Arbitrage and asset market equilibrium in infinite dimensional economies with short-selling and risk-averse expected utilities

Abstract

International audienceWe consider a model with an infinite number of states of nature, von Neumann–Morgenstern utilities, where agents have different probability beliefs and where short sells are allowed. We show that no-arbitrage conditions, defined for finite dimensional asset markets models, are not sufficient to ensure existence of equilibrium in presence of an infinite number of states of nature. However, if the individually rational utility set U is compact, we obtain an equilibrium. We give conditions which imply the compactness of U. We give examples of non-existence of equilibrium when these conditions do not hold

Existence of equilibrium on asset markets with a countably infinite number of states

“…Proposition 1 states that the set A is bounded in l 1 (π). In [15], the authors prove that A is l 1 (π) compact when b i = +∞ for any i. Hence U is compact since l ∞ (π) ⊂ l p (π) ⊂ l 1 (π).…”

“…Assumption (NA) deserves an explanation. In [15], we have an example with two agents, where (NA0) holds and we have no equilibrium because the agents do not have No Half-Line of the same line property. In our paper we introduce condition A1 (No Half Line of the same type).…”

“…6. In [15], if a i = 0, ∀i, or b i = +∞, ∀i and if the beliefs are equivalent, there exists an equilibrium. One can see immediately that the conditions a i = 0, ∀i, or b i = +∞, ∀i imply our conditions A1.…”

“…We have proved that (NA) holds. However, [15] does not assume A0 which is stronger than the equivalence of the beliefs. In our paper, A0 is required since a i may be non null and b i may be finite.…”

Risk-oriented approach to managing the social market economy

“…Literature review Preconditions and perspectives of creation of the social market economy and the accumulated practical experience are discussed in the works of Agunovich et al (2018), Andronova andRiazantsev (2006), D'Aleo and, Goyal and Sergi (2015a), Goyal and Sergi (2015b), Goyal et al (2015), Goyal et al (2017), Haabazoka et al (2019), Ivanov et al (2019), Pichkov (2016), Popkova and French (2017), Popkova and Sergi (2020), , , Popkova et al (2018), Popkova and Zmiyak (2019), Pritvorova et al (2018), Qerimi and Sergi (2015), Ragulina (2019), Sergi et al (2019), Shevyakova and Petrenko (2018), Shevyakova et al (2019) and Zavyalova et al (2018). Risk management and related crisis management of market economy ("pure" market) are described in the works of Adetiloye et al (2017), Amin (2019), Apergis (2015), Balima et al (2017), Benda et al (2019), Bertolini and Goglio (2019), Boonman (2019), Bouri (2015), Br€ ulle et al (2019), Deb et al (2020), Demiralay (2019), Floyd (2019), Gencer and Demiralay (2016), Ha-Huy et al (2016), Jiang et al (2017), Kourachanis (2019), Laitsou et al (2017), Matsuoka (2018), Mul e (2019), Naess and B ardsen (2015), ...…”