Modelling Under Risk and Uncertainty: An Introduction to Statistical, Phenomenological and Computational Methods

7.2 The Generic Monte-Carlo Simulation Method and Associated Error Control

The section introduces the Monte-Carlo method which is a universal solution for the computation of probabilistic risk measures as well as for mixed deterministic-probabilistic ones to some extent. However, computational constraints generate a fundamental difficulty with complex phenomenological models in practice, the answer to which comes with, beyond the preliminary closed-form simplifications introduced in Section 7.1.3, a series of alternative computational methods that will be introduced in the rest of this chapter.

7.2.1 Undertaking Monte-Carlo Simulation on a Computer

Monte-Carlo simulation may be seen informally as a technique designed to simulate random variables on the input side of a numerical model, run the model for each sample and thus post-treat the resulting model outputs to approximate any desirable model output feature: either a probabilistic risk measure (expectation, variance or any moment; quantile, exceedance probability, etc.) or a deterministic one (e.g. an approximate maximal or minimal output) or any combination. It is more generally a quantity q defined as a functional of the distribution function of Z:

(7.46)

that is another functional of the numerical model G(.) and the input distribution of X assuming at this stage a single-probabilistic uncertainty model:

(7.47)

While early ...