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

Yuchen Li, Zongxia Liang, Xiang Yu

TL;DR

The paper studies a relative-performance portfolio problem in a two-population mean-field game with Poisson jump risk, deriving the mean-field equilibrium (MFE) and the Nash equilibria for the finite $N_1+N_2$-player game. It adopts a fixed-point, verification-based approach analogous to prior work, solving an auxiliary deterministic problem to obtain best-response controls and then enforcing consistency to characterize the MFE. It proves that the Nash equilibria converge to the MFE as $N_1,N_2\to\infty$ and provides numerical illustrations of how Poisson idiosyncratic and common noise shape equilibrium strategies. The results illuminate nonlinear interactions between jump risk and relative performance and offer quantitative insights into portfolio behavior under large-population competition.

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.

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

TL;DR

The paper studies a relative-performance portfolio problem in a two-population mean-field game with Poisson jump risk, deriving the mean-field equilibrium (MFE) and the Nash equilibria for the finite -player game. It adopts a fixed-point, verification-based approach analogous to prior work, solving an auxiliary deterministic problem to obtain best-response controls and then enforcing consistency to characterize the MFE. It proves that the Nash equilibria converge to the MFE as and provides numerical illustrations of how Poisson idiosyncratic and common noise shape equilibrium strategies. The results illuminate nonlinear interactions between jump risk and relative performance and offer quantitative insights into portfolio behavior under large-population competition.

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 -player game. Furthermore, we prove the convergence of the Nash equilibrium in the -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.

Paper Structure

This paper contains 12 sections, 14 theorems, 105 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Finding equilibrium
  4. Solution of the auxiliary Problem \ref{['prob: deter invest prob']}
  5. Solution of the MFG
  6. Finite Population Game with $N_1+N_2$ Players
  7. Solution of Problem \ref{['prob: invest N']}
  8. Finding Nash equilibrium
  9. Relation between MFG and the $N_1+N_2$-Player Game
  10. Discussions on MFE and the Impact of Poisson jump risk
  11. MFG and N player game with one population
  12. Numerical examples

Key Result

Theorem 1

Suppose that there exist a function $V\in C^{2,1}$R[0,T)$\cap C$R[0,T]$$ and a control $\pi^*\in \mathcal{A}_{0}$ such that Then and $\pi^*$ is the optimal control and $V(x,t)$ is the value function of Problem prob MFE determin Z.

Figures (6)

  • Figure 1: Plots of $\pi^*$.
  • Figure 2: Plots of $\pi^*$ with positive $\gamma$.
  • Figure 3: Convergence of $(x,y)$ and the equilibrium strategy $\pi^*$.
  • Figure 4: Sensitivity with respect to $\gamma$
  • Figure 5: Sensitivity with respect to $\sigma^0$
  • ...and 1 more figures

Theorems & Definitions (25)

  • Definition 1
  • Definition 2
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Proposition 1
  • Proposition 2
  • Theorem 3
  • proof
  • ...and 15 more