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07-09-2020 | Issue 2/2021

Mathematics and Financial Economics 2/2021

Preferences over rich sets of random variables: on the incompatibility of convexity and semicontinuity in measure

Journal:
Mathematics and Financial Economics > Issue 2/2021
Authors:
Alexander Zimper, Hirbod Assa
Important notes
We are grateful to comments and suggestions from Frank Riedel, Alex Ludwig, an anonymous referee as well as from seminar participants at the Universities of Bielefeld, Frankfurt, Padova, Udine, and the ETH Zürich.

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Abstract

This paper considers a decision maker whose preferences are locally upper- or/and lower-semicontinuous in measure. We introduce the notion of a rich set which encompasses any standard vector space of random variables but also much smaller sets containing only random variables with at most two different outcomes in their support. Whenever preferences are complete on a rich set of random variables, lower- (resp. upper-) semicontinuity in measure becomes incompatible with convexity of strictly better (resp. worse) sets. We discuss implications for utility representations and risk-measures. In particular, we show that the value-at-risk criterion violates convexity exactly because it is lower-semicontinuous in measure.

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