Stop-loss and limited loss random variables are two important transforms of a loss random variable and appear in many modeling problems in insurance, finance, and other fields. Risk levels of a loss variable and its transforms are often measured by risk measures. When only partial information on a loss variable is available, risk measures of the loss variable and its transforms cannot be evaluated effectively. To deal with the situation of distribution uncertainty, the worst-case values of risk measures of a loss variable over an uncertainty set, describing all the possible distributions of the loss variable, have been extensively used in robust risk management for many fields. However, most of these existing results on the worst-case values of risk measures of a loss variable cannot be applied directly to the worst-case values of risk measures of its transforms. In this paper, we derive the expressions of the worst-case values of distortion risk measures of stop-loss and limited loss random variables over an uncertainty set introduced in Bernard et al. (2023). This set represents a decision maker’s belief in the distribution of a loss variable. We find the distributions under which the worst-case values are attainable. These results have potential applications in a variety of fields. To illustrate their applications, we discuss how to model optimal stop-loss reinsurance problems and how to determine optimal stop-loss retentions under distribution uncertainty. Explicit and closed-form expressions for the worst-case TVaRs of stop-loss and limited loss random variables and optimal stop-loss retentions are given under special forms of the uncertainty set. Numerical results are presented under more general forms of the uncertainty set.

中文翻译：

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分布不确定性下止损和有限损失随机变量的最坏情况风险度量及其在稳健再保险中的应用

止损随机变量和有限损失随机变量是损失随机变量的两种重要变换，出现在保险、金融等领域的许多建模问题中。损失变量及其变换的风险水平通常通过风险度量来衡量。当仅可获得损失变量的部分信息时，无法有效评估损失变量及其变换的风险度量。为了处理分布不确定性的情况，损失变量在不确定性集合上的风险度量的最坏情况值描述了损失变量的所有可能分布，已广泛应用于许多领域的稳健风险管理。然而，大多数关于损失变量的风险度量的最坏情况值的现有结果不能直接应用于其变换的风险度量的最坏情况值。在本文中，我们推导了 Bernard 等人引入的不确定性集上止损和有限损失随机变量的扭曲风险度量的最坏情况值的表达式。 （2023）。该集合代表决策者对损失变量分布的信念。我们找到了可以达到最坏情况值的分布。这些结果在各个领域都有潜在的应用。为了说明它们的应用，我们讨论如何对最佳止损再保险问题进行建模以及如何确定分布不确定性下的最佳止损保留。在不确定性集的特殊形式下，给出了止损和有限损失随机变量的最坏情况 TVaR 以及最佳止损保留的显式封闭式表达式。数值结果以不确定性集的更一般形式呈现。