For a random variable, superdistribution has emerged as a valuable probability concept. Similar to cumulative distribution function (CDF), it uniquely defines the random variable and can be evaluated with a simple one-dimensional minimization formula. This work leverages the structure of that formula to introduce buffered CDF (bCDF) and reduced CDF (rCDF) for random vectors. bCDF and rCDF are shown to be the minimal Schur-convex upper bound and the maximal Schur-concave lower bound of the multivariate CDF, respectively. Special structure of bCDF and rCDF is used to construct an algorithm for solving optimization problems with bCDF and rCDF in objective or constraints. The efficiency of the algorithm is demonstrated in a case study on optimization of a collateralized debt obligation with bCDF functions in constraints.

对于随机变量，超分布已经成为一个有价值的概率概念。与累积分布函数（CDF）类似，它唯一地定义了随机变量，并且可以用简单的一维最小化公式进行评估。这项工作利用该公式的结构为随机向量引入缓冲 CDF (bCDF) 和简化 CDF (rCDF)。bCDF 和 rCDF 分别显示为多元 CDF 的最小 Schur 凸上界和最大 Schur 凹下界。利用bCDF和rCDF的特殊结构构造了求解目标或约束下的bCDF和rCDF优化问题的算法。该算法的效率在关于在约束条件下使用 bCDF 函数优化债务抵押债券的案例研究中得到了证明。