Distributionally robust shortfall risk optimization model and its approximation
Distributionally robust shortfall risk optimization model and its approximation
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DOI：
10.1007/s10107-018-1307-z
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发表时间：
2018-06
影响因子：
2.7
通讯作者：
Shaoyan Guo;Huifu Xu
中科院分区：
文献类型：
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作者：
Shaoyan Guo;Huifu Xu
基于效用的短缺风险度量（SR）在过去几年中受到越来越多的关注，因为它们比条件风险价值更有效地量化了大尾部损失的风险。在本文中，我们考虑一个分布鲁棒版本的短缺风险度量（DRSR）的真实概率分布是未知的，最坏的分布从一个模糊的分布集被用来计算SR。我们开始表明，DRSR是一个凸风险度量，并在某些特殊情况下的相干风险度量。然后，我们继续研究一个优化问题的目标是最大限度地减少随机函数的DRSR和调查的模糊集通过发散球和康托洛维奇球构造的优化问题的数值易处理性。在球的名义分布是通过iid样本构造的经验分布的情况下，我们量化的模糊集的收敛到真实的概率分布作为样本大小的Kantorovich度量下增加，因此相应的DRSR问题的最佳值。具体来说，我们证明了最佳值的误差是线性有界的误差的近似模糊度集，并随后推导出的置信区间的最佳值下的每个近似计划。一些初步的数值试验结果报告所提出的建模和计算方案。
Utility-based shortfall risk measures (SR) have received increasing attention over the past few years for their potential to quantify the risk of large tail losses more effectively than conditional value at risk. In this paper, we consider a distributionally robust version of the shortfall risk measure (DRSR) where the true probability distribution is unknown and the worst distribution from an ambiguity set of distributions is used to calculate the SR. We start by showing that the DRSR is a convex risk measure and under some special circumstance a coherent risk measure. We then move on to study an optimization problem with the objective of minimizing the DRSR of a random function and investigate numerical tractability of the optimization problem with the ambiguity set being constructed through-divergence ball and Kantorovich ball. In the case when the nominal distribution in the balls is an empirical distribution constructed through iid samples, we quantify convergence of the ambiguity sets to the true probability distribution as the sample size increases under the Kantorovich metric and consequently the optimal values of the corresponding DRSR problems. Specifically, we show that the error of the optimal value is linearly bounded by the error of each of the approximate ambiguity sets and subsequently derive a confidence interval of the optimal value under each of the approximation schemes. Some preliminary numerical test results are reported for the proposed modeling and computational schemes.