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Unlike classic risk sharing problems based on expected utilities or convex risk measures, quantile-based risk sharing problems exhibit two special features. First, quantile-based risk measures (such as the Value-at-Risk) are often not convex, and second, they ignore some part of the distribution of the risk. These features create technical challenges in establishing a full characterization of optimal allocations, a question left unanswered in the literature. In this paper, we address the issues on the existence and the characterization of (Pareto-)optimal allocations in risk sharing problems for the Range-Value-at-Risk family. It turns out that negative dependence, mutual exclusivity in particular, plays an important role in the optimal allocations, in contrast to positive dependence appearing in classic risk sharing problems. As a by-product of our main finding, we obtain some results on the optimization of the Value-at-Risk (VaR) and the Expected Shortfall, as well as a new result on the inf-convolution of VaR and a general distortion risk measure. 相似文献
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It is known that there are feasible algorithms for minimizing convex functions, and that for general functions, global minimization
is a difficult (NP-hard) problem. It is reasonable to ask whether there exists a class of functions that is larger than the
class of all convex functions for which we can still solve the corresponding minimization problems feasibly. In this paper,
we prove, in essence, that no such more general class exists. In other words, we prove that global optimization is always
feasible only for convex objective functions. 相似文献
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