A contribution in multivariate risk measurement

This work takes origin by the aim to give a representation of the theory of risk (from an actuarial and financial point of view) in the multivariate static setting, and thus when we deal with random vectors instead of random variables. After describing distorted risk measures in the scalar case, we propose a definition for vector distorted risk measures, with their major properties and, through this definition, we introduce also a multivariate Value at Risk and Conditional Value at Risk. Another result is worth mentioning: we can represent subadditive distorted risk measures in the multivariate case as combination of vector CVaRs. Finally, we introduce a new multivariate measure, called Product Stop-loss Premium (PSP) and then we look at the similarities between this and the distorted CVaR with independent components in the bivariate case; but we also detect the peculiarities of a vector risk measure over concordance measures. Our PSP born from the attempt to find a multivariate risk measure very intuitive and useful in describing the joint risk. It can remember also a kind of tail covariance between the components of the random vector.