Multivariate Simulation

In many simulation problems, we need to generate not just one random variable, but a collection of variables that interact with one another. For example, a financial model may involve several asset returns observed together, a reliability study may involve lifetimes of related components, and a service system may require the joint simulation of arrivals, service durations, and customer behaviour. In such settings, simulating each variable separately is not enough: we must also account for the dependence structure between them.

This section extends the ideas of univariate simulation to the multivariate setting. We begin with transformations of multiple random variables and show how the Jacobian generalises to higher dimensions. We then study linear transformations, which provide the foundation for constructing and understanding the multivariate normal distribution. This leads naturally to Cholesky decomposition, one of the most important practical tools for simulating correlated random vectors.

However, not all dependence can be described adequately by linear correlation or by the multivariate normal model. To address this, we introduce copulas, which allow us to model dependence separately from the marginal distributions. This provides a more flexible framework for simulating dependent variables with potentially very different individual behaviours.

Overall, this topic forms an important bridge between probability theory and practical simulation.

By the end of this section, you should be able to:

• understand how transformations extend to multiple random variables,
• apply linear transformations to construct correlated random vectors,
• explain and simulate from the multivariate normal distribution,
• use Cholesky decomposition to generate dependent variables efficiently,
• understand the role of copulas in modelling dependence,
• simulate multivariate data with specified marginals and dependence structures.