Optimal Loss Allocation in a Mean-Field Model of Systemic Risk

We study a systemic-risk control problem in which a central planner allocates losses generated by bank defaults across the surviving institutions. Banks are modeled through their distances to default, evolving as absorbed Brownian motions with downward jumps induced by redistributed default losses. Unlike bailout models, the planner cannot inject external capital or reduce the aggregate loss, and the only admissible intervention is to decide how each endogenous loss is assigned among solvent banks. The objective is to maximize terminal system health, including survival mass as a leading special case and, more generally, increasing concave welfare functionals of the terminal distribution. Our main result identifies an optimal allocation rule with a simple economic interpretation: losses should be concentrated on the currently healthiest institutions. In discrete time, this rule takes the form of a cutoff or taxing-the-richest policy, which reduces banks above an endogenous threshold down to that threshold while leaving weaker banks untouched. We prove convergence of the time-discretized mean-field control problem as the allocation time step tends to zero and characterize the limiting problem as a singular mean-field control problem. The optimally controlled law is described by a reflected free-boundary formulation, in which the cutoff becomes the moving upper edge of the support, and the associated value function satisfies a Hamilton-Jacobi equation on Wasserstein space. Finally, we formulate the corresponding finite-particle control problem and show, under suitable assumptions, that the cutoff-controlled particle system converges to the continuous-time mean-field model. This provides a finite-system foundation for the optimal mean-field loss-allocation rule.