Bounds for functions of multivariate risks

• Published in: Journal of multivariate analysis Vol. 97; no. 2; pp. 526 - 547
• Main Authors: Embrechts, Paul, Puccetti, Giovanni
• Format: Journal Article
• Language: English
• Published: San Diego, CA Elsevier Inc 01.02.2006; Elsevier; Taylor & Francis LLC
• Series: Journal of Multivariate Analysis
• Subjects: Coupling; Distribution theory; Dual bounds; Exact sciences and technology; Mathematics; Measurement techniques; Multivariate analysis; Multivariate marginals; Multivariate marginals Coupling Dual bounds Value-at-Risk Risk measures; Probability and statistics; Probability theory and stochastic processes; Risk assessment; Risk measures; Sciences and techniques of general use; Studies; Value-at-Risk; Vector space; 60E15; Multivariate marginals; Value-at-Risk; Coupling; Dual bounds; Risk measures; 60E05; 60E05 Multivariate marginals; Risk function; Random vector; Dual bound; Multivariate analysis; Statistical method; Risk measure; N variable function; Hypercube; Marginal distribution; Distribution function; Distribution tail; Random function
• ISSN: 0047-259X; 1095-7243
• DOI: 10.1016/j.jmva.2005.04.001

Li et al. [Distributions with Fixed Marginals and Related Topics, vol. 28, Institute of Mathematics and Statistics, Hayward, CA, 1996, pp. 198–212] provide bounds on the distribution and on the tail for functions of dependent random vectors having fixed multivariate marginals. In this paper, we correct a result stated in the above article and we give improved bounds in the case of the sum of identically distributed random vectors. Moreover, we provide the dependence structures meeting the bounds when the fixed marginals are uniformly distributed on the k-dimensional hypercube. Finally, a definition of a multivariate risk measure is given along with actuarial/financial applications.