Factorisation properties of the strong product

We investigate a number of factorisation conditions in the frame- work of sets of probability measures, or coherent lower previsions, with finite referential spaces. We show that the so-called strong product constitutes one way to combine a number of marginal coherent lower previsions into an independent joint lower prevision, and we prove that under some conditions it is the only independent product that satisfies the factorisation conditions

Financial Risk Measurement with Imprecise Probabilities

Although financial risk measurement is a largely investigated research area, its relationship with imprecise probabilities has been mostly overlooked. However, risk measures can be viewed as instances of upper (or lower) previsions, thus letting us apply the theory of imprecise previsions to them. After a presentation of some well known risk measures, including Value-at-Risk or VaR, coherent and convex risk measures, we show how their definitions can be generalized and discuss their consistency properties. Thus, for instance, VaR may or may not avoid sure loss, and conditions for this can be derived. This analysis also makes us consider a very large class of imprecise previsions, which we termed convex previsions, generalizing convex risk measures. Shortfall-based measures and Dutch risk measures are also investigated. Further, conditional risks can be measured by introducing conditional convex previsions. Finally, we analyze the role in risk measurement of some important notions in the theory of imprecise probabilities, like the natural extension or the envelope theorems

Financial Risk Measurement

Chapter 12 (Financial Risk Measurement) presents strong arguments for the need to use imprecise previsions in finance. It is shown that theory of imprecise probabilities provides many tools that are closely linked to popular concepts in financial risk measurement, and it enables modelling based on fewer and simpler assumptions than the standard approaches

A Gambler's Gain Prospects with Coherent Imprecise Previsions

We explore some little investigated aspects of the well known betting scheme defining coherent lower or upper previsions in terms of admissible gains. A limiting situation (lose-or-draw) where the supremum of some gain is zero is discussed, deriving a gambler\u2019s gain evaluations and comparing the differences between the imprecise and precise prevision cases. Then, the correspondence of the betting scheme for imprecise previsions with real-world situations is analysed, showing how the gambler\u2019s profit objectives may compel him to select certain types of bets