Marginal and Dependence Uncertainty: Bounds, Optimal Transport, and Sharpness

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 convex increasing 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 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 [ J. Appl. Probab., 48 (2011), pp. 389--403] has showed that, under certain conditions, these bounds are sharp in dimension 2.