<?xml version="1.0" encoding="UTF-8"?><xml><records><record><source-app name="Biblio" version="7.x">Drupal-Biblio</source-app><ref-type>17</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Fangda Liu</style></author><author><style face="normal" font="default" size="100%">Jun Cai</style></author><author><style face="normal" font="default" size="100%">Christiane Lemieux</style></author><author><style face="normal" font="default" size="100%">Ruodu Wang</style></author></authors></contributors><titles><title><style face="normal" font="default" size="100%">Convex risk functionals: representation and applications.</style></title><secondary-title><style face="normal" font="default" size="100%">Insurance: Mathematics &amp; Economics</style></secondary-title></titles><dates><year><style  face="normal" font="default" size="100%">2020</style></year></dates><urls><web-urls><url><style face="normal" font="default" size="100%">https://doi.org/10.1016/j.insmatheco.2019.10.007</style></url></web-urls></urls><volume><style face="normal" font="default" size="100%">90</style></volume><pages><style face="normal" font="default" size="100%">66-79</style></pages><language><style face="normal" font="default" size="100%">eng</style></language><abstract><style face="normal" font="default" size="100%">We introduce the family of law-invariant convex risk functionals, which includes a wide majority of practically used convex risk measures and deviation measures. We obtain a unified representation theorem for this family of functionals. Two related optimization problems are studied. In the first application, we determine worst-case values of a law-invariant convex risk functional when the mean and a higher moment such as the variance of a risk are known. Second, we consider its application in optimal reinsurance design for an insurer. With the help of the representation theorem, we can show the existence and the form of optimal solutions.</style></abstract></record></records></xml>