Two sided ergodic singular control and mean field game for diffusions

In a probabilistic mean-field game driven by a linear diffusion an individual player aims to minimize an ergodic long-run cost by controlling the diffusion through a pair of -- increasing and decreasing -- c\`adl\`ag processes, while he is interacting with an aggregate of players through the expectation of a similar diffusion controlled by another pair of c\`adl\`ag processes. In order to find equilibrium points in this game, we first consider the control problem, in which the individual player has no interaction with the aggregate of players. In this case, we prove that the best policy is to reflect the diffusion process within two thresholds. Based on these results, we obtain criteria for the existence of equilibrium points in the mean-field game in the case when the controls of the aggregate of players are of reflection type, and give a pair of nonlinear equations to find these equilibrium points. In addition, we present an approximation result for Nash equilibria of erdogic games with finitely many players to the mean-field game equilibria considered above when the number of players tends to infinity. These results are illustrated by several examples where the existence and uniqueness of the equilibrium points depend on the coefficients of the underlying diffusion.