Appears in collection : Schlumberger workshop on Topics in Applied Probability

I will discuss recent advances in large population stochastic control, in the spirit of the pioneering by Lasry and Lions and by Caines and Malhamé in 2006. The basic point is to seek approximate equilibria over families of interacting players when the number of players tends to the infinity, by taking benefit of some underlying propagation of chaos. The framework I will consider is twofold: the first one is the standard mean-field game problem, for which equilibria are investigated as Nash equilibria, and the second one is the control of McKean-Vlasov diffusion processes, for which equilibria are investigated as cooperative equilibria. In both cases, I will assume that players are driven by correlated noises. In this setting, I will discuss a probabilistic approach for proving the existence of equilibria. I will also investigate the connection with an infinite dimensional partial differential equation, set on the space of probability measures, that describes the equilibria in an analytic way. Joint work with R. Carmona and D. Lacker (Princeton).