A robust distorted Orlicz premium: modeling, computational scheme and applications

In 2018, Bellini and coauthors proposed a robust version of the Orlicz premium for when there is an ambiguity in the subjective probability distribution of the underlying uncertainty or a multiplicity of Young functions. Inspired by their work, we take a step further and consider the case where the probability distribution is distorted. Distortion is widely used in behavioral economics and actuarial science and reflects the fact that decision makers often underweight medium and high probabilities of extreme outcomes but overweight low probabilities of such outcomes. We introduce the novel notion of a distorted Orlicz premium (DOP) and a robust distorted Orlicz premium (RDOP) when the information on distortion is incomplete. We demonstrate how Knightian ambiguity and distortion ambiguity can be synthesized in a single robust model and investigate the properties of DOP and RDOP. Moreover, we discuss how an ambiguity set of distortion functions may be constructed, and we propose tractable computational schemes for computing the DOP and RDOP. In the case when the Young function is piecewise linear, we demonstrate that calculating RDOP comes down to solving a linear program. The distortion and robust arguments are extended to the Haezendonck–Goovaerts risk measure with an application in portfolio optimization. Some numerical test results are reported.