Random Fixed Points, Limits and Systemic Risk

We consider random vector valued fixed point (FP) equations in large dimensional spaces, and study their almost sure solutions. An underlying directed random graph 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 aid to approximate the solution of the random FP equations for almost all realizations, in the asymptotic limit (as the number of components become large). 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. In contrast our focus is on almost sure FP solutions. We apply the results to study systemic risk in a stylized large financial network captured by one big institution and many small ones, where our analysis reveal structural insights in a simple manner.