Mean field games. II: Finite horizon and optimal control. (Jeux à champ moyen. II: Horizon fini et contrôle optimal.) (French. English summary) Zbl 1153.91010
Summary: We continue our study of the notion of mean field games that we introduced in a previous Note [Part I, see C. R., Math., Acad. Sci. Paris 343, No. 9, 619–625 (2006; Zbl 1153.91009)]. We consider here the case of Nash equilibria for stochastic control type problems in finite horizon. We present general existence and uniqueness results for the partial differential equations systems that we introduce. We also give a possible interpretation of these systems in term of optimal control.
MSC:
| 91A15 | Stochastic games, stochastic differential games |
| 91A10 | Noncooperative games |
| 91A07 | Games with infinitely many players |
| 93E20 | Optimal stochastic control |
| 35K50 | Systems of parabolic equations, boundary value problems (MSC2000) |
| 35K55 | Nonlinear parabolic equations |
| 49J55 | Existence of optimal solutions to problems involving randomness |
Citations:
Zbl 1153.91009References:
| [1] | Guionnet, A., First order asymptotics of matrix integrals: a rigorous approach towards the understanding of matrix models, Comm. Math. Phys., 244, 527-569 (2003) · Zbl 1076.82026 |
| [2] | Guionnet, A.; Zeitouni, O., Large deviations asymptotics for spherical integrals, J. Funct. Anal., 188, 461-515 (2001) · Zbl 1002.60021 |
| [3] | Lasry, J.-M.; Lions, P.-L., Jeux à champ moyen. I - Le cas stationnaire, C. R. Acad. Sci. Paris, Ser. I, 343 (2006) · Zbl 1153.91009 |
| [4] | Lions, P.-L., Mathematical Topics in Fluid Mechanics, Oxford Science Publications, vol. 1 (1996), Clarendon Press: Clarendon Press Oxford, vol. 2, 1998 · Zbl 0866.76002 |
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