In our previous paper “A unified approach to systemic risk measures via acceptance sets” (Mathematical Finance, 2018), we have introduced a general class of systemic risk measures that allow random allocations to individual banks before aggregation of their risks. In the present paper, we prove a dual representation of a particular subclass of such systemic risk measures and the existence and uniqueness of the optimal allocation related to them. We also introduce an associated utility maximisation problem which has the same solution as the minimisation problem associated to the systemic risk measure. In addition, the optimiser in the dual formulation provides a risk allocation which is fair from the point of view of the individual financial institutions. The case with exponential utilities which allows explicit computation is treated in detail.

中文翻译：

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论系统性风险措施的公平性

在我们之前的论文“通过接受集对系统风险度量采取统一方法”（数学金融，2018年）中，我们引入了一般的系统风险度量类别，该方法允许在个别银行汇总其风险之前对其进行随机分配。在本文中，我们证明了此类系统风险度量的特定子类的双重表示以及与它们相关的最优分配的存在和唯一性。我们还介绍了一个相关的效用最大化问题，该问题的解决方案与与系统性风险度量相关的最小化问题具有相同的解决方案。此外，双重配方中的优化器可提供风险分配从单个金融机构的角度来看，这是公平的。对于允许显式计算的指数实用程序，将进行详细处理。