Say that I have a bivariate random variable $X=(X_1,X_2)$ with known marginal distributions $F_1$ and $F_2$ and a known covariance matrix $S$. However, I do not know the joint distribution of $(X_1,X_2)$ and therefore not its copula.
If I decide to use a Gaussian copula to simulate from the joint distribution, how do I choose which Gaussian copula to use and what exactly is the simulation algorithm?
Is it correctly understood that because I do not know the joint distribution and therefore not the "correct" copula, that this simulation algorithm will only result in random variables that have the correct marginal distributions and covariance matrix, but not necessarily the correct joint distribution?