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

4.3 Handling Time-Cumulated Risk Measures Through Frequencies and Probabilities

This section explores the time dimension underlying any consideration of modelling in the context of risk and uncertainty. As was mentioned in Chapter 2, the state ω of the system considered, conditional upon taking some given actions d, is imperfectly known at a given time of interest: either at a given date t_f or during a given period [t_f, t_f + ΔT]. Abandoning the ambition of accurate prediction, modelling will consider a risk measure or quantity of interest as a means to control the lack of knowledge in ω at that time, as viewed by the analyst to the best of his knowledge. A thorough specification of the risk measure should therefore specify the time basis upon which the state of the system is studied. Except in the case of inverse calibration or data assimilation studies (see Chapter 6) where a prior analysis of the system involves a past time of interest to model indirect observations of the past states of the system, it generally refers to the future. Multiple definitions may be considered time-wise, such as frequencies or probabilities of average or maximal events or performance levels. As a number of consequences concern the associated risk measures, it is worth considering them carefully before possibly reformulating with easier-to-handle time-implicit formulations.

4.3.1 The Underlying Time Basis of the State of the System

The states of the system considered generally undergo a dynamic variation ...