Marginal and Dependence Uncertainty

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

Bartl, Daniel (University of Vienna)
Kupper, Michael (Universität Konstanz)
Lux, Thibaut (Helvetia Insurance Group)
Papapantoleon, A. (TU Delft Applied Probability)

2022

Motivated by applications in model-free finance and quantitative risk management, we consider Frechet 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 Frechet 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 = 1B, where B is a rectangular subset of Rd, and provide an intuitive geometric interpretation of this result. The improved Frechet-Hoeffding bounds provide ad hoc bounds for these Frechet classes. We show that the improved Frechet-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 Frechet-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.

dependence uncertainty
Frechet classes
improved Frechet-Hoeffding bounds
marginal uncertainty
optimal transport duality
relaxed duality
sharpness of bounds

http://resolver.tudelft.nl/uuid:81625d08-2b0b-4b2e-8d59-f707fa16430a

https://doi.org/10.1137/21M144709X

0363-0129

SIAM Journal on Control and Optimization, 60 (1), 410-434

Institutional Repository

journal article

© 2022 Daniel Bartl, Michael Kupper, Thibaut Lux, A. Papapantoleon