边际和相关不确定性：界限、最优输运和锐度
Marginal and dependence uncertainty: bounds, optimal transport, and sharpness
摘要：
出于在无模型金融和定量风险管理的应用，我们考虑Fréchet类的多元分布函数的额外信息的联合分布是假定的，而在边际的不确定性也是可能的。我们推导出这些Fréchet类的最优输运对偶结果，推广了相关文献中的结果。这些证明是基于增凸泛函的表示结果和共轭的显式计算。我们表明，双重运输问题承认一个明确的解决方案的功能$f=1_B$，其中$B$是一个矩形子集的$\mathbb R^d$，并提供了一个直观的几何解释这一结果。改进的Fréchet-Hoeffding界为这些Fréchet类提供了ad-hoc上界。我们表明，改进的Fréchet-Hoeffding界是逐点尖锐的这些类中存在的不确定性的边缘，而一个反例产量，他们不是逐点尖锐的边缘中没有不确定性，即使在2维。后一个结果为改进的Fréchet-Hoeffding界提供了新的线索，因为Tankov [30]已经证明，在一定条件下，这些界在2维上是尖锐的。
英文摘要：
Motivated by applications in model-free finance and quantitative risk management, we consider Fréchet classes of multivariate distribution functions where additional information on the joint distribution is assumed, while uncertainty in the marginals is also possible. We derive optimal transport duality results for these Fréchet classes that extend previous results in the related literature. These proofs are based on representation results for increasing convex functionals and the explicit computation of the conjugates. We show that the dual transport problem admits an explicit solution for the function $f=1_B$, where $B$ is a rectangular subset of $\mathbb R^d$, and provide an intuitive geometric interpretation of this result. The improved Fréchet--Hoeffding bounds provide ad-hoc upper bounds for these Fréchet classes. We show that the improved Fréchet--Hoeffding bounds are pointwise sharp for these classes in the presence of uncertainty in the marginals, while a counterexample yields that they are not pointwise sharp in the absence of uncertainty in the marginals, even in dimension 2. The latter result sheds new light on the improved Fréchet--Hoeffding bounds, since Tankov [30] has showed that, under certain conditions, these bounds are sharp in dimension 2.
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