Optimization in a Random Environment

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.