From the master equation to mean field game limit theory: A central limit theorem

Electronic Journal of Probability

Volumn 24, Issue , 2019, Pages

  Delarue, François ^a   Lacker, Daniel ^b   Ramanan, Kavita ^c  

^a UNIVERSITÉ CÔTE D'AZUR   (France)

^b Department of Biomedical Engineering   (United States)

^c BROWN UNIVERSITY   (United States)

Author keywords

Central limit theorem; Common noise; Fluctuations; Interacting particle systems; Master equation; McKean vlasov; Mean field games; Systemic risk

Indexed keywords

EID: 85072069003     PISSN: None     EISSN: 10836489     Source Type: Journal    
DOI: 10.1214/19-EJP298     Document Type: Article

References (47)

1. S. Ahuja, W. Ren, and T.-W. Yang, Asymptotic analysis of mean field games with small common noise, arXiv preprint arXiv:1707.09090 (2017). MR-3764155
2. E. Bayraktar and A. Cohen, Analysis of a finite state many player game using its master equation, arXiv preprint arXiv:1707.02648 (2017). MR-3860894
3. A. Bensoussan and J. Frehse, Nonlinear elliptic systems in stochastic game theory, Universität Bonn. SFB 72. Approximation und Optimierung, 1983.
4. A. Bensoussan, J. Frehse, and S.C.P. Yam, The master equation in mean field theory, Journal de Mathématiques Pures et Appliquées 103 (2015), no. 6, 1441–1474. MR-3343705
5. A. Bensoussan, J. Frehse, and S.C.P. Yam, On the interpretation of the master equation, Stochastic Processes and their Applications 127 (2017), no. 7, 2093–2137. MR-3652408
6. P. Cardaliaguet, F. Delarue, J.-M. Lasry, and P.-L. Lions, The master equation and the convergence problem in mean field games, Annals of Maths Studies, Princeton University Press, 2019. MR-3616319
7. R. Carmona and F. Delarue, The master equation for large population equilibriums, Stochastic Analysis and Applications 2014, Springer, 2014, pp. 77–128. MR-3332710
8. R. Carmona and F. Delarue, Probabilistic Theory of Mean Field Games: Vol. I, Mean Field FBSDEs, Control, and Games, Stochastic Analysis and Applications, Springer Verlag, 2017. MR-3752669
9. R. Carmona and F. Delarue, Probabilistic Theory of Mean Field Games: Vol. II, Mean Field Games with Common Noise and Master Equations, Stochastic Analysis and Applications, Springer Verlag, 2017. MR-3572323
10. R. Carmona, J.-P. Fouque, and L.-H. Sun, Mean field games and systemic risk, Communications in Mathematical Sciences 13 (2015), no. 4, 911–933. MR-3325083
11. R. Carmona and X. Zhu, A probabilistic approach to mean field games with major and minor players, The Annals of Applied Probability 26 (2016), no. 3, 1535–1580. MR-3513598
12. A. Cecchin and M. Fischer, Probabilistic approach to finite state mean field games, arXiv preprint arXiv:1704.00984 (2017).
13. A. Cecchin and G. Pelino, Convergence, fluctuations and large deviations for finite state mean field games via the master equation, arXiv preprint arXiv:1707.01819 (2017).
14. J.-F. Chassagneux, D. Crisan, and F. Delarue, A probabilistic approach to classical solutions of the master equation for large population equilibria, arXiv preprint arXiv:1411.3009 (2014).
15. M. Coghi and F. Flandoli, Propagation of chaos for interacting particles subject to environmental noise, Ann. Appl. Probab. 26 (2016), no. 3, 1407–1442. MR-3513594
16. Giuseppe Da Prato and Jerzy Zabczyk, Stochastic equations in infinite dimensions, second ed., Encyclopedia of Mathematics and its Applications, vol. 152, Cambridge University Press, Cambridge, 2014. MR-3236753
17. D. Dawson and J. Vaillancourt, Stochastic McKean-Vlasov equations, Nonlinear Differential Equations and Applications NoDEA 2 (1995), no. 2, 199–229. MR-1328577
18. F. Delarue, D. Lacker, and K. Ramanan, From the master equation to mean field game limit theory: Large deviations and concentration of measure, 2018.
19. M. Fischer, On the connection between symmetric n-player games and mean field games, The Annals of Applied Probability 27 (2017), no. 2, 757–810. MR-3655853
20. N. Fournier and A. Guillin, On the rate of convergence in Wasserstein distance of the empirical measure, Probability Theory and Related Fields 162 (2015), no. 3-4, 707–738. MR-3383341
21. W. Gangbo and A. Swiech, Existence of a solution to an equation arising from the theory of mean field games, Journal of Differential Equations 259 (2015), no. 11, 6573–6643. MR-3397332
22. J. Gärtner, On the McKean-Vlasov limit for interacting diffusions, Mathematische Nachrichten 137 (1988), no. 1, 197–248. MR-0968996
23. M. Hitsuda and I. Mitoma, Tightness problem and stochastic evolution equation arising from fluctuation phenomena for interacting diffusions, Journal of Multivariate Analysis 19 (1986), no. 2, 311–328. MR-0853061
24. M. Huang, R. Malhamé, and P. Caines, Large population stochastic dynamic games: closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle, Communications in Information & Systems 6 (2006), no. 3, 221–252. MR-2346927
25. B. Jourdain and S. Méléard, Propagation of chaos and fluctuations for a moderate model with smooth initial data, 34 (1998), 727–766. MR-1653393
26. V.N. Kolokoltsov, J. Li, and W. Yang, Mean field games and nonlinear Markov processes, arXiv preprint arXiv:1112.3744 (2011). MR-2780345
27. H. Kunita, Stochastic flows and stochastic differential equations, Cambridge Studies in Advanced Mathematics, Cambridge University Press, 1990. MR-1070361
28. T.G. Kurtz and J. Xiong, Numerical solutions for a class of SPDEs with application to filtering, Stochastics in finite and infinite dimensions, Trends Math., Birkhäuser Boston, Boston, MA, 2001, pp. 233–258. MR-1797090
29. T.G. Kurtz and J. Xiong, A stochastic evolution equation arising from the fluctuations of a class of interacting particle systems, Communications in Mathematical Sciences 2 (2004), no. 3, 325–358. MR-2118848
30. D. Lacker, A general characterization of the mean field limit for stochastic differential games, Probability Theory and Related Fields 165 (2016), no. 3-4, 581–648. MR-3520014
31. D. Lacker, On the convergence of closed-loop Nash equilibria to the mean field game limit, arXiv preprint arXiv:1808.02745 (2018).
32. O.A. Ladyzhenskaia, V.A. Solonnikov, and N.N. Ural’tseva, Linear and quasi-linear equations of parabolic type, vol. 23, American Mathematical Soc., 1988. MR-0241822
33. J.-M. Lasry and P.-L. Lions, Jeux à champ moyen. i–le cas stationnaire, Comptes Rendus Mathématique 343 (2006), no. 9, 619–625. MR-2269875
34. J.-M. Lasry and P.-L. Lions, Jeux à champ moyen. ii–horizon fini et contrôle optimal, Comptes Rendus Mathématique 343 (2006), no. 10, 679–684. MR-2271747
35. J.-M. Lasry and P.-L. Lions, Mean field games, Japanese journal of mathematics 2 (2007), no. 1, 229–260. MR-2295621
36. P.-L. Lions, Cours au Collège de France.
37. Huang M, R. Malhamé, and P. Caines, Large-population cost-coupled LQG problems with nonuniform agents: individual-mass behavior and decentralized ε-Nash equilibria, IEEE transactions on automatic control 52 (2007), no. 9, 1560–1571. MR-2352434
38. J. Ma, P. Protter, and J. Yong, Solving forward-backward stochastic differential equations explicitly – a four step scheme, Probab. Theory Related Fields 98 (1994), 339–359. MR-1262970
39. H.P. McKean, A class of Markov processes associated with nonlinear parabolic equations, Proceedings of the National Academy of Sciences 56 (1966), no. 6, 1907–1911. MR-0221595
40. S. Méléard, Asymptotic behaviour of some interacting particle systems; McKean-Vlasov and Boltzmann models, Probabilistic models for nonlinear partial differential equations (Montecatini Terme, 1995), Lecture Notes in Math., vol. 1627, Springer, 1996, pp. 42–95. MR-1431299
41. K. Oelschlager, A martingale approach to the law of large numbers for weakly interacting stochastic processes, The Annals of Probability (1984), 458–479. MR-0735849
42. I. Pinelis, Optimum bounds for the distributions of martingales in Banach spaces, Ann. Probab. 22 (1994), no. 4, 1679–1706. MR-1331198
43. T. Strömberg, A note on the differentiability of conjugate functions, Archiv der Mathematik 93 (2009), no. 5, 481. MR-2563594
44. A.-S. Sznitman, Topics in propagation of chaos, Ecole d’Eté de Probabilités de Saint-Flour XIX—1989 (1991), 165–251. MR-1108185
45. A.S. Sznitman, A fluctuation result for nonlinear diffusions, Infinite Dimensional Analysis and Stochastic Processes (1985), 145–160. MR-0865024
46. H. Tanaka and M. Hitsuda, Central limit theorem for a simple diffusion model of interacting particles, Hiroshima Mathematical Journal 11 (1981), no. 2, 415–423. MR-0620546
47. A. J. Veretennikov, On strong solutions and explicit formulas for solutions of stochastic integral equations, Mathematics of the USSR-Sbornik 39 (1981), no. 3, 387.