Comonotone lower probabilities with robust marginal distributions functions

Abstract

AbstractOne of the usual dependence structures between random variables is comononicity, which refers to random variables that increase or decrease simultaneously. Besides the good mathematical properties, comonotonicity has been applied in choice theory under risk or in finance, among many other fields. The problem arises when the marginal distribution functions are only partially known, hence we only know bounds of their values. This can be mathematically modelled using p-boxes, allowing us to build a bridge with the theory of imprecise probabilities. This paper investigates the existence, construction and uniqueness of a joint (imprecise) comonotone model with the given marginal p-boxes. In particular, given that the joint comonotone model is not unique when it exists, we follow the philosophy of the imprecise probability theory and we characterise under which conditions there exists a least-committal comonotone model, called the comonotone natural extension.