Faculty of Actuarial Science and Insurance with Peng Liu from University of Essex

Abstract: We establish sharp upper and lower bounds for distortion risk metrics under distributional uncertainty. The uncertainty sets are characterized by four key features of the underlying distribution: mean, variance, unimodality, and Wasserstein distance to a reference distribution.

We first examine a broad class of distortion functions, assuming only finite variation and imposing neither continuity nor monotonicity. This class includes important examples such as the Gini deviation, the mean–median deviation, and inter-quantile differences. When the uncertainty set is defined by a fixed mean, variance, and Wasserstein distance, we derive the worst- and best-case values of the distortion risk metric and identify the corresponding extremal distributions. then impose an additional unimodality constraint. In this case, for absolutely continuous distortion functions, we again characterize the worst- and best-case values and explicitly determine the optimal distributions attaining these bounds.

Finally, we illustrate the practical relevance of our theoretical results in the context of robust portfolio optimization. (This talk is based on a joint work with Steven Vanduffel and Yi Xia).

Short Bio:

Dr. Peng Liu has been a Lecturer at the University of Essex since 2020. He obtained his PhD in Probability and Statistics from Nankai University in 2015. He subsequently held postdoctoral positions at the University of Lausanne and the University of Waterloo. His research primarily focuses on quantitative risk management, actuarial science, financial mathematics, and extreme value theory. His work has been published in leading journals in the field, including Mathematics of Operations Research, Mathematical Finance, Finance and Stochastics, SIAM Journal on Financial Mathematics, European Journal of Operational Research, Stochastic Processes and Their Applications, and Insurance: Mathematics and Economics.