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论文类型：期刊论文

发表时间：2019-03-01

发表刊物：MATHEMATICAL PROGRAMMING

收录刊物：SCIE、SSCI、EI

卷号：174

期号：1-2,SI

页面范围：473-498

ISSN号：0025-5610

关键字：DRSR; Kantorovich metric; phi-divergence ball; Kantorovich ball; Quantitative convergence analysis

摘要：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 phi-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.