Solving MFGs with a Common Noise

The lion’s share of this chapter is devoted to the construction of equilibria for mean field games with a common noise. We develop a general two-step strategy for the search of weak solutions. The first step is to apply Schauder’s theorem in order to prove the existence of strong solutions to mean field games driven by a discretized version of the common noise. The second step is to make use of a general stability property of weak equilibria in order to pass to the limit along these discretized equilibria. We also present several criteria for strong uniqueness, in which cases weak equilibria are known to be strong.