Characterization of Invariance, Periodic Solutions and Optimization of Dynamic Financial Networks

TL;DR

This paper analyzes the dynamic behavior of financial networks, exploring invariance, periodic solutions, and optimal interventions to enhance systemic resilience beyond static models.

Contribution

It provides a systematic analysis of the system's invariance regions, demonstrates the existence of complex periodic solutions, and formulates an optimization problem for minimal cash injections.

Findings

Identified invariant regions of the financial system's dynamics.

Discovered the possibility of periodic solutions with periods greater than 2.

Formulated an optimization approach for minimal cash injections to stabilize the system.

Abstract

Cascading failures, such as bankruptcies and defaults, pose a serious threat for the resilience of the global financial system. Indeed, because of the complex investment and cross-holding relations within the system, failures can occur as a result of the propagation of a financial collapse from one organization to another. While this problem has been studied in depth from a static angle, namely, when the system is at an equilibrium, we take a different perspective and study the corresponding dynamical system. The contribution of this paper is threefold. First, we carry out a systematic analysis of the regions of attraction and invariance of the system orthants, defined by the positive and negative values of the organizations' equity. Second, we investigate periodic solutions and show through a counterexample that there could exist periodic solutions of period greater than 2. Finally, we study the problem of finding the smallest cash injection that would bring the system to the maximal invariant region of the positive orthant.