Modeling and Stabilizing Financial Systemic Risk Using Optimal Control Theory

Paper: arXiv 2511.11909

Authors: Jiacheng Wu

Abstract

A theoretical model of systemic-risk propagation of financial market is analyzed for stability. The state equation is an unsteady diffusion equation with a nonlinear logistic growth term, where the diffusion process captures the spread of default stress between interconnected financial entities and the reaction term captures the local procyclicality of financial stress. The stabilizing controller synthesis includes three steps: First, the algebraic Riccati equation is derived for the linearized system equation, the solution of which provides an exponentially stabilizing controller. Second, the nonlinear system is treated as a linear system with the nonlinear term as its forcing term. Based on estimation of the solutions for linearized equations and the contraction mapping theorem, unique existence of the solution for the nonlinear system equation is proved. Third, local asymptotic stability of the nonlinear system is obtained by considering the corresponding Hamilton-Jacobi equation. In both the linearized and nonlinear systems, the resulting controllers ensure that the $H^{\infty}$ norms of the mappings from disturbance to the output are less than a predefined constant. Stabilizing conditions provide a new framework of achieving system-level financial risk managing goals via the synergy of decentralized components, which offers policy-relevant insights for governments, regulators and central banks to mitigate financial crises.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 1.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a highly theoretical, mathematically dense framework applying partial differential equations and H-infinity optimal control theory to systemic risk stabilization. However, it completely lacks empirical validation, real-world financial data, or backtesting to demonstrate the practical viability of the proposed controller synthesis.

Research Flowchart

  flowchart TD
    A[Goal: Model & Stabilize Financial Systemic Risk] --> B[System Model: Unsteady Diffusion Eq + Nonlinear Logistic Term]
    B --> C[Linearized System & Algebraic Riccati Equation]
    B --> D[Nonlinear System Analysis via Hamilton-Jacobi Eq]
    C --> E[Prove Existence via Contraction Mapping Theorem]
    D --> F[Synthesize H-infinity Controllers]
    E --> F
    F --> G[Outcome: Exponential & Local Asymptotic Stability for Systemic Risk]

Cited References (14)

1. Systemic risk in banking ecosystems (2011)
2. Debtrank: Too central to fail? financial networks, the fed and systemic risk (2012)
3. Systemic risk and stability in financial networks (2015)
4. Contagion in financial networks (2010)
5. Financial contagion (2000)
6. Liquidity and leverage (2010)
7. Market liquidity and funding liquidity (2009)
8. Bank runs, deposit insurance, and liquidity (1983)

0 of 14 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.