Mean Field Game of Optimal Tracking Portfolio

ArXiv ID: 2505.01858 “View on arXiv”

Authors: Lijun Bo, Yijie Huang, Xiang Yu

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 constraint. In the n-agent model, each agent can strategically inject capital to ensure that the total wealth outperforms the benchmark process, which is modeled as a linear combination of the population’s average wealth process and a market index process. That is, each agent is concerned about the performance of her competitors captured by the floor constraint. With a continuum of agents, we formulate the constrained MFG problem and transform it into an equivalent unconstrained 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.

Keywords: mean field games, benchmark tracking constraint, constrained optimization, PDE approach, reflected diffusion process, Asset Management

Complexity vs Empirical Score

• Math Complexity: 9.2/10
• Empirical Rigor: 2.5/10
• Quadrant: Lab Rats
• Why: The paper is highly mathematically advanced, featuring complex mean field game theory, reflected diffusion processes, dual transforms, and PDE analysis to characterize the equilibrium. However, it is purely theoretical with no backtests, datasets, or implementation details provided for the proposed strategies.

  flowchart TD
    A["Research Goal:<br>Model MFG in Asset<br>Management with Tracking Constraint"] --> B["Formulate N-Agent Model:<br>Strategic Capital Injection<br>to Outperform Benchmark"]
    B --> C["Formulate Constrained MFG:<br>Continuum of Agents<br>with Floor Constraint"]
    C --> D["Transform to Unconstrained MFG:<br>Dual Transform &<br>Reflected State Process"]
    D --> E["Compute Best Response:<br>Analytical Control via<br>Dual Reflected Diffusion"]
    E --> F["Establish MFE:<br>Verify Consistency<br>in Separated Domains"]
    F --> G{"Key Findings/Outcomes"}
    G --> H["Analytical Solution<br>for Optimal Tracking Portfolio"]
    G --> I["Existence of Mean Field<br>Equilibrium via PDE Approach"]
    G --> J["Novel Consistency<br>Verification Method"]