1,721,026 research outputs found
Financial Risk Measurement with Imprecise Probabilities
No full textAlthough 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
Epistemic independence for imprecise probabilities
No full textAbstractThe aim of this paper is that of studying a notion of independence for imprecise probabilities which is essentially based on the intuitive meaning of this concept. This is expressed, in the case of two events, by the reciprocal irrelevance of the knowledge of the value of each event for evaluating the other one, and has been termed epistemic independence. In order to consider more general situations in the framework of coherent imprecise probabilities, a definition of (epistemic) independence is introduced referring to arbitrary sets of logically independent partitions. Logical independence is viewed as a natural prerequisite for epistemic independence. It is then proved that the definition is always consistent, its relationship with the factorization rule is analysed, and some of its more relevant implications are discussed
A Note on the Equivalence of Coherence and Constrained Coherence
Full text linkConstrained coherence is compared to coherence and its role in the behavioural interpretation of coherence is discussed. The equivalence of these two notions is proven for coherent conditional previsions, showing that the same course of reasoning applies to several similar concepts developed in the realm of imprecise probability theory
Bayes Theorem Bounds for Convex Lower Previsions
No full textIn this paper we consider some bounds for lower previsions that are either coherent or, more generally, centered convex. We focus on bounds concerning the classical product and Bayes’ rules, discussing first weak product rules and some of their implications for coherent lower previsions. We then generalise a well-known lower bound, which is a (weak) version for events and coherent lower probabilities of Bayes’ theorem, to the case of random variables and (centered) convex previsions. We
obtain a family of bounds and show that one of them is undominated in all cases. Some applications are outlined, and it is shown that 2-monotonicity, which ensures that the bound is sharp in the case of events, plays a much more limited role in this general framework
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