Abstract
Comonotonicity has become popular in actuarial science and finance.The notion of upper comonotonicity has recently been proposed.Using distributional representation we provide a unified method to extend the notion of comonotonicity further to lower comonotonicity,lower and upper comonotonicity,and interval comonotonicity.Numerical illustrations are provided to make a comparison among the six types of dependence structure:Independence,upper comonotonicity,lower comonotonicity,lower and upper comonotonicity,interval comonotonicity,comonotonicity.The numerical results are related to the sum of uniform(0,1) random variables,for which we obtain the explicit formula for the density function of the sum of two random variables in every case.For higher dimension,it becomes complicated to find the corresponding explicit formulas.
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