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Computation of mean field equilibria in economics. (English) Zbl 1193.91018

The authors present some applications of the mean field games introduced by J. M. Lasry and P. L. Lions [Jpn. J. Math. (3) 2, No. 1, 229–260 (2007; Zbl 1156.91321)]. It is known that a Nash equilibrium in such games leads to a system coupling a Hamiltonian-Jacobi-Bellman equation and a Kolmogorov equation. In the paper an efficient algorithm to find such an equilibrium is presented and tested. The algorithm is based on a procedure used in the field of quantum chemistry (see e.g. [Y. Maday and G.Turinici, New formulations of monotonically convergent quantum control algorithms, J. Chem. Phys. 118, 8191–8196 (2003)]) following the line of reasoning proposed by V. F. Krotov [Izv. Akad. Nauk SSSR, Tekh. Kibern. 1975, No. 5, 3–15 (1975; Zbl 0316.65009) and ibid. 1975, No. 6, 3–13 (1975; Zbl 0339.65040)]. The algorithm contains two parts: the first is related to discretization of the cost functional and the system of coupled partial differential equations, the second is an optimization monotonic algorithm. The efficiency of the proposed algorithm is illustrated by some numerical examples.

MSC:

91A07 Games with infinitely many players
91A15 Stochastic games, stochastic differential games
49L99 Hamilton-Jacobi theories
93C35 Multivariable systems, multidimensional control systems
81S20 Stochastic quantization
49K20 Optimality conditions for problems involving partial differential equations
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