This system can be also obtained directly, and the Master equation is then a way to decouple the coupled equations. In the case of mean field games, there is no Bellman equation, so we introduce the system of HJB-FP equations directly, with a fixed point approach. The Master equation is then obtained by trying to decouple the HJB equation from the Fokker Planck equation. Carmona and Delarue try to derive the Master equation from a common optimality principle of Dynamic Programming, with constraints. The differences between the Mean field games and Mean field type control cases stem from the way the minimization is performed. It results that for the Mean field type control, our Master equation is different from that of Carmona-Delarue. To some extent, our approach is more analytic (sufficient conditions) whereas the approach of Carmona-Delarue is more probabilistic (necessary conditions). In particular, Carmona-Delarue rely on the lift-up approach introduced by P.L. Lions in his lectures. This method connects functions of random variables to functions of probability measures. Both works are largely formal, and a full comparison is a daunting work, which is left for future work. It may be interesting to also combine these ideas.

LINEAR QUADRATIC PROBLEMS
