Probabilistic Theory of Mean Field Games with Applications II

This chapter is a preparation for the analysis of mean field games with a common noise, to which we dedicate the entire first half of this second volume. By necessity, we revisit the basic tools introduced in Chapters (Vol I)-3 and (Vol I)-4 for mean field games without common noise, and in particular, the theory of forward-backward stochastic differential equations and its connection with optimal stochastic control. Our goal is to investigate optimal stochastic control problems based on stochastic dynamics and cost functionals depending on an additional random environment. To that effect, we provide a general discussion of forward-backward systems in a random environment. In the framework of mean field games, this random environment will account for the random state of the population in equilibrium given the (random) realization of the systemic noise source common to all the players.

We introduce the concept of master field within the framework of mean field games with a common noise. We present it as the decoupling field of an infinite dimensional forward-backward system of stochastic partial differential equations characterizing the equilibria. The forward equation is a stochastic Fokker-Planck equation and the backward equation a stochastic Hamilton-Jacobi-Bellman equation. We show that whenever existence and uniqueness of equilibria hold for any initial condition, the master field is a viscosity solution of Lions’ master equation.

This chapter is concerned with existence and uniqueness of classical solutions to the master equation. The importance of classical solutions will be demonstrated in the next chapter where they play a crucial role in proving the convergence of games with finitely many players to mean field games. We propose constructions based on the differentiability properties of the flow generated by the solutions of the forward-backward system of the McKean-Vlasov type representing the equilibrium of the mean field game on an L ²-space. Existence of a classical solution is first established for small time. It is then extended to arbitrary finite time horizons under the additional Lasry-Lions monotonicity condition.

The rationale of this chapter is the same as for the last chapter of the first volume of the book. We leverage the technology developed in the second volume to revisit some of the examples introduced in Chapter (Vol I)-1, and complete their mathematical analysis. We use some of the tools introduced for the analysis of mean field games with a common noise to study important game models which are not amenable to the theory covered by the first volume. These models include extensions to games with minor and major players, games of timing, and some finite state space models. We believe that these mean field game models have a great potential for the quantitative analysis of very important practical applications, and we show how the technology developed in the second volume of the book can be brought to bear on their solutions.