Sharp convex bounds on the aggregate sums--An alternative proof

It is well known that a random vector with given marginal distributions is comonotonic if and only if it has the largest sum with respect to the convex order [ Kaas, Dhaene, Vyncke, Goovaerts, Denuit (2002), A simple geometric proof that comonotonic risks have the convex-largest sum, ASTIN Bulletin 32, 71-80. Cheung (2010), Characterizing a comonotonic random vector by the distribution of the sum of its components, Insurance: Mathematics and Economics 47(2), 130-136] and that a random vector with given marginal distributions is mutually exclusive if and only if it has the minimal convex sum [Cheung and Lo (2014), Characterizing mutual exclusivity as the strongest negative multivariate dependence structure, Insurance: Mathematics and Economics 55, 180-190]. In this note, we give a new proof of this two results using the theories of distortion risk measure and expected utility.Comment: 11page

Joint Mixability of Elliptical Distributions and Related Families

In this paper, we further develop the theory of complete mixability and joint mixability for some distribution families. We generalize a result of R\"uschendorf and Uckelmann (2002) related to complete mixability of continuous distribution function having a symmetric and unimodal density. Two different proofs to a result of Wang and Wang (2016) which related to the joint mixability of elliptical distributions with the same characteristic generator are present. We solve the Open Problem 7 in Wang (2015) by constructing a bimodal-symmetric distribution. The joint mixability of slash-elliptical distributions and skew-elliptical distributions is studied and the extension to multivariate distributions is also investigated.Comment: 15page

Two Sufficient Conditions for Convex Ordering on Risk Aggregation

We define new stochastic orders in higher dimensions called weak correlation orders. It is shown that weak correlation orders imply stop-loss order of sums of multivariate dependent risks with the same marginals. Moreover, some properties and relations of stochastic orders are discussed