Mean Field Game of Optimal Tracking Portfolio

TL;DR

This paper develops a mean field game model for large-scale fund management competition, focusing on relative performance benchmarking and deriving equilibrium strategies through PDE and duality methods.

Contribution

It introduces a novel MFG framework with reflected state processes and analytically characterizes the equilibrium control using dual diffusion processes.

Findings

Established existence of the mean field equilibrium using PDE methods.

Derived explicit best response controls via dual transform and reflected diffusions.

Constructed approximate Nash equilibria for large finite-player games.

Abstract

This paper studies the mean field game (MFG) problem arising from a large population competition in fund management, featuring a new type of relative performance via the benchmark tracking. In the $n$-player model, each agent aims to minimize the expected largest shortfall of the wealth with reference to the benchmark process, which is modeled by a linear combination of the population's average wealth process and a market index process. With a continuum of agents, we formulate the MFG problem with a reflected state process. We establish the existence of the mean field equilibrium (MFE) using the partial differential equation (PDE) approach. Firstly, by applying the dual transform, the best response control of the representative agent can be characterized in analytical form in terms of a dual reflected diffusion process. As a novel contribution, we verify the consistency condition of the MFE in separated domains with the help of the duality relationship and properties of the dual process. Moreover, based on the MFE, we construct an approximate Nash equilibrium for the $n$-player game when the number $n$ is sufficiently large.