AI 中文总结
该研究针对系统性风险控制问题，提出将损失集中于最健康银行的最优分配规则，证明离散时间规则可收敛为连续时间奇异平均场控制问题，为有限系统的最优损失分配提供了基础。
AI 中文摘要
我们研究一个系统性风险控制问题，其中中央计划者将银行违约产生的损失分配给存续的机构。银行通过其违约距离建模，该距离作为带向下跳跃的吸收布朗运动演化，向下跳跃由重新分配的违约损失引发。与救助模型不同，计划者无法注入外部资本或减少总损失，唯一允许的干预是决定如何将每个内生损失分配给有偿付能力的银行。目标是最大化终端系统健康，包括以生存质量为主要特例，更一般地，最大化终端分布的递增凹福利泛函。我们的主要结果确定了一个具有简单经济解释的最优分配规则：损失应集中在当前最健康的机构。在离散时间中，该规则表现为截断或对最富者征税的政策，即高于内生阈值的银行被削减至该阈值，而较弱的银行不受影响。我们证明了当分配时间步长趋于零时，离散时间平均场控制问题收敛，并将极限问题表征为奇异平均场控制问题。最优受控律由反射自由边界公式描述，其中截断成为支撑的移动上边界，相关值函数满足Wasserstein空间上的Hamilton-Jacobi方程。最后，我们构建了相应的有限粒子控制问题，并证明在适当假设下，受截断控制的粒子系统收敛到连续时间平均场模型，这为最优平均场损失分配规则提供了有限系统基础。
英文摘要
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.
