Abstract
We discuss the system of Fokker-Planck and Hamilton-Jacobi-Bellman equations arisingfrom the finite horizon control of McKean-Vlasov dynamics. We give examples of existence and uniqueness results. Finally, we propose some simple models for the motion of pedestrians and report about numerical simulations in which we compare mean filed games and mean field type control.
Highlights
An important research activity has been devoted to the study of stochastic differential games with a large number of players
Lions have introduced the notion of mean field games, which describe the asymptotic behavior of stochastic differential games (Nash equilibria) as the number N of players tends to infinity
In the limit when N → +∞, a given agent feels the presence of the other agents through the statistical distribution of the states of the other players
Summary
An important research activity has been devoted to the study of stochastic differential games with a large number of players. Lions have introduced the notion of mean field games, which describe the asymptotic behavior of stochastic differential games (Nash equilibria) as the number N of players tends to infinity In these models, it is assumed that the agents are all identical and that an individual agent can hardly influence the outcome of the game. Given a common feedback strategy, the asymptotics are given by the McKean-Vlasov theory, [16, 20] : the dynamics of a given agent is found by solving a stochastic differential equation with coefficients depending on a mean field, namely the statistical distribution of the states, which may affect the objective function. A∈Rn where p · q denotes the scalar product in Rd
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