Selection of equilibria in a linear quadratic mean-field game

Stochastic Processes and their Applications

Volumn 130, Issue 2, 2020, Pages 1000-1040

  Delarue, François ^a   Foguen Tchuendom, Rinel ^a  

^a UNIVERSITÉ CÔTE D'AZUR   (France)

Author keywords

Common noise; Mean field game; N player game; Peano phenomenon; Selection of equilibria; Vanishing viscosity

Indexed keywords

COMPUTER SIMULATION; STATISTICS; STOCHASTIC SYSTEMS;

COMMON NOISE; MEAN FIELD GAMES; N-PLAYER GAMES; PEANO PHENOMENON; SELECTION OF EQUILIBRIA; VANISHING VISCOSITY;

VISCOSITY;

EID: 85066289105     PISSN: 03044149     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.spa.2019.04.005     Document Type: Article

References (52)

1. Attanasio, S., Flandoli, F., Zero-noise solutions of linear transport equations without uniqueness: an example. C. R. Math. Acad. Sci. Paris 347:13–14 (2009), 753–756.
2. Bafico, R., Baldi, P., Small random perturbations of Peano phenomena. Stochastics 6 (1982), 279–292.
3. Bardi, M., Explicit solutions of some linear quadratic mean field games. Netw. Heterogeneous Media 7 (2012), 243–261.
4. C. Bertucci, J.M. Lasry, P.L. Lions, Some remarks on Mean Field Games, Technical report, 2018.
5. Bossy, M., Talay, D., A stochastic particle method for the Mckean-Vlasov and the Burgers Equation. Math. Comp. 66 (1997), 157–192.
6. P. Cardaliaguet, Notes from P.L. Lions’ lectures at the Collège de France, Technical report,, 2012.
7. Cardaliaguet, P., Delarue, F., Lasry, J.-M., Lions, P.-L., The Master Equation and the Convergence Problem in Mean Field Games. 2019, Annals Maths Studies 9780691193717 in press.
8. Carmona, R., Delarue, F., Probabilistic analysis of mean field games. SIAM J. Control Optim. 51 (2013), 2705–2734.
9. Carmona, R., Delarue, F., Probabilistic Theory of Mean Field Games: vol. I, Mean Field FBSDEs, Control, and Games Stochastic Analysis and Applications, 2018, Springer Verlag.
10. Carmona, R., Delarue, F., Probabilistic Theory of Mean Field Games: vol. II, Mean Field Games with Common Noise and Master Equations Stochastic Analysis and Applications, 2018, Springer Verlag.
11. Carmona, R., Delarue, F., Lachapelle, A., Control of McKean-Vlasov versus Mean Field Games. Math. Financ. Econom. 7 (2013), 131–166.
12. Carmona, R., Delarue, F., Lacker, D., Ann. Probab. 44:6 (2016), 3740–3803.
13. Carmona, R., Fouque, J.P., Sun, L.H., Mean field games and systemic risk: a toy model. Commun. Math. Sci. 13 (2015), 911–933.
14. A. Cecchin, P. Dai Pra, M. Fischer, G. Pelino, On the convergence problem in mean field games: A two state model without uniqueness, Technical report, 2018,.
15. J.F. Chassagneux, D. Crisan, F. Delarue, McKean-Vlasov FBSDEs and related Master Equation, Technical report, 2014.
16. Davie, A.M., Uniqueness of solutions of stochastic differential equations. Int. Math. Res. Not. IMRN, 24, 2007 Art. ID rnm124, 26 pp.
17. Delarue, F., On the existence and uniqueness of solutions to FBSDEs in a non-degenerate Case. Stochastic Process. Appl. 99 (2002), 209–286.
18. Delarue, F., Restoring uniqueness to mean-field games by randomizing the equilibria. Stoch. Partial Differential Equations: Anal. Comput., 2019.
19. Delarue, F., Flandoli, F., The transition point in the zero noise limit for a 1D Peano example. Discrete Contin. Dyn. Syst. 34:10 (2014), 4071–4083.
20. Delarue, F., Flandoli, F., Vincenzi, D., Noise prevents collapse of Vlasov-Poisson point charges. Comm. Pure Appl. Math. 67:10 (2014), 1700–1736.
21. Delarue, F., Guatteri, G., Weak existence and uniqueness for FBSDEs. Stochastic Process. Appl. 116 (2006), 1712–1742.
22. Fischer, M., On the connection between symmetric N-player games and mean field games. Ann. Appl. Probab. 27:2 (2017), 757–810.
23. Flandoli, F., Random Perturbation of PDEs and Fluid Dynamics: Ecole d’été de probabilités de Saint-Flour XL. Lecture Notes in Mathematics, 2011, Springer Verlag.
24. Flandoli, F., Gubinelli, M., Priola, E., Well posedness of the transport equation by stochastic perturbation. Invent. Math. 180 (2010), 1–53.
25. Gangbo, W., Swiech, A., Existence of a solution to an equation arising from the theory of mean field games. J. Differential Equations 259:11 (2015), 6573–6643.
26. Gomes, D.A., Mohr, J., Souza, R.R., Discrete time, finite state space mean field games. J. Math. Pures Appl. (9) 93 (2010), 308–328.
27. Gradinaru, M., Herrmann, S., Roynette, B., A singular large deviations phenomenon. Ann. Inst. H. Poincaré Probab. Statist. 37:5 (2001), 555–580.
28. Guéant, O., Lasry, J.M., Lions, P.L., Mean field games and applications. et al., R. Carmona, (eds.) Paris Princeton Lectures in Mathematical Finance IV Lecture Notes in Mathematics, vol. 2003, 2010, Springer Verlag.
29. Hörmander, L., Lectures on Nonlinear Hyperbolic Differential Equations Mathématiques & Applications (Berlin) [Mathematics & Applications], vol. 26, 1997, Springer-Verlag, Berlin.
30. Huang, M., Caines, P.E., Malhamé, R.P., Large population stochastic dynamic games: closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle. Commun. Inf. Syst. 6 (2006), 221–252.
31. Jourdain, B., Reygner, J., The small noise limit of order-based diffusion processes. Electron. J. Probab., 19, 2014, no. 29, 36.
32. Krylov, N.V., Röckner, M., Strong solutions of stochastic equations with singular time dependent drift. Probab. Theory Related Fields 131 (2005), 154–196.
33. Kuznetsov, N.N., The accuracy of certain approximate methods for the computation of weak solutions of a first order quasilinear equation. Ž. Vyčisl. Mat. i Mat. Fiz. 16:6 (1976), 1489–1502, 1627.
34. D. Lacker, On the convergence of closed-loop Nash equilibria to the mean field game limit, Technical report,.
35. Lacker, D., A general characterization of the mean field limit for stochastic differential games. Probab. Theory Related Fields 165 (2016), 581–648.
36. Ladyzenskaja, O.A., Solonnikov, V.A., Ural'ceva, N.N., Linear and Quasi-linear Equations of Parabolic Type (Translations of Mathematical Monographs Reprint), 1968, American Math. Society.
37. Lasry, J.M., Lions, P.L., Jeux à champ moyen I. Le cas stationnaire. C. R. Math. Acad. Sci. Paris A, 343(9), 2006.
38. Lasry, J.M., Lions, P.L., Jeux à champ moyen II. Horizon fini et contrôle optimal. C. R. Math. Acad. Sci. Paris A, 343(10), 2006.
39. Lasry, J.M., Lions, P.L., Mean field games. Jpn. J. Math. 2:1 (2007), 229–260.
40. Lax, P., Hyperbolic Differential Equations Courant Institute Lecture Notes, 2006, American Mathematical Society, Providence, RI.
41. P.L. Lions, Théorie des jeux à champs moyen et applications, Lectures at the Collège de France., 2007-2008.
42. Ma, J., Protter, P., Yong, J., Solving forward-backward stochastic differential equations explicitly – a four step scheme. Probab. Theory Related Fields 98 (1994), 339–359.
43. Ma, J., Wu, Z., Zhang, D., Zhang, J., On well-posedness of forward-backward SDEs - A unified approach. Ann. Appl. Prob. 25 (2015), 2168–2214.
44. Stroock, D., Varadhan, S.R.S., Multidimensional Diffusion Processes. 2005, Springer Verlag.
45. Sznitman, A.S., Topics in propagation of chaos. Burkholder, D.L., et al. (eds.) Ecole de Probabilités de Saint Flour, XIX-1989 Lecture Notes in Mathematics, vol. 1464, 1989, 165–251.
46. Tadmor, E., Tang, T., Pointwise convergence rate for nonlinear conservation laws. Hyperbolic Problems: Theory, Numerics, Applications, Vol. II (ZÜrich, 1998) Internat. Ser. Numer. Math., vol. 130, 1999, Birkhäuser, Basel, 925–934.
47. Tadmor, E., Tang, T., Pointwise error estimates for scalar conservation laws with piecewise smooth solutions. SIAM J. Numer. Anal. 36:6 (1999), 1739–1758.
48. Tang, T., Zhen, T.H., Viscosity methods for piecewise smooth solutions to scalar conservation laws. Math. Comp. 66:218 (1997), 495–526.
49. Tchuendom, R. Foguen, Uniqueness for linear-quadratic mean field games with common noise. Dynam. Games Appl. 8 (2018), 199–210.
50. Trevisan, D., Zero noise limits using local times. Electron. Commun. Probab., 18, 2013, no. 31, 7.
51. Veretennikov, A.Y., Strong solutions and explicit formulas for solutions of stochastic integral equations. Mat. Sb. 111 (1980), 434–452.
52. Zvonkin, A.K., A transformation of the phase space of a diffusion process that will remove the drift. Mat. Sb. 93 (1974), 129–149.