Tight Bounds for a Class of Data-Driven Distributionally Robust Risk Measures

TL;DR

This paper develops tight bounds for distributionally robust risk measures using Wasserstein ambiguity sets, extending classical inequalities and analyzing their impact on data-driven decision-making in inventory and finance.

Contribution

It introduces a framework for deriving tight bounds on risk measures under Wasserstein distributional ambiguity, generalizing classical inequalities and providing computational methods.

Findings

Derived finite-dimensional dual problems for robust moment bounds

Analyzed the effect of distributional ambiguity on risk measures

Conducted computational experiments in inventory and portfolio management

Abstract

This paper expands the notion of robust moment problems to incorporate distributional ambiguity using Wasserstein distance as the ambiguity measure. The classical Chebyshev-Cantelli (zeroth partial moment) inequalities, Scarf and Lo (first partial moment) bounds, and semideviation (second partial moment) in one dimension are investigated. The infinite dimensional primal problems are formulated and the simpler finite dimensional dual problems are derived. A principal motivating question is how does data-driven distributional ambiguity affect the moment bounds. Towards answering this question, some theory is developed and computational experiments are conducted for specific problem instances in inventory control and portfolio management. Finally some open questions and suggestions for future research are discussed.