  We consider Mean Field Games without idiosyncratic but with Brownian type
common noise. We introduce a notion of solutions of the associated
backward-forward system of stochastic partial differential equations. We show
that the solution exists and is unique for monotone coupling functions. This
the first general result for solutions of the Mean Field Games system with
common and no idiosynctratic noise. We also use the solution to find
approximate optimal strategies (Nash equilibria) for N-player differential
games with common but no idiosyncratic noise. An important step in the analysis
is the study of the well-posedness of a stochastic backward Hamilton-Jacobi
equation.
