Handbook of Heavy Tailed Distributions in Finance

$\begin{array}{l}\underset{C_L\left(F_1\left(x_1\right),\dots ,F_n\left(x_n\right)\right)}{\underbrace{\max \left\{0,F_1\left(x_1\right)+\cdots +F_n\left(x_n\right)-\left(n-1\right)\right\}}}\le \underset{C_L\left(F_1\left(x_1\right),\dots ,F_n\left(x_n\right)\right)}{\underbrace{F_1\left(x_1,\dots ,x_n\right)}}\\ \le \underset{C_U\left(F_1\left(x_1\right),\dots ,F_n\left(x_n\right)\right)}{\underbrace{\min \left\{F_1\left(x_1\right),\dots F_n\left(x_n\right)\right\}.}}\end{array}$

The function $C_U(u_1\text{...},u_n)$ is a copula for all $n\ge 2$, but the function $C_L(u_1,\text{...},u_n)$ is a copula for $n=2$ only. If $n=2$, the ...