SIMPOL Model for Solving Continuous-Time Heterogeneous Agent Problems

ArXiv ID: 2509.23557 “View on arXiv”

Authors: Ricardo Alonzo Fernández Salguero

Abstract

This paper presents SIMPOL (Simplified Policy Iteration), a modular numerical framework for solving continuous-time heterogeneous agent models. The core economic problem, the optimization of consumption and savings under idiosyncratic uncertainty, is formulated as a coupled system of partial differential equations: a Hamilton-Jacobi-Bellman (HJB) equation for the agent’s optimal policy and a Fokker-Planck-Kolmogorov (FPK) equation for the stationary wealth distribution. SIMPOL addresses this system using Howard’s policy iteration with an upwind finite difference scheme that guarantees stability. A distinctive contribution is a novel consumption policy post-processing module that imposes regularity through smoothing and a projection onto an economically plausible slope band, improving convergence and model behavior. The robustness and accuracy of SIMPOL are validated through a set of integrated diagnostics, including verification of contraction in the Wasserstein-2 metric and comparison with the analytical solution of the Merton model in the no-volatility case. The framework is shown to be not only computationally efficient but also to produce solutions consistent with economic and mathematical theory, offering a reliable tool for research in quantitative macroeconomics.

Keywords: heterogeneous agent models, Hamilton-Jacobi-Bellman (HJB) equation, Fokker-Planck-Kolmogorov (FPK) equation, policy iteration, finite difference scheme, Macroeconomics

Complexity vs Empirical Score

• Math Complexity: 9.2/10
• Empirical Rigor: 3.5/10
• Quadrant: Lab Rats
• Why: The paper presents a highly advanced mathematical framework involving continuous-time Hamilton-Jacobi-Bellman and Fokker-Planck PDEs, viscosity solutions, and the Wasserstein-2 metric, indicating very high mathematical density. However, it focuses on numerical framework design and theoretical validation without providing backtests, specific financial datasets, or implementation-heavy empirical analysis for trading strategies.

  flowchart TD
    Goal["Research Goal<br>Solve Continuous-Time<br>Heterogeneous Agent Models"] --> Inputs["Data/Inputs<br>Model Parameters<br>Grid Specifications"]
    Inputs --> Methodology["Methodology<br>SIMPOL Framework<br>Howard's Policy Iteration"]
    Methodology --> Solver["Computational Process<br>Upwind Finite Difference Scheme<br>Coupled HJB & FPK Solver"]
    Solver --> PostProc["Computational Process<br>Consumption Policy<br>Post-Processing (Smoothing & Projection)"]
    PostProc --> Outcomes["Key Findings/Outcomes<br>Stable Numerical Solution<br>Convergence in Wasserstein-2 Metric<br>Validated against Merton Model"]