In an n-dimensional scenario, the Gaussian copula, denoted as C_G, is given by:

C_G[G₁(u₁), …, Gₙ(uₙ)] = Mₙ[N⁻¹(G₁(u₁)), …, N⁻¹(Gₙ(uₙ)); ρ_M]

Here, Mₙ represents the joint, n-variate cumulative standard normal distribution with ρ_M as the n × n symmetric, positive-definite correlation matrix of the n-variate normal distribution Mₙ. N⁻¹ is the inverse function of a univariate standard normal distribution.

The Gaussian default time copula, C_GD:

C_GD[Qᵢ(t), …, Qₙ(t)] = Mₙ[N⁻¹(Q₁(t)), …, N⁻¹(Qₙ(t)); ρ_M]

In this equation, the marginal distributions represent the cumulative default probabilities Q for entities i = 1 to n at times t, Qᵢ(t). A Gaussian copula function CGD exists, which enables the mapping of these marginal distributions Qᵢ(t) via the inverse standard normal function N⁻¹ to standard normal and amalgamating the mapped values N⁻¹Qᵢ(t) into a single n-variate standard normal distribution Mₙ. This is done by maintaining the correlation structure denoted by ρ_M.

To derive the default time t of asset i, Tᵢ, which is correlated to the default times of all other assets i = 1…n, we would first derive a sample Mₙ(.) from a multivariate copula, in the Gaussian case, Mₙ(.) ∈ [0, 1].