## On the convergence rate for the finite player approximation of mean field control problems

We study mean field control problems for diffusion-based dynamics, with an idiosyncratic noise. These can be seen as limit problems for cooperative N-player games, when the number of players goes to infinity. The quantitative rate for the convergence of the corresponding value functions is understood either when the limit value function is smooth on the Wasserstein space, so that the optimal trajectory is unique, or in some case when the volatility is constant and non degenerate. Here we establish a rate for the convergence in absence of any convexity assumption, so that the limit value function is just Lipschitz and the optimal trajectory is not unique, and in case the volatility is controlled and possibly degenerate. The strategy is to perform a sup-convolution of the value function of the mean field control problem and show that it remains a viscosity sub-solution of the dynamic programming equation set in a suitable Hilbert space.

Area: IS7 - Stochastic optimal control of McKean-Vlasov equations (Elena Bandini)

Keywords: Mean field control problems, Cooperative N-player game, Equation on the Wasserstein space, sup-convolution

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