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Mean Field Game of Optimal Tracking Portfolio

Lijun Bo, Yijie Huang, Xiang Yu

TL;DR

This work models a large population of fund managers who must beat a benchmark that blends the population's average wealth with a market index, allowing fictitious capital injections to enforce a floor constraint. It develops a mean-field game with a reflected state to handle the hard constraint, and uses a Legendre-Fenchel dual to linearize the HJB and obtain explicit best-response controls in two regions. A fixed-point, duality-based approach then establishes the existence of a mean-field equilibrium and yields a semi-explicit characterization of the equilibrium strategies. The analysis combines PDE methods with stochastic control and duality to provide a rigorous framework for optimal tracking with public-utility-like capital injections in large-scale portfolio competition, along with numerical insights into how parameters shape injections and investment behavior.

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.

Mean Field Game of Optimal Tracking Portfolio

TL;DR

This work models a large population of fund managers who must beat a benchmark that blends the population's average wealth with a market index, allowing fictitious capital injections to enforce a floor constraint. It develops a mean-field game with a reflected state to handle the hard constraint, and uses a Legendre-Fenchel dual to linearize the HJB and obtain explicit best-response controls in two regions. A fixed-point, duality-based approach then establishes the existence of a mean-field equilibrium and yields a semi-explicit characterization of the equilibrium strategies. The analysis combines PDE methods with stochastic control and duality to provide a rigorous framework for optimal tracking with public-utility-like capital injections in large-scale portfolio competition, along with numerical insights into how parameters shape injections and investment behavior.

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.
Paper Structure (6 sections, 12 theorems, 82 equations, 9 figures)

This paper contains 6 sections, 12 theorems, 82 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Finite-Agent and MFG Problem
  3. Equivalent Problem Formulation for Best Response Control
  4. Mean Field Equilibrium
  5. Discussions on MFE
  6. Proofs

Key Result

Lemma 3.2

Let $(t,z)\in[0,T]\times \mathds{R}_+$ be fixed. Then, the value function $x\mapsto u(t,x,z)$ given by eq_mfg-2 is non-decreasing. Moreover, for all $(t,x_1,x_2,z)\in[0,T]\times \mathds{R}_+^3$, we have

Figures (9)

  • Figure 1: The road map of solving the MFG problem.
  • Figure 2: The function $r\mapsto x(r)$.
  • Figure 3: The fixed point function $t\mapsto f^*(t)$.
  • Figure 4: The portfolio feedback function $x\mapsto \theta^{*,f^*}(t,x,z)$. The parameters are set to be $(x_0,z_0)=(2.0308,20)$, $(t,z)=(0.5,20)$.
  • Figure 5: The value function $x\mapsto v(t,x,z)$. The parameters are set to be $(x_0,z_0)=(2.0308,20)$, $(t,z)=(0.5,20)$.
  • ...and 4 more figures

Theorems & Definitions (20)

  • Remark 2.1
  • Definition 2.1: MFE
  • Remark 3.1
  • Lemma 3.2
  • Proposition 3.3
  • Theorem 3.4: Verification Theorem
  • Corollary 3.5
  • Remark 3.6
  • Lemma 4.1
  • Lemma 4.2
  • ...and 10 more