Optimal Loss Allocation and Reflected Free Boundaries in Systemic Risk

Abstract

We study a singular mean-field control problem in which a central planner allocates losses generated by bank defaults to maximize terminal system health. Every generated loss must be borne by the surviving banks, without external capital. We prove that a cutoff, or "taxing-the-richest" rule is optimal for every non-decreasing concave terminal payoff, including the survival indicator. We construct the continuous-time optimizer by solving a time-discretized problem and passing to the limit, and establish its uniqueness in law for every non-constant payoff. The optimal flow is characterized by a novel reflected free-boundary problem. We establish well-posedness in a probabilistic solution class and prove positive-time smoothness of both the free boundary and the density of surviving banks. The main difficulty is the dependence of the limiting feedback on a moving support endpoint, which is unstable under Wasserstein convergence. We address this through stochastic-order comparisons, compactness estimates for the cutoff boundaries, and an augmented call-function formulation.