It is known that singular copulas with prescribed support do not have a density, i.e. $\frac{\partial^2 \mathbf { C } \left( u _ { 1 } , u _ { 2 } \right)}{\partial u_1 \partial u_2} = 0$. For example, the upper Fréchet–Hoeffding bound is a singular copula, and is defined as: $$\mathbf { C }\left( u _ { 1 } , \ldots , u _ { d } \right) = \min \left\{ u _ { 1 } , \ldots , u _ { d } \right\}$$
Scatterplots of Comonotonicity Copula
which concentrates the probability mass uniformly on $\left\{ (u_1, u_2) \in [0,1]^2 : u _ { 1 } = u _ { 2 } \right\}$
My question is: How the copula distribution $\mathbf { C } \left( u _ { 1 } , u _ { 2 } \right)$ is then obtained, if we cannot integrate over the singular support. 3d graph of Comonotonicity Copula
