Existence and uniqueness results for a mean-field game of optimal investment

Abstract

We establish the existence and uniqueness of the equilibrium for a stochastic mean-field game 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 identical and indistinguishable firms is modeled through the time-dependent price at which the produced good is sold. At equilibrium, this price is given in terms of a nonlinear function of the expected (optimally controlled) production capacity of the representative company at each time. The proof of the existence and uniqueness of the mean-field equilibrium 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.

Figures (6)

• Figure 1: The solution $z$ to the integro-differential equation \ref{['eq:zIDE']}, the equilibrium average production capacity $\widehat{q}$, and the equilibrium optimal investment strategy $\widehat{{\mathbf u} }$, in the short, medium, and long time horizon cases. Initial condition $x = 10$.
• Figure 2: The solution $z$ to the integro-differential equation \ref{['eq:zIDE']}, the equilibrium average production capacity $\widehat{q}$, and the equilibrium optimal investment strategy $\widehat{{\mathbf u} }$, in the short, medium, and long time horizon cases. Initial condition $x = y_\infty \approx 13.5721$.
• Figure 3: Trajectories of $\widehat{X}$ and the equilibrium average production capacity $\widehat{q}$, in the short, medium, and long time horizon cases, and for $\sigma = 0.001, 0.01, 0.1$. Initial condition $x = 10$.
• Figure 4: Trajectories of $\widehat{X}$ and the equilibrium average production capacity $\widehat{q}$, in the short, medium, and long time horizon cases, and for $\sigma = 0.001, 0.01, 0.1$. Initial condition $x = y_\infty \approx 13.5721$.
• Figure 5: The solution $z$ to the integro-differential equation \ref{['eq:zIDE']}, the equilibrium average production capacity $\widehat{q}$, and the equilibrium optimal investment strategy $\widehat{{\mathbf u} }$. Initial conditions: $x = 10$ in the first row; $x = y_\infty + 0.01 \approx 13.5821$ in the second row.

Theorems & Definitions (43)

• Lemma 2.2
• proof
• Definition 2.3
• Lemma 3.1
• proof
• Remark 3.2
• Proposition 3.3
• proof
• Lemma 3.4
• Remark 3.5