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.
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BibTeX
@misc{guo2026optimal,
title = {{Optimal Loss Allocation and Reflected Free Boundaries in Systemic Risk}},
author = {Yucheng Guo and Qinxin Yan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.18113},
eprint = {2608.18113},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.18113},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Optimal Loss Allocation and Reflected Free Boundaries in Systemic Risk
AU - Yucheng Guo
AU - Qinxin Yan
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2608.18113
UR - https://arxiv.org/abs/2608.18113
AB - 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.
ER -