Optimal Loss Allocation and Reflected Free Boundaries in Systemic Risk

This study focuses on the allocation of bank default losses without external capital injection, proving that for all non-decreasing concave terminal payoffs, the "tax the richest" truncation rule is optimal. The study constructs a continuous-time optimal solution through time discretization, characterizes the optimal flow as a novel reflected free boundary problem, and proves the positive time smoothness of the free boundary and the density of surviving banks.