EXPLICIT SOLUTIONS OF SOME LINEAR-QUADRATIC MEAN FIELD GAMES

被引:105
作者
Bardi, Martino [1 ]
机构
[1] Univ Padua, Dipartimento Matemat, I-35121 Padua, Italy
关键词
Mean field games; stochastic control; linear-quadratic problems; differential games; models of population distribution;
D O I
10.3934/nhm.2012.7.243
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
We consider N-person differential games involving linear systems affected by white noise, running cost quadratic in the control and in the displacement of the state from a reference position, and with long-time-average integral cost functional. We solve an associated system of Hamilton-Jacobi-Bellman and Kolmogorov-Fokker-Planck equations and find explicit Nash equilibria in the form of linear feedbacks. Next we compute the limit as the number N of players goes to infinity, assuming they are almost identical and with suitable scalings of the parameters. This provides a quadratic-Gaussian solution to a system of two differential equations of the kind introduced by Lasry and Lions in the theory of Mean Field Games [22]. Under a natural normalization the uniqueness of this solution depends on the sign of a single parameter. We also discuss some singular limits, such as vanishing noise, cheap control, vanishing discount. Finally, we compare the L-Q model with other Mean Field models of population distribution.
引用
收藏
页码:243 / 261
页数:19
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