Existence of Optimal Stationary Singular Controls and Mean Field Game Equilibria

In this paper, we examine the stationary relaxed singular control problem within a multidimensional framework for a single agent as well as its mean field game equivalent. We demonstrate that optimal relaxed controls exist for two problem classes: one driven by queueing control and the other by harvesting models. These relaxed controls are defined by random measures across the state and control spaces with the state process described as a solution to the associated martingale problem. By leveraging findings from Kurtz and Stockbridge (2001), we establish the equivalence between the martingale problem and the stationary forward equation. This allows us to reformulate the relaxed control problem into a linear programming problem within the measure space. We prove the sequential compactness of these measures, thereby confirming the feasibility of achieving an optimal solution. Subsequently, our focus shifts to mean field games. Drawing on insights from the single-agent problem and employing the Kakutani–Glicksberg–Fan fixed point theorem, we derive the existence of a mean field game equilibria.