The improved Fr\'echet--Hoeffding bounds are ad-hoc estimates on the Fr\'echet class of probability distributions with given marginals and prescribed values on a subset of R d . Using an optimal transport approach, we first provide an alternative derivation of the improved upper bound. To this end, we establish a dual representation of a constrained transport problem over distributions in the aforementioned Fr\'echet class and show that the improved upper Fr\'echet--Hoeffding bound belongs to the class of admissible functions for the dual optimization problem. The proof of strong duality is based on a general representation result for increasing convex functionals and the explicit computation of the conjugates. We show further that the improved upper Fr\'echet--Hoeffding bound is not the dual optimizer of this transport problem . In turn we prove that the improved upper bound is the dual optimizer of a relaxed version of the initial transport problem, thus proving sharpness of the improved upper Fr\'echet--Hoeffding bound for the relaxed Fr\'echet class. This is achieved by direct construction of the dual optimizers for a certain class of objective functions. ] 
Motivated by applications in model-free finance and quantitative risk management, we consider Fr\'echet 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 Fr\'echet 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 Fr\'echet--Hoeffding bounds provide ad-hoc upper bounds for these Fr\'echet classes. We show that the improved Fr\'echet--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 Fr\'echet--Hoeffding bounds, since Tankov 30] has showed that, under certain conditions, these bounds are sharp in dimension 2.
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The improved Fr\'echet--Hoeffding bounds are ad-hoc estimates on the
Motivated by applications in model-free finance and quantitative risk management, we consider
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class of probability distributions with given marginals and prescribed values on a subset of R d . Using an optimal transport approach, we first provide an alternative derivation of the improved upper bound. To this end, we establish a dual representation of a constrained transport problem over distributions in the aforementioned
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
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class and show that the improved upper Fr\'echet--Hoeffding bound belongs to the class of admissible functions for the dual optimization problem. The proof of strong duality is based on a general representation result
classes that extend previous results in the related literature. These proofs are based on representation results
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further that the improved upper
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
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bound is not the dual optimizer of this transport problem . In turn we prove that the improved upper bound is the dual optimizer of a relaxed version of the initial transport problem, thus proving sharpness of the improved upper Fr\'echet--Hoeffding bound for the relaxed
bounds provide ad-hoc upper bounds for these
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class. This is achieved by direct construction of the dual optimizers for a certain class of objective functions.
classes. We show that the improved Fr\'echet--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 Fr\'echet--Hoeffding bounds, since Tankov
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30
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has showed that, under certain conditions, these bounds are sharp in dimension 2.