Some Results on Multivariate Dependence Modeling

Abstract

The goal of this thesis is to solve some problems in dependence modeling. Under special assumptions, we use Tankov [2011]’s result to give sharp bounds on variance of the sum of two random variables with partial information available and point out some drawbacks in his method. Thus, two different methods based on convex ordering are proposed. We show the one inspired by Bernard and Vanduffel [2014] may fail and provide an improved method. This thesis then discusses the compatible matrix problem. We characterize the covariance matrix for sums of normal distributed random variables to reach the minimum variance in dimensions three and four. This result is supported with application on variance bounds with background risk. The last part reviews some existing dependence measures and a new multivariate dependence measure focusing on the sum of random variables is introduced with properties and estimation method. Each chapter ends with a conclusion and future research directions.