Mean-Field Game of Relative Performance Portfolio for Two Populations with Poisson Common Noise

TL;DR

This paper develops a mean-field game model for two populations managing portfolios with Poisson jump risks and common noise, analyzing equilibria and convergence, with numerical and financial insights.

Contribution

It introduces a novel MFG framework incorporating Poisson jump risks and establishes convergence of Nash equilibria to the mean-field equilibrium.

Findings

Nash equilibrium converges to MFE as populations grow large.

Poisson risks significantly influence equilibrium strategies.

Numerical examples illustrate the impact of Poisson noise on portfolio performance.

Abstract

This paper studies the mean field game (MFG) and N-player game on relative performance portfolio management with two heterogeneous populations. In addition to the Brownian idiosyncratic and common noise, the first population invests in assets driven by idiosyncratic Poisson jump risk, while the second population invests in assets subject to Poisson common noise. We establish the characterization of the mean-field equilibrium (MFE) in MFG with two populations as well as the Nash equilibrium in the $N_1+N_2$-player game. Furthermore, we prove the convergence of the Nash equilibrium in the $N_1+N_2$-player game to the MFE as the number of players in two populations tends to infinity. We also discuss some impacts on MFE by the Poisson idiosyncratic risk and Poisson common noise in the context of relative performance, compensated by some numerical examples and financial implications.