On The Mixability Of Elliptical Distributions

This thesis is a direct extension of the complete and the joint mixability.The concept of complete mixability and joint mixability has been studied by many researchers in recent years,it helps to determine all possible distributions of S=X1...+Xn,where X1,...,Xn are random variables and Xi～Fi,but the dependence structure among X1,...,Xn is un-known.At the same time,complete and joint mixability are widely used in solving opti-mization of convex functions and probability inequalities with marginal constraints.Aspired by the concept of complete mixability and joint mixability,Bignozzi and Puccetti(2015)proposed the notion of φ-mixability.Similarly,many optimization problems in quantitative risk management can be solved through the study of the φ-mixability.In this thesis,we consider two examples of supermodular functions and investigate theφ-mixability for elliptical distributions and logarithmic elliptical distributions.At the same time,this thesis studys bounds for risk aggregation with respect to stop-loss order.Be-sides,this thesis investigates the set of center of φ-mixability and show the relation between the center of mixability and that of φ-mixability of elliptical distributions and log-elliptical distributions,which is the main result of this paper.The thesis is organized as follows.First of all,this thesis introduces the research back-grounds of the φ-mixability.Secondly,we introduce the basic definitions,such as the com-plete mixability,the joint mixability,φ-joint mixability,φ-complete mixability,supermodular functions and so on.What’s more,this thesis considers the φ-complete mixability and φ-joint mixability for the class of elliptical distributions.Finally,this thesis considers the φ-complete mixability and φ-joint mixability for the class of logarithmic elliptical distributions.At the same time,we consider the bounds of two class of risk aggregation with respect to stop-loss order.