Differential sensitivity measures provide valuable tools for interpreting complex computational models used in applications ranging from simulation to algorithmic prediction. Taking the derivative of the model output in direction of a model parameter can reveal input-output relations and the relative importance of model parameters and input variables. Nonetheless, it is unclear how such derivatives should be taken when the model function has discontinuities and/or input variables are discrete. We present a general framework for addressing such problems, considering derivatives of quantile-based output risk measures, with respect to distortions to random input variables (risk factors), which impact the model output through step-functions. We prove that, subject to weak technical conditions, the derivatives are well-defined and derive the corresponding formulas. We apply our results to the sensitivity analysis of compound risk models and to a numerical study of reinsurance credit risk in a multi-line insurance portfolio.

翻译：微分敏感性测度为解释从仿真模拟到算法预测等各类应用中使用的复杂计算模型提供了有价值的工具。沿模型参数方向对模型输出求导能够揭示输入-输出关系以及模型参数与输入变量的相对重要性。然而，当模型函数存在间断性和/或输入变量为离散变量时，此类导数的定义方式尚不明确。本文提出一个通用框架来解决此类问题，该框架考虑基于分位数的输出风险测度相对于随机输入变量（风险因子）扰动的导数，这些风险因子通过阶跃函数影响模型输出。我们证明在较弱的技术条件下，这些导数是良定义的，并推导出相应的计算公式。我们将研究结果应用于复合风险模型的敏感性分析，并对多险种保险组合中的再保险信用风险进行了数值研究。