We study the McKean-Vlasov optimal control problem with common noise in various formulations, namely the strong and weak formulation, as well as the Markovian and non-Markovian formulations, and allowing for the law of the control process to appear in the state dynamics. By interpreting the controls as probability measures on an appropriate canonical space with two filtrations, we then develop the classical measurable selection, conditioning and concatenation arguments in this new context, and establish the dynamic programming principle under general conditions. * Mao Fabrice Djete gratefully acknowledges support from the région Île-de-France. This work also benefited from support of the ANR project PACMAN ANR-16-CE05-0027.

In this paper, we study the extended mean field control problem, which is a class of McKean-Vlasov stochastic control problem where the state dynamics and the reward functions depend upon the joint (conditional) distribution of the controlled state and the control process. By considering an appropriate controlled Fokker-Planck equation, we can formulate an optimization problem over a space of measure-valued processes and, under suitable assumptions, prove the equivalence between this optimization problem and the extended mean-field control problem. Moreover, with the help of this new optimization problem, we establish the associated limit theory i.e. the extended mean field control problem is the limit of a large population control problem where the interactions are achieved via the empirical distribution of state and control processes.

In this paper, we investigate a class of mean field games where the mean field interactions are achieved through the joint (conditional) distribution of the controlled state and the control process. The strategies are of open loop type, and the volatility coefficient σ can be controlled. Using (controlled) Fokker-Planck equations, we introduce a notion of measure-valued solution of mean-field games of controls, and through convergence results, prove a relation between these solutions on the one hand, and the ǫN -Nash equilibria on the other hand. It is shown that ǫN -Nash equilibria in the N -player games have limits as N tends to infinity, and each limit is a measure-valued solution of the mean-field games of controls. Conversely, any measure-valued solution can be obtained as the limit of a sequence of ǫN -Nash equilibria in the N -player games. In other words, the measure-valued solutions are the accumulating points of ǫN -Nash equilibria. Similarly, by considering an ǫ-strong solution of mean field games of controls which is the classical strong solution where the optimality is obtained by admitting a small error ǫ, we prove that the measure-valued solutions are the accumulating points of this type of solutions when ǫ goes to zero. Finally, existence of measure-valued solution of mean-field games of controls are proved in the case without common noise. * The author is grateful to Dylan Possamaï and Xiaolu Tan for helpful comments and suggestions.

In this paper, we study the extended mean field control problem, which is a class of McKean-Vlasov stochastic control problem where the state dynamics and the reward functions depend upon the joint (conditional) distribution of the controlled state and the control process. By considering an appropriate controlled Fokker-Planck equation, we can formulate an optimization problem over a space of measure-valued processes and prove, under suitable assumptions, the equivalence between this optimization problem and the extended mean-field control problem. Moreover, with the help of this new optimization problem, we establish the associated limit theory i.e. the extended mean-field control problem is the limit of a large population control problem where the interactions are achieved via the empirical distribution of state and control processes. * The author is grateful to Dylan Possamaï and Xiaolu Tan for helpful comments and suggestions.