Existence and Uniqueness Results for a Mean-Field Game of Optimal Investment

We establish the existence and uniqueness of the equilibrium for a stochastic mean-fieldgame of optimal investment. The analysis covers both finite and infinite time horizons,and the mean-field interaction of the representative company with a mass of identicaland indistinguishable firms is modeled through the time-dependent price at which theproduced good is sold. At equilibrium, this price is given in terms of a nonlinear func-tion of the expected (optimally controlled) production capacity of the representativecompany at each time. The proof of the existence and uniqueness of the mean-fieldequilibrium relies on a priori estimates and the study of nonlinear integral equations,but employs different techniques for the finite and infinite horizon cases. Additionally, we investigate the deterministic counterpart of the mean-field game under study.