Mean-Field Games with two-sided singular controls for L\'evy processes

In a probabilistic mean field game driven by a L\'evy process an individual player aims to minimize a long run discounted/ergodic cost by controlling the process 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 controlled process by another pair of c\`adl\`ag processes. With the Brouwer fixed point theorem, we provide easy to check conditions for the existence of mean field game equilibrium controls for both the discounted and ergodic control problem, characterize them as the solution of an integro-differential equation and show with a counterexample that uniqueness does not always holds. Furthermore, we study the convergence of equilibrium controls in the abelian sense. Finally, we treat the convergence of a finite-player game to this problem to justify our approach.