Random Fixed Points, Limits and Systemic Risk

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 'neighbour' 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. 1 Note that i Wj,i + W j,b = 1 for all j.

Financial replicator dynamics: emergence of systemic-risk-averting strategies

“…The modeller may not have access to the exact connections of the network, but random graph model is a good approach to analyse such a complex system. In particular we consider the graphs that satisfy the assumptions of [1].…”

“…Each round is composed of two time periods, the agents invest during the initial investment period and they obtain their returns after some given time gap. The two time period model is borrowed from [5,4,1] etc. The new agents make their choice for the next (and the future) round(s), based on their observations of these returns of the previous round.…”

“…Clearing vectors using results of [1]: Our financial framework can be analysed using the results of [1], as the details of the model match 4 the assumptions of the paper. By [1, Theorem 1], the aggregate claims converge almost surely to constant values (as the network size increases to infinity):…”

“…In such a system the agents not only face random economic shocks (received via significantly smaller returns of their risky investments), they are also affected by the percolation of the shocks faced by their neighbours (creditors), neighbours of their neighbours etc. In the recent years from 2007 − 2008 onwards, there is a surge of activity to study the financial and systemic level risks caused by such a percolation of shocks ( [6,5,4,1]). Systemic risk is the study of the risks related to financial networks, when individual or entity level shocks can trigger severe instability at system level that can collapse the entire economy (e.g., [6,5,4]).…”

“…The analysis of these complex networks (in each round) necessitated the study of random fixed point equations (defined sample path-wise in large dimensional spaces), which represent the clearing vectors of all the agents ([1,6,5,4] etc). The study is made possible because of the recent result in [1], which provided an asymptotically accurate one dimensional equivalent solution.…”