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Uncertainty modelling and conditioning with convex imprecise previsions. (English) Zbl 1123.62077
Summary: Two classes of imprecise previsions, which we termed convex and centered convex previsions, are studied in this paper in a framework close to P. Walley’s [Statistical reasoning with imprecise probabilities. (1991; Zbl 0732.62004)] and P. M. Williams’ [Notes on conditional previsions. Res. Rep. School Math. Phys. Sci., Univ. Sussex (1975)] theory of imprecise previsions. We show that convex previsions are related with a concept of convex natural extension, which is useful in correcting a large class of inconsistent imprecise probability assessments, characterised by a condition of avoiding unbounded sure loss. Convexity further provides a conceptual framework for some uncertainty models and devices, like unnormalised supremum preserving functions. Centered convex previsions are intermediate between coherent previsions and previsions avoiding sure loss, and their not requiring positive homogeneity is a relevant feature for potential applications. We discuss in particular their usage in (financial) risk measurement. In a final part we introduce convex imprecise previsions in a conditional environment and investigate their basic properties, showing how several of the preceding notions may be extended and the way the generalised Bayes rule applies.

62P05 Applications of statistics to actuarial sciences and financial mathematics
91B28 Finance etc. (MSC2000)
Full Text: DOI
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