Upper comonotonicity and convex upper bounds for sums of random variables

It is well-known that if a random vector with given marginal distributions is comonotonic, it has the
largest sum with respect to convex order. However, replacing the (unknown) copula by the comonotonic
copula will in most cases not reflect reality well. For instance, in an insurance context we may have
partial information about the dependence structure of different risks in the lower tail. In this paper, we
extend the aforementioned result, using the concept of upper comonotonicity, to the case where the
dependence structure of a random vector in the lower tail is already known. Since upper comonotonic
random vectors have comonotonic behavior in the upper tail, we are able to extend several well-known
results of comonotonicity to upper comonotonicity. As an application, we construct different increasing
convex upper bounds for sums of random variables and compare these bounds in terms of increasing
convex order