Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Bounds for Trivariate Copulas with Given Bivariate Marginals | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008 Article ID 161537 9 pages doi 2008 161537 Research Article Bounds for Trivariate Copulas with Given Bivariate Marginals Fabrizio Durante 1 Erich Peter Klement 1 and Jose Juan Quesada-Molina2 1 Department of Knowledge-Based Mathematical Systems Johannes Kepler University 4040 Linz Austria 2 Departamento de Matemdtica Aplicada Universidad de Granada 18071 Granada Spain Correspondence should be addressed to Jose Juan Quesada-Molina jquesada@ Received 26 September 2008 Accepted 27 November 2008 Recommended by Paolo Ricci We determine two constructions that starting with two bivariate copulas give rise to new bivariate and trivariate copulas respectively. These constructions are used to determine pointwise upper and lower bounds for the class of all trivariate copulas with given bivariate marginals. Copyright 2008 Fabrizio Durante et al. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. 1. Introduction In recent literature several researchers have focused the attention on constructions and stochastic orders among probability distribution functions with given marginals. These problems are interesting especially for their relevance in finance and quantitative risk management like models of multivariate portfolios and bounding functions of dependent risks see . 1 . If a random vector X X1z. Xn is characterized by a distribution function . F with known univariate marginals then upper and lower bounds for F were given in early works by Frechet. When instead we have some information about the multivariate marginals of F then the problem has not been considered extensively in the literature although it seems natural that for some applications one needs to estimate the joint distribution F of X when the dependence .
