Mean Field Game of Optimal Tracking Portfolio

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.