Comonotone Pareto optimal allocations for law invariant robust utilities on \(L^1\)
type: Item
instance of: scholarly article
title: Comonotone Pareto optimal allocations for law invariant robust utilities on \(L^1\)
zbMATH Open document ID: 1307.91076
author: Claudia Ravanelli
author: Gregor Svindland
published in: Finance and Stochastics
publication date: 2014-11-14T00:00:00Z
review text: The authors prove the existence of comonotone Pareto optimal allocations satisfying utility constraints when decision makers have probabilistic sophisticated variational preferences and thus, representing criteria in the class of law invariant robust utilities. \textit{F. Maccheroni} et al. [Econometrica 74, No. 6, 1447--1498 (2006; Zbl 1187.91066)] first introduced variational preferences and axiomatically characterized them. The connection between variational preferences and robust utilities as representing choice criteria of variational preferences on random endowments corresponding to Savage acts was found in [\textit{A. Schied} et al., in: Handbook of numerical analysis. Vol XV. Special Volume: Mathematical modeling and numerical methods in finance. Amsterdam: Elsevier/North-Holland. 29--87 (2009; Zbl 1180.91274)]. The class of law invariant robust utilities which represent probabilistic sophisticated variational preferences are of the type \[ {\mathcal U}(X)= \inf_{Q\in {\mathcal Q}}( E^Q(u(X))+\alpha (Q)),\quad X\in L^1, \] where \(u: {\mathbb R} \to {\mathbb R}\cup \{-\infty\}\) is a concave increasing utility function, \({\mathcal Q}\) is a set of probability measures which is closed under densities with the same distribution, and \(\alpha (Q)\) is a suitable law invariant penalization on \(Q\in {\mathcal Q}\). The existence of Pareto optimal allocations has so far only been established for a few subclasses of law invariant robust utilities, and for the \(L^1\) condition from Yaari preferences represented by the robust utility \[ {\mathcal U}_1(X) = \frac{1}{\alpha} \int_0^{\alpha} q_X(s) ds,\quad X\in L^1, \] for the quantile function \(q_X(s) = \inf\{x: P(X\leq x)\geq s\}\) with \(s\in (0, 1)\). The Yaari preference \({\mathcal U}_1\) is the well-known average value at risk coherent risk measure (VaR) (see [\textit{M. E. Yaari}, Econometrica 55, 95--115 (1987; Zbl 0616.90005)]). Section 2 gives the setup and definitions for the probabilistic sophisticated…
Mathematics Subject Classification ID: 91B16
Mathematics Subject Classification ID: 91B30
Mathematics Subject Classification ID: 91B32
Mathematics Subject Classification ID: 91B50
zbMATH DE Number: 6369505
zbMATH Keywords: comonotone Pareto optimal allocations
zbMATH Keywords: variational preferences
zbMATH Keywords: robust utility
zbMATH Keywords: probabilistic sophisticated variational preference
zbMATH Keywords: law invariance
zbMATH Keywords: ambiguity aversion
zbMATH Keywords: weighted sup-convolution
MaRDI profile type: Publication
full work available at URL: s00780-013-0214-7
OpenAlex ID: W1993483736
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DOI: 10.1007/S00780-013-0214-7
reviewed by: Weiping Li
description: scientific article; zbMATH DE number 6369505