Paper: arXiv 2401.15659

Authors: Zongxia Liang, Keyu Zhang

Abstract

This paper studies an optimal investment-consumption problem for competitive agents with exponential or power utilities and a common finite time horizon. Each agent regards the average of habit formation and wealth from all peers as benchmarks to evaluate the performance of her decision. We formulate the n-agent game problems and the corresponding mean field game problems under the two utilities. One mean field equilibrium is derived in a closed form in each problem. In each problem with n agents, an approximate Nash equilibrium is then constructed using the obtained mean field equilibrium when n is sufficiently large. The explicit convergence order in each problem can also be obtained. In addition, we provide some numerical illustrations of our results.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 2.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a sophisticated theoretical framework using Mean Field Games to model relative investment-consumption with habit formation, featuring strong mathematical derivations and closed-form equilibria. However, it is purely theoretical with minimal empirical validation, relying only on basic numerical simulations rather than real-world data.

Research Flowchart

  flowchart TD
    A["Research Goal:<br>Solve Relative Investment-Consumption Games with Habit Formation"] --> B["Inputs:<br>Exponential/Power Utilities, Wealth & Habit Benchmarks, Market Parameters"]
    B --> C["N-Agent Game Formulation"]
    C --> D["Mean Field Game Approximation<br>(N → ∞ Limit)"]
    D --> E["Stochastic Control & PDE Methods<br>(HJB Equations)"]
    E --> F["Closed-Form Mean Field Equilibrium"]
    F --> G["Outcomes:<br>Approx. Nash Equilibrium for N-Agents, Explicit Convergence Rates & Numerical Simulations"]

Cited References (30)

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  4. Backward Stochastic Differential Equations : From Linear to Fully Nonlinear Theory (2017)
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  6. Equilibrium Pricing Under Relative Performance Concerns (2017) — link
  7. OPTIMAL INVESTMENT UNDER RELATIVE PERFORMANCE CONCERNS (2015) — link
  8. Mean Field Portfolio Games (2022) — link
and 22 more
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0 of 30 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.