4th International Symposium on Imprecise Probabilities and Their Applications, Pittsburgh, Pennsylvania, 2005 Envelope Theorems and Dilation with Convex Conditional Previsions Renato Pelessoni University of Trieste renato.pelessoni@econ.units.it Paolo Vicig University of Trieste paolo.vicig@econ.units.it Abstract This paper focuses on establishing envelope theorems for convex conditional lower previsions, a recently in- vestigated class of imprecise previsions larger than co- herent imprecise conditional previsions. It is in partic- ular discussed how the various theorems can be em- ployed in assessing convex previsions. We also con- sider the problem of dilation for these kinds of im- precise previsions, and point out the role of convex previsions in measuring conditional risks. Keywords. Convex Prevision, Envelope Theorem, Dilation, Convex Risk Measure. 1 Introduction Generally speaking, envelope theorems relate a func- tion in a certain set F to a set P of other functions with well specified features. These theorems either ensure that by performing the (pointwise) infimum or supremum on the elements of P we get a function f ∈F , or else guarantee that every f ∈F may be ex- pressed as an infimum or supremum over some set P , or both (thus characterising the functions in F ). En- velope theorems are found in many different research areas, like for instance cooperative games [10] or con- vex analysis [9]. In the theory of imprecise probabilities, a fundamen- tal envelope theorem [13] states that a real function P is a coherent lower prevision over a set D of (uncon- ditional) random variables (or gambles) if and only if P (X) = inf P ∈P {P (X)}, ∀X ∈D, where all P ∈P are coherent precise previsions. The theorem on one hand points out a way of assessing coherent lower pre- visions, on the other hand relates the behavioural ap- proach to imprecise previsions with the indirect ap- proach, which defines imprecise previsions or proba- bilities in terms of sets of other uncertainty measures (precise previsions, or probabilities). In the language of risk measures, a version of this theorem characterises coherent risk measures [1] and the precise previsions are called ‘scenarios’ (see also [6] and, for a unifying approach, [4]). Envelope theorems were introduced also for other kinds of imprecise previsions, including coherent lower previsions for unbounded random variables [12], con- vex previsions [7] and conditional coherent lower pre- visions [16]. The conditional framework is intrinsi- cally more complex, because the set P is generally not convex and because conditioning events may be allowed to have zero probability. This paper is concerned with establishing some enve- lope theorems for conditional convex previsions. Con- vex and centered convex previsions were introduced in [7] in a framework close to Walley’s approach to im- precise previsions. Centered convex previsions are a special subset of previsions that avoid sure loss, are close to coherent lower previsions, but do not require positive homogeneity. In particular, convex risk mea- sures [2] are a special case of convex (not necessarily centered) previsions. Conditional convex previsions and their basic properties were studied in [8]. The notions about convex and conditional convex previsions needed in the sequel are included in Sec- tions 2.1 and 2.2, following the approach in [7, 8], where proofs of the results may be found. Some alter- native approaches, like that in [5], are also discussed in [8]. Section 2.3 contains some preliminary material on conditional precise probabilities. Envelope theorems are stated and discussed in Sec- tion 3. In particular, Theorems 5 and 6 point out ways of assessing conditional convex previsions. It is assumed in Theorem 5 that conditioning events have non-zero probability under every prevision in P , while this assumption is dropped in Theorem 6. Theorem 7 characterises implicitly conditional convex previsions, while Theorem 8 gives an explicit characterisation. These results are then compared and their role in as- sessing or extending convex previsions is discussed.