Fundamental Aspects of Operational Risk and Insurance Analytics: A Handbook of Operational Risk

One important result of this representation is the ability to simulate exactly Archimedean copula random variates, as discussed in Marshall and Olkin (1988) and shown in Algorithm 11.5.

Algorithm 11.5 (Simulation from Archimedean Copula via Laplace Transform)

1. Sample a random variable V ~ F where the distribution F is given by the inverse Laplace transform of the generator ψ such that
2. Sample d i.i.d. draws from a uniform distribution U_i ~ Uniform(0, 1) for i ∈{1, 2, …, d};
3. Construct via transformation the d-variate random vector U = (U₁, …, U_d), which is drawn from the Archimedean copula characterized by generator ψ given by
   (11.80)

The following results in Table 11.1 from Hofert (2008, table 1) demonstrate examples of popular Archimedean copula models for which closed form distributions of such inverse Laplace transforms of the generator are known.

Table 11.1 Generators and inverse Laplace transforms for several copulas from Archimedean family

Archimedean Family	ρ Range	Generator ψ (x; ρ)	Distribution of –1 [ψ]
Ali–Mikhail–Haq	[0, 1]
Clayton	(0, )		Γ(1/ ...