Extensions of the notion of overall comonotonicity to partial comonotonicity

The overall comonotonicity has become popular in actuarial science and finance over the last decade. As a further step, the notion of upper comonotonicity has recently been proposed. Using the technique of distributional representation we provide a unified method to extend the notion of comonotonicity further to lower comonotonicity, tail comonotonicity, and interval comonotonicity respectively. Numerical illustrations are provided to make a comparison among these different types of dependence structures. The numerical results can be explained to some extent by the sum of uniform (0,1) random variables, for which we obtain explicit formulae for the probability density functions of the sum of two random variables in partial comonotonicity cases. For higher dimension cases, it becomes complicated to find the corresponding explicit formulae.