A measure approximation theorem for Wasserstein-robust expected values

TL;DR

This paper studies how to approximate the worst-case expected value of a bounded Lipschitz function under Wasserstein uncertainty, showing convergence of solutions when the underlying sigma-algebra is approximated.

Contribution

It establishes a convergence result for Wasserstein-robust expected value minimization problems under sigma-algebra approximations, providing theoretical foundations for approximation methods.

Findings

Solutions of approximated problems converge to the original problem's solution.

Provides a theoretical basis for approximating Wasserstein-robust expectations.

Applicable to bounded Lipschitz functions on $ extbf{R}^d$.

Abstract

We consider the problem of finding the infimum, over probability measures being in a ball defined by Wasserstein distance, of the expected value of a bounded Lipschitz random variable on $\mathbf{R}^d$. We show that if the $\sigma-$algebra is approximated in by a sequence of $\sigma$-algebras in a certain natural sense, then the solutions of the induced approximated minimization problems converge to that of the initial minimization problem.