阈值风险的隐藏方程
Hidden Equations of Threshold Risk
摘要：
我们考虑阈值风险的灵敏度问题，定义为一个随机变量的函数落在一个指定的阈值水平$δ>0.$以下的概率。我们证明了对于该随机变量的多项式和有理函数，最多存在100个风险临界点。后者是当$δ$接近这些门限值时风险变化率是无界的门限参数的那些特殊值。我们的特点是候选人的风险临界点为零的相关$δ-$扰动多项式的预解式，或其首项系数，或两者兼而有之。因此，需要求解的方程本身是$δ$中的多项式方程，其利用了基础多项式或有理函数的代数性质。我们将这些重要的方程称为“阈值风险隐藏方程”。
英文摘要：
We consider the problem of sensitivity of threshold risk, defined as the probability of a function of a random variable falling below a specified threshold level $δ>0.$ We demonstrate that for polynomial and rational functions of that random variable there exist at most finitely many risk critical points. The latter are those special values of the threshold parameter for which rate of change of risk is unbounded as $δ$ approaches these threshold values. We characterize candidates for risk critical points as zeroes of either the resolvent of a relevant $δ-$perturbed polynomial, or of its leading coefficient, or both. Thus the equations that need to be solved are themselves polynomial equations in $δ$ that exploit the algebraic properties of the underlying polynomial or rational functions. We name these important equations as "hidden equations of threshold risk".
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