Reliable Computing 9: 465–485, 2003.
2003 Kluwer Academic Publishers. Printed in the Netherlands.
Convex Imprecise Previsions
RENATO PELESSONI and PAOLO VICIG
University of Trieste, Dipartimento di Matematica Applicata “B. de Finetti”, Piazzale Europa 1,
I–34127 Trieste, Italy, e-mail: firstname.lastname@example.org, email@example.com
(Received: 30 August 2002; accepted: 14 April 2003)
Abstract. In this paper centered convex previsions are introduced as a special class of imprecise
previsions, showing that they retain or generalise most of the relevant properties of coherent imprecise
previsions but are not necessarily positively homogeneous. The broader class of convex imprecise
previsions is also studied and its fundamental properties are demonstrated, introducing in particular
a notion of convex natural extension which parallels that of natural extension but has a larger domain
of applicability. These concepts appear to have potentially many applications. In this paper they are
applied to risk measurement, leading to a general deﬁnition of convex risk measure which corresponds,
when its domain is a linear space, to the one recently introduced in risk measurement literature.
Theories of imprecise probabilities have been increasingly studied and developed
in recent years, due to their generality and greater ability to dependably handle
uncertainty with respect to more traditional tools. The advantages of an imprecise
probability approach are more patent when opinions are based on imprecise or partly
conﬂicting information or beliefs, as often happens in practical problems, or when
some measure of our degree of ignorance or uncertainty should be supplied.
The book by Walley  is a fundamental reference in this area. In , impre-
cise probability theory is developed in terms of imprecise previsions,andtwo
major classes of (unconditional) imprecise previsions are considered, relying upon
reasonable consistency requirements: previsions that avoid sure loss and coherent
previsions. The condition of avoiding sure loss is less restrictive than coherence but
many of its properties are often too weak.
Because of their generality, coherent imprecise previsions encompass several
other uncertainty measures as special cases , including belief functions, pos-
sibility or necessity measures, 2-monotone probabilities, precise probabilities and
others. Many of these uncertainty formalisms have been by now developed and
studied extensively, and were often introduced prior to the theory of imprecise
Previsions avoiding sure loss received less attention, and it is an interesting
question to state whether some special class of previsions that avoid sure loss can
be identiﬁed, which is such that:
(a) its properties are not too far from those of coherent previsions;