Systemic Risk Illustrated

We study the behavior of diffusions coupled through their drifts in a way that each component mean-reverts to the mean of the ensemble. In particular, we are interested in the number of components reaching a “default” level in a given time. This coupling creates stability of the system in the sense that there is a large probability of “nearly no default” as opposed to the case of independent Brownian motions for which the distribution of number of defaults is of binomial type. However, we show that this “swarming” behavior also creates a small probability that a large number of components default corresponding to a “systemic risk event”. The goal of this work is to illustrate systemic risk with a toy model of lending and borrowing banks, using mean-field limit and large deviation estimates for a simple linear model.