摘要翻译：
本文比较了两个不同的框架，最近在文献中介绍的风险度量在一个多周期设置。第一种方法对应于对累积的未来成本应用单一的一致风险度量，而第二种方法涉及应用一步一致风险映射的组合。我们总结了这两种方法的相对优势，刻画了其中一种度量总是支配另一种度量的几个充要条件，并引入了一个度量来量化两种风险度量的接近程度。利用这一概念，我们解决了一个给定的相干测度可以用上下界合成测度逼近到多紧的问题。我们在这两种情况之间展示了一个有趣的不对称性：最紧可能的上界可以精确地表征，并对应于文献中的一个流行结构，而最紧可能的下界却不是现成的。我们表明，即使风险度量是共单调和律不变的，检验控制和计算逼近因子通常是NP困难的。然而，我们刻画了条件，并讨论了多项式时间算法可能的几个例子。其中一个例子是著名的条件风险价值度量，在我们的同伴论文[Huang,Iancu,Petrik和Subramanian，“静态和动态条件风险价值”(2012)]中对此进行了进一步的探讨。我们的理论和算法构造开发了风险度量的研究与子模块化和组合优化理论之间有趣的联系，这可能是独立的兴趣。
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英文标题：
《Tight Approximations of Dynamic Risk Measures》
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作者：
Dan A. Iancu, Marek Petrik, Dharmashankar Subramanian
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最新提交年份：
2013
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分类信息：

一级分类：Quantitative Finance        数量金融学
二级分类：Risk Management        风险管理
分类描述：Measurement and management of financial risks in trading, banking, insurance, corporate and other applications
衡量和管理贸易、银行、保险、企业和其他应用中的金融风险
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一级分类：Mathematics        数学
二级分类：Optimization and Control        优化与控制
分类描述：Operations research, linear programming, control theory, systems theory, optimal control, game theory
运筹学，线性规划，控制论，系统论，最优控制，博弈论
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英文摘要：
  This paper compares two different frameworks recently introduced in the literature for measuring risk in a multi-period setting. The first corresponds to applying a single coherent risk measure to the cumulative future costs, while the second involves applying a composition of one-step coherent risk mappings. We summarize the relative strengths of the two methods, characterize several necessary and sufficient conditions under which one of the measurements always dominates the other, and introduce a metric to quantify how close the two risk measures are.   Using this notion, we address the question of how tightly a given coherent measure can be approximated by lower or upper-bounding compositional measures. We exhibit an interesting asymmetry between the two cases: the tightest possible upper-bound can be exactly characterized, and corresponds to a popular construction in the literature, while the tightest-possible lower bound is not readily available. We show that testing domination and computing the approximation factors is generally NP-hard, even when the risk measures in question are comonotonic and law-invariant. However, we characterize conditions and discuss several examples where polynomial-time algorithms are possible. One such case is the well-known Conditional Value-at-Risk measure, which is further explored in our companion paper [Huang, Iancu, Petrik and Subramanian, "Static and Dynamic Conditional Value at Risk" (2012)]. Our theoretical and algorithmic constructions exploit interesting connections between the study of risk measures and the theory of submodularity and combinatorial optimization, which may be of independent interest.
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PDF链接：
https://arxiv.org/pdf/1106.6102