Comonotonicity for sets of probabilities

Two variables are called comonotone when there is an increasing relation between them, in the sense that when one of them increases (decreases), so does the other one. This notion has been widely investigated in probability theory, and is related to copulas. This contribution studies how the notion of comonotonicity can be extended to an imprecise setting on discrete spaces, where probabilities are only known to belong to a convex set. We define comonotonicity for such sets and investigate its characterizations in terms of lower probabilities, as well as its connection with copulas. As this theoretical characterization can be tricky to apply to general lower probabilities, we also investigate specific models of practical importance. In particular, we provide some sufficient conditions for a comonotone belief function with fixed marginals to exist, and characterize comonotone bivariate p-boxes.