Extensions of Ohlin's lemma with applications to optimal reinsurance structures
type: Item
instance of: scholarly article
title: Extensions of Ohlin's lemma with applications to optimal reinsurance structures
zbMATH Open document ID: 0795.62093
DOI: 10.1016/0167-6687(93)90538-Z
author: Ole Hesselager
published in: Insurance Mathematics \& Economics
publication date: 1993-01-01T00:00:00Z
review text: From the author's introduction and abstract: Let \(X\) and \(Y\) be random variables with cumulative distribution functions \(F\) and \(G\). Ohlin's lemma states that if \(E X= E Y\), and \((F-G)(t)\) has exactly one sign change from \(-\) to \(+\), then \(E u(x)<E u(Y)\) for any convex function \(u\). With \(X= f(Z)\) and \(Y=g(Z)\) denoting the reinsured amounts in respect to some risk \(Z\), this result makes it possible to identify extremal compensation functions \(f\) and \(g\) corresponding to a fixed expected reinsurance compensation. Ohlin's lemma also plays a role in the theory of (stop-loss) ordering of risks. A risk \(Y\) is said to be `more dangerous' than \(X\) if \(E X\leq E Y\) and the distribution functions \(F\) and \(G\) intersect exactly once as stated in Ohlin's lemma. It is a well-known result that if \(Y\) is more dangerous than \(X\), than \(X\) procedes \(Y\) in stop-loss order. We extend Ohlin's lemma in two directions and show how this leads to an extension of known results on optimal reinsurance structures. Among these are the Arrow-Ohlin theorem and the Vajda-Ohlin theorem.
Mathematics Subject Classification ID: 62P05
Mathematics Subject Classification ID: 91B30
Mathematics Subject Classification ID: 65K99
zbMATH DE Number: 540718
zbMATH Keywords: Chebyshev systems
zbMATH Keywords: cumulative distribution functions
zbMATH Keywords: Ohlin's lemma
zbMATH Keywords: convex function
zbMATH Keywords: extremal compensation functions
zbMATH Keywords: expected reinsurance compensation
zbMATH Keywords: ordering of risks
zbMATH Keywords: more dangerous
zbMATH Keywords: stop-loss order
zbMATH Keywords: optimal reinsurance structures
zbMATH Keywords: Arrow-Ohlin theorem
zbMATH Keywords: Vajda-Ohlin theorem
reviewed by: Wolf-Rüdiger Heilmann
MaRDI profile type: Publication
Wikidata QID: Q124805793
cites work: Ordering of risks: a review
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cites work: Optimal reinsurance in relation to ordering of risks
full work available at URL: 0167-6687(93)90538-z
OpenAlex ID: W2055706252
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description: scientific article; zbMATH DE number 540718