Abstract
We examine mean field control problems on a finite state space, in continuous time and over a finite time horizon. We characterize the value function of the mean field control problem as the unique viscosity solution of a Hamilton-Jacobi-Bellman equation in the simplex. In absence of any convexity assumption, we exploit this characterization to prove convergence, asNgrows, of the value functions of the centralizedN-agent optimal control problem to the limit mean field control problem value function, with a convergence rate of order [see formula in PDF]. Then, assuming convexity, we show that the limit value function is smooth and establish propagation of chaos,i.e.convergence of theN-agent optimal trajectories to the unique limiting optimal trajectory, with an explicit rate.
Highlights
Mean field control problems (MFCP), called control of McKean-Vlasov equations, can be interpreted as limit of cooperative N -agent games, as the number of players tends to infinity
We investigate N -agent optimization and mean field control problems in continuous time over a finite time horizon, with dynamics belonging to a finite state space {1, . . . , d}
Our goal is to study in detail the N -agent optimization and the mean field control problem and prove convergence of the former to the latter, the main result being to provide an explicit convergence rate
Summary
Mean field control problems (MFCP), called control of McKean-Vlasov equations, can be interpreted as limit of cooperative N -agent games, as the number of players tends to infinity. Players are non-cooperative in the prelimit N -player game and the notion of optimality is that of Nash equilibrium, which highly depends on the set of admissible startegies that is considered This makes the convergence analysis more difficult, expecially in case limiting mean field game solutions are non-unique; some references are [9, 19, 23, 28, 30] for diffusion-based models and [2, 3, 14, 17, 26] for finite state space.
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