Paper: arXiv 2512.04704

Authors: Toshiaki Yamanaka

Abstract

We develop a robust linear-quadratic mean-field control framework for systemic risk under model uncertainty, in which a central bank jointly optimizes interest rate policy and supervisory monitoring intensity against adversarial distortions. Our model features multiple policy instruments with interactive dynamics, implemented via a variance weight that depends on the policy rate, generating coupling effects absent in single-instrument models. We establish viscosity solutions for the associated HJB–Isaacs equation, prove uniqueness via comparison principles, and provide verification theorems. The linear-quadratic structure yields explicit feedback controls derived from a coupled Riccati system, preserving analytical tractability despite adversarial uncertainty. Simulations reveal distinct loss-of-control regimes driven by robustness-breakdown and control saturation, alongside a pronounced asymmetry in sensitivity between the mean and variance channels. These findings demonstrate the importance of instrument complementarity in systemic risk modeling and control.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 3.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a rigorous theoretical formulation combining mean-field control, Hamilton-Jacobi-Bellman-Isaacs (HJBI) viscosity solutions, and robust control for systemic risk. While mathematically dense and conceptually novel in its multi-instrument setup, it relies purely on numerical simulations rather than real-world empirical data validation.

Research Flowchart

  flowchart TD
    A[Research Goal: Robust Systemic Risk Control<br>with Multiple Policy Instruments & Model Uncertainty] --> B[Model Setup & Inputs:<br>Linear-Quadratic Mean-Field Game,<br>Policy Rate & Monitoring Intensity Inputs]
    B --> C[Theoretical Analysis:<br>Formulate HJB-Isaacs Equation &<br>Prove Viscosity Solution/Uniqueness]
    C --> D[Computational Derivation:<br>Solve Coupled Riccati System for<br>Explicit Feedback Controls]
    D --> E[Simulations & Stress Testing:<br>Evaluate Adversarial Distortions &<br>Instrument Coupling Dynamics]
    E --> F[Key Outcomes:<br>Explicit Policy Rules, Loss-of-Control Regimes,<br>& Instrument Complementarity Insights]