不连续模型中基于分位数的微分敏感性
Differential Quantile-Based Sensitivity in Discontinuous Models
AI总结：
本文提出在不连续模型和离散输入下基于分位数风险度量的微分敏感性通用框架，证明导数存在并推导公式，应用于复合风险模型和再保险信用风险分析。
摘要：
微分敏感性度量为解释从模拟到算法预测等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可以揭示输入-输出关系以及模型参数和输入变量的相对重要性。然而，当模型函数存在不连续点和/或输入变量为离散变量时，应如何求取此类导数尚不清楚。我们提出了一个解决此类问题的通用框架，考虑基于分位数的输出风险度量对随机输入变量（风险因子）畸变的导数，这些风险因子通过阶跃函数影响模型输出。我们证明，在较弱的数学条件下，这些导数是有良好定义的，并推导出相应的公式。我们将结果应用于复合风险模型的敏感性分析，以及多业务线保险组合中再保险信用风险的数值研究。
英文摘要：
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.
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