Almost Worst Case Distributions in Multiple Priors Models

TL;DR

This paper investigates the properties and localization of almost worst case distributions in multiple priors models, providing insights into their structure and how to compute them for robust risk management.

Contribution

It introduces the concept of almost worst case distributions, proves their densities cluster near a worst case localiser, and discusses their computation and practical relevance.

Findings

Densities of almost worst case distributions cluster in Bregman neighborhoods.

The worst case localiser may not be a density when the worst case density does not exist.

The localization depends on the plausibility constraint threshold.

Abstract

A worst case distribution is a minimiser of the expectation of some random payoff within a family of plausible risk factor distributions. The plausibility of a risk factor distribution is quantified by a convex integral functional. This includes the special cases of relative entropy, Bregman distance, and $f$-divergence. An ($\epsilon$-$\gamma$)-almost worst case distribution is a risk factor distribution which violates the plausibility constraint at most by the amount $\gamma$ and for which the expected payoff is not better than the worst case by more than $\epsilon$. From a practical point of view the localisation of almost worst case distributions may be useful for efficient hedging against them. We prove that the densities of almost worst case distributions cluster in the Bregman neighbourhood of a specified function, interpreted as worst case localiser. In regular cases, it coincides with the worst case density, but when the latter does not exist, the worst case localiser is perhaps not even a density. We also discuss the calculation of the worst case localiser, and its dependence on the threshold in the plausibility constraint.