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Finance and Stochastics

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01-01-2014

Comonotone Pareto optimal allocations for law invariant robust utilities on L ¹

Authors: Claudia Ravanelli, Gregor Svindland

Published in: Finance and Stochastics | Issue 1/2014

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Abstract

We 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. The total endowment is only required to be integrable.

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Probabilistic sophisticated preferences were introduced by Machina and Schmeidler [22], and further studied by Strzalecki [24] for the case of variational preferences.

Since \(\mathcal {U}\) is defined via the law invariant penalization α, it is law invariant. This follows as in [12], Theorem 4.54. Conversely, the dual function in the convex duality sense of a law invariant concave function \(\mathcal {U}:L^{1}\to \mathbb {R}\cup\{-\infty\}\), which in particular can serve as a penalization α in the sense of (2.1), is always law invariant; again see [12], Theorem 4.54.

c _i=∞ is understood as the restriction \(\mathcal {U}_{i}(X_{i})\geq \mathcal {U}_{i}(W_{i})-\infty:=-\infty\) being redundant.

See Example 5.4.

The individual constraints may for instance imply that the closed set \(\mathbb {A}_{c}(W)\) is essentially bounded. Let us think of it as compact. Then (3.3) will allow a solution for any positive weights, whereas the optimization (3.2) over all allocations only works for certain weights.

Note that λ _j=0 implies that the decision maker j is not considered in the social welfare maximization problem (3.3).

When \(d_{L}^{i}=d_{H}^{i}=d^{i}\) and s _i=−∞ (i.e., \(\operatorname {dom}\,u_{i}=\mathbb{R}\)), then the corresponding robust utility \(\mathcal{U}_{i}\) is cash additive in the sense that \(\mathcal{U}_{i}(X+m)=\mathcal{U}_{i}(X)+d^{i} m\) for all \(m\in \mathbb {R}\) and X∈L ¹ and thus corresponds to a convex risk measure (if d ⁱ=1 and when multiplied by −1).

Here \(\frac{0}{\infty}:=0\) and \(\frac{\infty}{0}:=\infty\).

Note that if u is a concave function, then it is always dominated by x↦u′(y)(x−y)+u(y).

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Title: Comonotone Pareto optimal allocations for law invariant robust utilities on L 1
Authors: Claudia Ravanelli
Gregor Svindland
Publication date: 01-01-2014
Publisher: Springer Berlin Heidelberg
Published in: Finance and Stochastics / Issue 1/2014
Print ISSN: 0949-2984
Electronic ISSN: 1432-1122
DOI: https://doi.org/10.1007/s00780-013-0214-7

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