Random Fixed Points, Limits and Systemic risk

TL;DR

This paper analyzes fixed point equations in large random systems with network dependencies, deriving finite-dimensional approximations that reveal systemic risk phenomena in financial networks.

Contribution

It introduces a novel approach to approximate realization-wise fixed points in large random systems, differing from traditional mean-field methods.

Findings

Finite-dimensional limit equations approximate solutions for large systems.

Application to financial networks reveals systemic risk phenomena.

Method provides realization-wise solutions, not distribution-based.

Abstract

We consider vector fixed point (FP) equations in large dimensional spaces involving random variables, and study their realization-wise solutions. We have an underlying directed random graph, that defines the connections between various components of the FP equations. Existence of an edge between nodes i, j implies the i th FP equation depends on the j th component. We consider a special case where any component of the FP equation depends upon an appropriate aggregate of that of the random neighbor components. We obtain finite dimensional limit FP equations (in a much smaller dimensional space), whose solutions approximate the solution of the random FP equations for almost all realizations, in the asymptotic limit (number of components increase). Our techniques are different from the traditional mean-field methods, which deal with stochastic FP equations in the space of distributions to describe the stationary distributions of the systems. In contrast our focus is on realization-wise FP solutions. We apply the results to study systemic risk in a large financial heterogeneous network with many small institutions and one big institution, and demonstrate some interesting phenomenon.