Bivariate sampling for distribution expressed in Sklar's copula theorem?

Question

In the univariate case, one can easily sample a distribution via random numbers $u\sim[0,1]$ and plugging into $F^{-1}(u)$.

I have a bivariate distribution constructed via Sklar's theorem on Copulas:

$$F(x,y)=C(F(x),F(y))$$

The distribution is not in closed form.

How do I sample from this bivariate distribution $F(x,y)$?

Answer 1

Uniformly sample from the copula's domain $[0,1]\times [0,1]$ using acceptance-rejection method. Then use the inverse t-distribution on the resulting points to get a point in $F(x,y)$. That's the basics of how copulas work.