A probabilistic weak formulation of mean field games and applications

Annals of Applied Probability

Volumn 25, Issue 3, 2015, Pages 1189-1231

  Carmona, René ^a   Lacker, Daniel ^a  

^a PRINCETON UNIVERSITY   (United States)

Author keywords

Flocking models; Mean field games; Price impact; Weak formulation

Indexed keywords

EID: 84925426147     PISSN: 10505164     EISSN: None     Source Type: Journal    
DOI: 10.1214/14-AAP1020     Document Type: Article

References (41)

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. MR2642960
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. MR2784835
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. MR2822408
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. MR3091726
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. MR3072222
12. CARMONA, R., DELARUE, F. and LACHAPELLE, A. (2013). Control of McKean-Vlasov dynamics versus mean field games. Math. Financ. Econ. 7 131-166. MR3045029
13. CELLINA, A. (1969). Approximation of set valued functions and fixed point theorems. Ann. Mat. Pura Appl. (4) 82 17-24. MR0263046
14. CUCKER, F. and SMALE, S. (2007). Emergent behavior in flocks. IEEE Trans. Automat. Control 52 852-862. MR2324245
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. MR0547467
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. MR1009162
18. EL KAROUI, N., PENG, S. andQUENEZ, M. C. (1997). Backward stochastic differential equations in finance. Math. Finance 7 1-71. MR1434407
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). MR2257222
20. GÄRTNER, J. (1988). On the McKean-Vlasov limit for interacting diffusions. Math. Nachr. 137 197-248. MR0968996
21. GATHERAL, J., SCHIED, A. and SLYNKO, A. (2012). Transient linear price impact and Fredholm integral equations. Math. Finance 22 445-474. MR2943180
22. GOMES, D. A. and VOSKANYAN, V. K. (2013). Extended mean field games. Izv. Nats. Akad. Nauk Armenii Mat. 48 63-76. MR3112690
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. MR2762362
24. HAMADÈNE, S. and LEPELTIER, J. P. (1995). Backward equations, stochastic control and zero-sum stochastic differential games. Stoch. Stoch. Rep. 54 221-231. MR1382117
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. MR1451251
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. MR2346927
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. MR0622586
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. MR0959133
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. MR1986266
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. MR2295621
33. LASRY, J. M., LIONS, P. L. and GUÉANT, O. (2008). Application of mean field games to growth theory. Preprint, available at http://hal.archives-ouvertes.fr/hal-00348376/.
34. MEYER-BRANDIS, T.,ØKSENDAL, B. and ZHOU, X. Y. (2012). A mean-field stochastic maximum principle via Malliavin calculus. Stochastics 84 643-666. MR2995516
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. MR1037747
37. PENG, S. (2004). Filtration consistent nonlinear expectations and evaluations of contingent claims. Acta Math. Appl. Sin. Engl. Ser. 20 191-214. MR2064000
38. RANGA RAO, R. (1962). Relations between weak and uniform convergence of measures with applications. Ann. Math. Statist. 33 659-680. MR0137809
39. VARADARAJAN, V. S. (1958). On the convergence of sample probability distributions. Sankhya 19 23-26. MR0094839
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. MR1964483