Existence of Optimal Stationary Singular Controls and Mean Field Game Equilibria

In this paper, we examine the stationary relaxed singular control problem within a multi-dimensional framework for a single agent, as well as its Mean Field Game (MFG) equivalent. We demonstrate that optimal relaxed controls exist for both maximization and minimization cases. 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-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 Kakutani--Glicksberg--Fan fixed point theorem, we derive the existence of a mean field game equilibria.