I am doing an assignment on copula functions. In my assignment, copula function is defined as follows:
Def A copula function is simply a multivariate uniform distribution over $[0, 1]$ hypercube. For bivariate function $C(u, v)$, $[0, 1]\times[0, 1] → [0, 1]$ we can call $C(u, v)$ is a bivariate copula function if it meets the following three conditions:
a. $C(0, u) = C(u, 0) = 0$
b. $C(1, u) = C(u, 1) = u$
c. $C(u2, v2) − C(u2, v1) − C(u1, v2) + C(u1, v1) ≥ 0$ for all $0 ≤ u1 ≤ u2 ≤ 1$ and $0 ≤ v1 ≤ v2 ≤ 1$,
This is the definition quoted as analytic term version, versus probability term version.
What I'd like to check is the probability term version, i.e.:
if $F_1(x_1)$ and $F_2(x_2)$ are univariate distribution functions, then $F(X_1,X_2):= C(F_1(x_1), F_2(x_2))$ is a bivariate distribution function.
Does anyone know how to deal with this conversion from analytic version to probability version? Specifically, I need to verify that this is a right-continuous and monotonically non-decreasing function with respect to each dimension. How would I prove this? Thanks!