Runtime (Copula)

Copulas are a way to model dependence between random variables while keeping marginal distributions unchanged.

In Monte Carlo simulation, a common pattern is:

1. Sample correlated uniforms $(U_1, \dots, U_d)$.
2. Transform each uniform into a marginal sample using the inverse CDF of the marginal distribution.

---

Gaussian copula (concept)

A Gaussian copula uses a multivariate normal correlation structure:

1. Sample $Z \sim \mathcal{N}(0, \Sigma)$
2. Convert to uniforms: $U_i = \Phi(Z_i)$

where $\Phi$ is the standard normal CDF.

---

What CRML uses copulas for

The CRML language does not mandate a universal “copula field” for all dependency modeling. Dependency constructs are generally engine/tool-specific.

In this repo, the reference engine supports correlating control-state sampling (binary up/down states) via a Gaussian copula specified in portfolio.dependency.copula.

See:

• Engine capabilities: Portfolio execution
• Engine capabilities: Controls