Mean Field Exponential Utility Game: A Probabilistic Approach

TL;DR

This paper analyzes a complex multi-player and mean field game involving stocks affected by individual and common shocks, using probabilistic methods to characterize equilibria through novel FBSDEs with quadratic growth.

Contribution

It introduces a probabilistic approach to characterize equilibria in multi-player and mean field exponential utility games via new multi-dimensional FBSDEs.

Findings

Unique equilibrium characterized by novel FBSDEs

Established well-posedness of the FBSDEs

Proved convergence from N-player to mean field game

Abstract

We study an $N$-player and a mean field exponential utility game. Each player manages two stocks; one is driven by an individual shock and the other is driven by a common shock. Moreover, each player is concerned not only with her own terminal wealth but also with the relative performance of her competitors. We use the probabilistic approach to study these two games. We show the unique equilibrium of the $N$-player game and the mean field game can be characterized by a novel multi-dimensional FBSDE with quadratic growth and a novel mean-field FBSDEs, respectively. The well-posedness result and the convergence result are established.