Tight Approximations of Dynamic Risk Measures
Tight Approximations of Dynamic Risk Measures
复制标题
DOI：
10.1287/moor.2014.0689
复制
发表时间：
2015-08-01
影响因子：
1.7
通讯作者：
Subramanian, Dharmashankar
中科院分区：
文献类型：
--
作者：
Iancu, Dan A.;Petrik, Marek;Subramanian, Dharmashankar
本文比较了两种衡量多时期环境下风险的框架。第一个对应于对累积的未来成本应用单个连贯的风险度量，第二个涉及应用一步连贯风险映射的组合。我们描述了一个测量总是优于另一个测量的几个必要和充分条件，并引入了一个度量来量化两个测量的接近程度。使用这个概念，我们解决了一个给定的相干测度如何紧密地被下限或上限组合测度所近似的问题。我们在这两种情况之间展示了一种有趣的不对称性：最紧上界可以精确地表征，并与文献中流行的结构相对应，而最紧下界则不容易得到。我们表明，即使当风险度量是共频和定律不变时，测试支配和计算近似因子通常也是np困难的。然而，我们描述条件和讨论的例子，其中多项式时间算法是可能的。其中一个例子是众所周知的条件风险值度量，我们将对此进行更详细的探讨。我们的理论和算法构建利用了风险度量研究与子模块化和组合优化理论之间的有趣联系，这可能是独立的兴趣。
This paper compares two frameworks for measuring risk in a multiperiod setting. The first corresponds to applying a single coherent risk measure to the cumulative future costs, and the second involves applying a composition of one-step coherent risk mappings. We characterize several necessary and sufficient conditions under which one measurement always dominates the other and introduce a metric to quantify how close the two 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 upper bound can be exactly characterized and corresponds to a popular construction in the literature, whereas the tightest lower bound is not readily available. We show that testing domination and computing the approximation factors are generally NP-hard, even when the risk measures are comonotonic and law-invariant. However, we characterize conditions and discuss examples where polynomial-time algorithms are possible. One such case is the well-known conditional value-at-risk measure, which we explore in more detail. 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.