On the comonotone natural extension of coherent lower probabilities

Random variables coupled by the dependence structure of comonotonicity posses the property of increasing or decreasing simultaneously. Moreover, comonotone random variables share a number of interesting properties. One of them is that given two marginal probability distributions, there exists a joint comonotone model with the given marginals, such a joint model can be easily constructed, and it unique. This paper explores comonotonicity within the context of uncertainty modelled using imprecise probability models. Specifically, we assume that the uncertainty about the marginal models is described in terms of coherent lower probabilities and we seek a comonotone lower probability with the given marginals, referred to as the comonotone extension. In this setting, we analyse whether a comonotone extension can be built (existence), how to build it (construction) and if it unique (uniqueness). The potential lack of uniqueness leads us to examine the same questions in relation to the most conservative comonotone extension, known as the comonotone natural extension.