Some proposal about multivariate risk measurement

In actuarial literature the properties of risk measures or insurance premium principles have been extensively studied. In our work we propose a characterization of some particular classes of multivariate and bivariate risk measures. Given two random variables we can define an univariate integral stochastic ordering by considering a set of functions, that, through their peculiar properties, originate different stochastic orderings. These stochastic order relations of integral form may be extended to cover also the case of random vectors. It is, in fact, proposed a kind of stop-loss premium, and then a stop-loss order in the multivariate setting and some equivalent conditions. We propose an axiomatic approach based on a minimal set of properties which characterizes an insurance premium principle. In the univariate case we know that Conditional Value at Risk can be represented through distortion risk measures and a distortion risk measure can be viewed as a combination of CVaRs, we propose a generalization of this result in a multivariate framework. In the bivariate case we want to compare the concept of risk measure to that one of concordance measure when the marginals are given