying and constructing distributed strategies which provide approximate Nash equlibria for finite-player games. Our results are applied to a new class of multi-agent price impact models and a class of flocking models for which we prove existence of equilibria.

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Ann. Appl. Probab., Volume 25, Number 3 (2015), 1189-1231.

First available in Project Euclid: 23 March 2015

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Zentralblatt MATH identifier

Primary: 60H30: Applications of stochastic analysis (to PDE, etc.)
Secondary: 93E20: Optimal stochastic control 91A13: Games with infinitely many players

Mean field games weak formulation price impact flocking models


Carmona, René; Lacker, Daniel. A probabilistic weak formulation of mean field games and applications. Ann. Appl. Probab. 25 (2015), no. 3, 1189--1231. doi:10.1214/14-AAP1020.

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  • [1] Alfonsi, A., Fruth, A. and Schied, A. (2010). Optimal execution strategies in limit order books with general shape functions. Quant. Finance 10 143–157.
  • [2] Aliprantis, C. and Border, K. (2007). Infinite Dimensional Analysis: A Hitchhiker’s Guide, 3rd ed. Springer, Berlin.
  • [3] Almgren, R. and Chriss, N. (2000). Optimal execution of portfolio transactions. J. Risk 3 5–39.
  • [4] Andersson, D. and Djehiche, B. (2011). A maximum principle for SDEs of mean-field type. Appl. Math. Optim. 63 341–356.
  • [5] Ballerini, M., Cabibbo, N., Candelier, R., Cavagna, A., Cisbani, E., Giardina, I., Lecomte, V., Orlandi, A., Parisi, G. and Procaccini, A. et al. (2008). Interaction ruling animal collective behavior depends on topological rather than metric distance: Evidence from a field study. Proc. Natl. Acad. Sci. USA 105 1232–1237.
  • [6] Bensoussan, A., Sung, K. C. J., Yam, S. C. P. and Yung, S. P. (2011). Linear-quadratic mean field games. Preprint.
  • [7] Buckdahn, R., Djehiche, B. and Li, J. (2011). A general stochastic maximum principle for SDEs of mean-field type. Appl. Math. Optim. 64 197–216.
  • [8] Carlin, B. I., Lobo, M. S. and Viswanathan, S. (2007). Episodic liquidity crises: Cooperative and predatory trading. J. Finance 62 2235–2274.
  • [9] Carmona, R. and Delarue, F. (2013). Mean field forward–backward stochastic differential equations. Electron. Commun. Probab. 18 no. 68, 15.
  • [10] Carmona, R. and Delarue, F. (2013). Forward–backward stochastic differential equations and controlled McKean Vlasov dynamics. Preprint. Available at arXiv:1303.5835.
  • [11] Carmona, R. and Delarue, F. (2013). Probabilistic analysis of mean-field games. SIAM J. Control Optim. 51 2705–2734.
  • [12] Carmona, R., Delarue, F. and Lachapelle, A. (2013). Control of McKean–Vlasov dynamics versus mean field games. Math. Financ. Econ. 7 131–166.
  • [13] Cellina, A. (1969). Approximation of set valued functions and fixed point theorems. Ann. Mat. Pura Appl. (4) 82 17–24.
  • [14] Cucker, F. and Smale, S. (2007). Emergent behavior in flocks. IEEE Trans. Automat. Control 52 852–862.
  • [15] Davis, M. H. A. (1979). Martingale methods in stochastic control. In Stochastic Control Theory and Stochastic Differential Systems (Proc. Workshop, Deutsch. Forschungsgemeinsch., Univ. Bonn, Bad Honnef, 1979) (M. Kohlmann and W. Vogel, eds.). Lecture Notes in Control and Information Sci. 16 85–117. Springer, Berlin.
  • [16] Dembo, A. and Zeitouni, O. (2009). Large Deviations Techniques and Applications. Springer, New York.
  • [17] Dunford, N. and Schwartz, J. T. (1988). Linear Operators. Part I: General Theory. Wiley, New York.
  • [18] El Karoui, N., Peng, S. and Quenez, M. C. (1997). Backward stochastic differential equations in finance. Math. Finance 7 1–71.
  • [19] Fuhrman, M., Hu, Y. and Tessitore, G. (2006). On a class of stochastic optimal control problems related to BSDEs with quadratic growth. SIAM J. Control Optim. 45 1279–1296 (electronic).
  • [20] Gärtner, J. (1988). On the McKean–Vlasov limit for interacting diffusions. Math. Nachr. 137 197–248.
  • [21] Gatheral, J., Schied, A. and Slynko, A. (2012). Transient linear price impact and Fredholm integral equations. Math. Finance 22 445–474.
  • [22] Gomes, D. A. and Voskanyan, V. K. (2013). Extended mean field games. Izv. Nats. Akad. Nauk Armenii Mat. 48 63–76.
  • [23] Guéant, O., Lasry, J.-M. and Lions, P.-L. (2011). Mean field games and applications. In Paris–Princeton Lectures on Mathematical Finance 2010. Lecture Notes in Math. 2003 205–266. Springer, Berlin.
  • [24] Hamadène, S. and Lepeltier, J. P. (1995). Backward equations, stochastic control and zero-sum stochastic differential games. Stoch. Stoch. Rep. 54 221–231.
  • [25] Hu, Y. and Peng, S. (1997). A stability theorem of backward stochastic differential equations and its application. C. R. Acad. Sci. Paris Sér. I Math. 324 1059–1064.
  • [26] Huang, M., Malhamé, R. and Caines, P. (2007). The Nash certainty equivalence principle and McKean–Vlasov systems: An invariance principle and entry adaptation. In Proceedings of the 46th IEEE Conference on Decision and Control 121–126. IEEE, New York.
  • [27] Huang, M., Malhamé, R. P. and Caines, P. E. (2006). Large population stochastic dynamic games: Closed-loop McKean–Vlasov systems and the Nash certainty equivalence principle. Commun. Inf. Syst. 6 221–251.
  • [28] Jacod, J. and Mémin, J. (1981). Sur un type de convergence intermédiaire entre la convergence en loi et la convergence en probabilité. In Seminar on Probability, XV (Univ. Strasbourg, Strasbourg, 1979/1980) (French). 529–546. Springer, Berlin.
  • [29] Jacod, J. and Shiryaev, A. N. (1987). Limit Theorems for Stochastic Processes. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 288. Springer, Berlin.
  • [30] Jadbabaie, A., Lin, J. and Morse, A. S. (2003). Coordination of groups of mobile autonomous agents using nearest neighbor rules. IEEE Trans. Automat. Control 48 988–1001.
  • [31] Lachapelle, A. and Wolfram, M. T. (2011). On a mean field game approach modeling congestion and aversion in pedestrian crowds. Transp. Res., Part B: Methodol. 45 1572–1589.
  • [32] Lasry, J.-M. and Lions, P.-L. (2007). Mean field games. Jpn. J. Math. 2 229–260.
  • [33] Lasry, J. M., Lions, P. L. and Guéant, O. (2008). Application of mean field games to growth theory. Preprint, available at
  • [34] Meyer-Brandis, T., Øksendal, B. and Zhou, X. Y. (2012). A mean-field stochastic maximum principle via Malliavin calculus. Stochastics 84 643–666.
  • [35] Nourian, M., Caines, P. and Malhamé, R. (2011). Mean field analysis of controlled Cucker–Smale type flocking: Linear analysis and perturbation equations. In Proceedings of the 18th IFAC World Congress, Milan, August 2011. 4471–4476. IFAC, New York.
  • [36] Pardoux, É. and Peng, S. G. (1990). Adapted solution of a backward stochastic differential equation. Systems Control Lett. 14 55–61.
  • [37] Peng, S. (2004). Filtration consistent nonlinear expectations and evaluations of contingent claims. Acta Math. Appl. Sin. Engl. Ser. 20 191–214.
  • [38] Ranga Rao, R. (1962). Relations between weak and uniform convergence of measures with applications. Ann. Math. Statist. 33 659–680.
  • [39] Varadarajan, V. S. (1958). On the convergence of sample probability distributions. Sankhyā 19 23–26.
  • [40] Vicsek, T., Czirók, A., Ben-Jacob, E., Cohen, I. and Shochet, O. (1995). Novel type of phase transition in a system of self-driven particles. Phys. Rev. Lett. 75 1226–1229.
  • [41] Villani, C. (2003). Topics in Optimal Transportation. Graduate Studies in Mathematics 58. Amer. Math. Soc., Providence, RI.