Stochastic efficiency and inefficiency in portfolio optimization with incomplete information: a set-valued probability a
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Stochastic efficiency and inefficiency in portfolio optimization with incomplete information: a set-valued probability approach D. La Torre1
· F. Mendivil2
Accepted: 19 November 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In this paper we extend the notion of stochastic efficiency and inefficiency in portfolio optimization to the case of incomplete information by means of set-valued probabilities. The notion of set-valued probability models the concept of incomplete information about the underlying probability space and the probability associated with each scenario. Unlike other approaches in literature, our notion of inefficiency is introduced by means of the Monge– Kantorovich metric. We provide some numerical examples to illustrate this approach. Keywords Portfolio optimization · Set-valued probability · Imprecise probability · Set-valued optimization · Stochastic efficiency · Stochastic inefficiency
1 Introduction In portfolio optimization the notion of optimality is based on the notion of stochastic dominance and on a knowledge of the underlying probability space (Markowitz 1952). However, often the probabilities associated with each scenario are not completely known. This leads to analyzing portfolio problems with incomplete information. The notion of set-valued probability seems to be the right tool to describe this lack of information. In this setting, each event is associated with a compact and convex set that models the uncertainty; as a result, the comparison of uncertainties can be based on a partial order and many such partial orders can be used. The consequence is a rich framework for modeling uncertainty. In this paper we focus on the problem of evaluating whether a given portfolio is stochastically efficient in the presence of a lack of information. More specifically, we consider the case where the Decision Maker (DM) knows the set of all possible scenarios or the underlying space of events but he/she does not exactly know the probability distribution of each event and so we describe this uncertainty with a set-valued probability.
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D. La Torre [email protected]
1
SKEMA Business School, Université Côte d’Azur, Sophia Antipolis Campus, Sophia Antipolis, France
2
Department of Mathematics and Statistics, Acadia University, Wolfville, NS, Canada
123
Annals of Operations Research
With respect to other papers in literature (see, for instance, Ben Abdelaziz and Masri 2005, 2010; Ben Abdelaziz et al. 1999; Ermoliev and Gaivoronski 1985; La Torre and Mendivil 2018a, b; Urli and Nadeau 1990, 2004) our approach strongly relies on the notion of set-valued probability. This definition extends the notion of imprecise probability that has been widely investigated in the literature as it represents a quite natural way to extend the traditional notion of probability (Dempster 1966; Shafer 1976; Walley 1991; Ferson et al. 2003; Weichselberger 2000; Dubois 1988). The notion of set-valued probability and its mathematical and statistical properties have been a
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