# Marginal and dependence uncertainty: bounds, optimal transport, and sharpness

arXiv: https://arxiv.org/abs/1709.00641 · HTML: https://mlsanity.com/paper/1709.00641/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Sep 2017.

## Built on by (most cited first, 4 of 4)

- [Weak Supervision Performance Evaluation via Partial Identification](https://mlsanity.com/paper/2312.04601/) (2023)
- [Moments of Causal Effects](https://mlsanity.com/paper/2505.04971/) (2025)
- [Fitted value iteration methods for bicausal optimal transport](https://mlsanity.com/paper/2306.12658/) (2023)
- [Improved model-free bounds for multi-asset options using option-implied information and deep learning](https://mlsanity.com/paper/2404.02343/) (2024)

## Abstract (arXiv)

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=1B, where B is a rectangular subset of mathbb Rd, 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.