Convex Imprecise Previsions for Risk Measurement Renato Pelessoni (renatop@econ.univ.trieste.it) and Paolo Vicig (paolov@econ.univ.trieste.it) University of Trieste, Dipartimento di Matematica Applicata ‘B. de Finetti’ Piazzale Europa 1, I-34127 Trieste, Italy Abstract. In this paper we introduce convex imprecise previsions 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 weakly convex imprecise previsions is also studied and its fundamental properties are demonstrated. The notions of weak convexity and convexity are then applied to risk measurement, leading to a more general definition of convex risk measure than the one already known in risk measurement literature. Keywords: imprecise previsions, risk measures, weakly convex imprecise previsions, convex imprecise previsions. Table of Contents 1 Introduction 2 2 Imprecise Previsions 4 3 Convex Lower Previsions 6 3.1 Weakly convex previsions and convex natural ex- tension 6 3.2 Convex previsions 13 3.3 Envelope theorems 15 4 Convex Risk Measures 17 References 23 1