Paper: arXiv 2603.22058

Authors: Masashi Sekine

Abstract

This thesis develops equilibrium asset pricing models in incomplete markets with a large number of heterogeneous agents using mean field game theory. The market equilibrium is characterized by a novel form of mean field backward stochastic differential equations (BSDEs). First, we propose a theoretical model that endogenously derives the equilibrium risk premium. Agents with exponential preferences are heterogeneous in initial wealth, risk aversion, and unspanned stochastic terminal liability. We solve the optimal investment problem using the optimal martingale principle. The equilibrium is characterized by a mean field BSDE whose driver has quadratic growth in both the stochastic integrands and their conditional expectations. We prove the existence of solutions and show that the risk premium clears the market in the large population limit. Second, we extend the model to include consumption and habit formation, relaxing the time-separability assumption of utility functions. A similar mean field BSDE is derived, and its well-posedness and asymptotic behavior are examined. We also introduce an exponential quadratic Gaussian (EQG) reformulation to obtain equilibrium solutions in semi-analytic form. Finally, the model is extended to partially observable markets where agents must infer the risk premium from stock price observations to determine trading strategies. We provide semi-analytic expressions for the equilibrium via the EQG framework, and the equilibrium risk-premium process is constructed endogenously using Kalman-Bucy filtering theory. Numerical simulations are included to visualize the resulting market dynamics.

Complexity vs Empirical Score

  • Math Complexity: 9.2/10
  • Empirical Rigor: 3.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The thesis provides a highly sophisticated mathematical framework utilizing mean field backward stochastic differential equations (BSDEs) and Kalman-Bucy filtering for equilibrium asset pricing. While theoretically rich and highly novel in its analytical modeling, it relies primarily on numerical simulations rather than empirical data validation.

Research Flowchart

  flowchart TD
    A[Research Goal: Model Equilibrium Asset Pricing in Incomplete Markets with Heterogeneous Agents] --> B[Model Inputs: Exponential Preferences, Wealth, Risk Aversion & Terminal Liability]
    B --> C[Methodology: Apply Optimal Martingale Principle & Mean Field Game Theory]
    C --> D[Derive Mean Field BSDEs with Quadratic Growth Drivers]
    D --> E[Extensions: Habit Formation, EQG Reformulation & Kalman-Bucy Filtering for Partial Observability]
    E --> F[Numerical Simulations of Market Dynamics]
    F --> G[Outcomes: Endogenous Equilibrium Risk Premium & Well-Posed Semi-Analytic Solutions]