On the Comonotone Natural Extension of Marginal p-Boxes

The relationship between several random variables is gathered by their joint distribution. While this distribution can be easily determined by the marginals when an assumption of independence is satisfied, there are situations where the random variables are connected by some dependence structure. One such structure that arises often in practice is comonotonicity. This type of dependence refers to random variables that increase or decrease simultaneously. This paper studies the property of comonotonicity when the uncertainty about the random variables is modelled using p-boxes and the induced coherent lower probabilities. In particular, we analyse the problem of finding a comonotone lower probability with given marginal p-boxes, focusing on the existence, construction and uniqueness of such a model. Also, we prove that, under some conditions, there is a most conservative comonotone lower probability with the given marginal p-boxes, that will be called the comonotone natural extension.