Modeling and Stabilizing Financial Systemic Risk Using Optimal Control Theory

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.