An algebraic convergence rate for the optimal control of mckean-vlasov dynamics

We establish an algebraic rate of convergence of the value functions of N-particle stochastic control problems towards the value function of the corresponding McKean-Vlasov problem, also known as mean field control. The rate is obtained in the presence of both idiosyncratic and common noises and in a setting where the value function for the McKean-Vlasov problem need not be smooth. Our approach relies crucially on uniform in N Lipschitz and semiconcavity estimates for the N-particle value functions as well as a certain concentration inequality.