Simulation

For each copula (Gaussian, Archimedean or Student), it is possible to generate samples according to copula’s structure.

Gaussian Copula Sampling

We suppose that our copula is a gaussian \(\mathcal{N}(0, \Sigma)\) where \(\Sigma\) is the covariance matrix. Before sampling new variables from quantile functions, we are required to compute the correlation matrix \(P\) as follow :

\[P=\sqrt{\text{Diag}(\Sigma)}^{-1} \Sigma \sqrt{\text{Diag}(\Sigma)}^{-1}\]

According to the process, we extract the Cholesky triangular matrix \(L\) such that \(P=L L^{\top}\). Finally, sampling gaussian copula is now easily possible.

• Sample \(Z \sim \mathcal{N}(0, I_d)\)
• Compute \(V=LZ\)
• Apply quantile function to \(V\)