Time Dynamics of Systemic Risk in Banking Networks: A UEDR-PDE Approach

Understanding the time dynamics of systemic risk in banking networks is crucial for preventing financial crises and ensuring economic stability. This paper aims to quantify key transition times in the evolution of distress within a banking system using a mathematical framework. We investigate the dynamics of systemic risk in a hypothetical, homogeneous banking network using the Undistressed–Exposed–Distressed–Recovered (UEDR) model. The UEDR model, inspired by compartmental epidemic frameworks, captures how financial distress propagates and recedes through interactions between banks. It is selected because of its tractability and its ability to distinguish between different stages of bank vulnerability. We focus on two critical times, denoted as t1 and t2, which play a fundamental role in understanding the behavior of the distressed compartment (representing the number of distressed banks) over time. The time t1 represents the first instance of a decrease in the number of distressed banks, indicating the containment of systemic risk. On the other hand, the time t2 marks the onset when the number of undistressed banks falls below a specified threshold, signifying the restoration of financial stability. We examine these time dependencies by considering the initial conditions of the UEDR model and assess their characteristics using partial differential equations. We establish the continuity, smoothness, and uniqueness of solutions for t1 and t2, along with their corresponding boundary conditions. Furthermore, we provide explicit representation formulas for t1 and t2, allowing for precise estimation when the initial population compartments are large. Our results provide practical insights for financial regulators and policymakers in determining time-sensitive interventions for mitigating systemic risk and accelerating recovery in banking systems. The findings highlight how mathematical modeling can inform real-time risk management strategies in financial networks.