Abstract. Sklar's theorem is an important tool that connects bidimensional distribution functions with their marginals by means of a copula. When there is imprecision about the marginals, we can model the available information by means of p-boxes, that are pairs of ordered distribution functions. Similarly, we can consider a set of copulas instead of a single one. We study the extension of Sklar's theorem under these conditions, and link the obtained results to stochastic ordering with imprecision.

Two classes of imprecise previsions, which we termed convex and centered convex previsions, are studied in this paper in a framework close to Walley’s and Williams’ 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

In this paper we discuss the consistency concept of Williams coherence for imprecise conditional previsions, presenting a variant of this notion, which we call W-coherence. It is shown that W-coherence ensures important consistency properties and is quite general and well-grounded. This is done comparing it with alternative or anyway similar known and less known consistency definitions. The common root of these concepts is that they variously extend to imprecision the subjective probability approach championed by de Finetti. The analysis in the paper is also helpful in better clarifying several little investigated aspects of these notions.

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We explore generalizations of the pari-mutuel model (PMM), a formalization of an intuitive way of assessing an upper probability from a precise one. We discuss a naive extension of the PMM considered in insurance, compare the PMM with a related model, the Total Variation Model, and generalize the natural extension of the PMM introduced by P. Walley and other pertained formulae. The results are subsequently given a risk measurement interpretation: in particular it is shown that a known risk measure, Tail Value at Risk (TVaR), is derived from the PMM, and a coherent risk measure more general than TVaR from its imprecise version. We analyze further the conditions for coherence of a related risk measure, Conditional Tail Expectation. Conditioning with the PMM is investigated too, computing its natural extension, characterising its dilation and studying the weaker concept of imprecision increase