Abstract
In this chapter we derive that comonotonicity is the worst case dependence structure concerning convex order of the joint portfolio \(\sum _{i=1}^{n}X_{i}\) or equivalently of the excess of loss. Two different approaches to this result are given. The first approach due to Meilijson and Nadas (1979) is based on a simple duality argument and uses just a monotonicity property of the inverse distribution function.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
B.C. Arnold, Majorization and the Lorenz Order: A Brief Introduction. Volume 43 of Lecture Notes in Statistics (Springer, Berlin, 1987)
K.-M. Chong, Some extensions of a theorem of Hardy, Littlewood and Pólya and their applications. Can. J. Math. 26, 1321–1340 (1974)
K.-M. Chong, N.M. Rice, Equimeasurable Rearrangements of Functions. Volume 28 of Queen’s Papers in Pure and Applied Mathematics (Queen’s University, Kingston, 1971)
P.W. Day, Rearrangement inequalities. Can. J. Math. 24, 930–943 (1972)
M. Fréchet, Sur les tableaux de corrélation dont les marges sont données. Annales de l’Université de Lyon, Section A, Series 3 14, 53–77 (1951)
G.H. Hardy, J.E. Littlewood, G. Pólya, Some simple inequalities satisfied by convex functions. Messenger 58, 145–152 (1929)
G.H. Hardy, J.E. Littlewood, G. Pólya, Inequalities, 2nd edn. (Cambridge University Press, Cambridge, 1952)
J. Karamata, Sur une inégalité relative aux fonctions convexes. Publ. Math. Univ. Belgrade 1, 145–148 (1932)
S. Karlin, A. Novikoff, Generalized convex inequalities. Pac. J. Math. 13, 1251–1279 (1963)
W.A.J. Luxemburg, Rearrangement invariant Banach function spaces. Queen’s Papers Pure Appl. Math. 10, 83–144 (1967)
A.W. Marshall, I. Olkin, Inequalities: Theory of Majorization and Its Applications. Number 143 in Mathematics in Science and Engineering (Academic, New York, 1979)
I. Meilijson, A. Nadas, Convex majorization with an application to the length of critical paths. J. Appl. Probab. 16, 671–677 (1979)
P.A. Meyer, Probability and Potentials (Blaisdell Publishing Company, a Division of Ginn and Company, Waltham, 1966)
K. Mosler, M. Scarsini, Some theory of stochastic dominance, in Stochastic Orders and Decision Under Risk: Papers from the International Workshop held in Hamburg, Germany, May 16–20, 1989. Volume 19 of Lecture Notes (Institute of Mathematical Statistics, Hayward, 1991a), pp. 261–284
A. Müller, D. Stoyan, Comparison Methods for Stochastic Models and Risks (Wiley, Chichester, 2002)
K.R. Parthasarathy, Probability Measures on Metric Spaces. Probability and Mathematical Statistics. A Series of Monographs and Textbooks (Academic, New York, 1967)
V.A. Rohlin, On the fundamental ideas of measure theory. Am. Math. Soc. Transl. 1952(71), 55p (1952)
L. Rüschendorf, Inequalities for the expectation of Δ-monotone functions. Z. Wahrscheinlichkeitstheorie Verw. Geb. 54, 341–354 (1980)
L. Rüschendorf, Ordering of distributions and rearrangement of functions. Ann. Probab. 9, 276–283 (1981b)
L. Rüschendorf, Solution of a statistical optimization problem by rearrangement methods. Metrika 30, 55–62 (1983)
L. Rüschendorf, Stochastic ordering of risks, influence of dependence and a.s. constructions, in Advances on Models, Characterizations and Applications, ed. by N. Balakrishnan, I.G. Bairamov, O.L. Gebizlioglu (Chapman & Hall/CRC, Boca Raton, 2005), pp. 19–56
J.V. Ryff, Orbits of L 1 functions under doubly stochastic transformations. Trans. Am. Math. Soc. 117, 92–100 (1965)
M. Shaked, J.G. Shanthikumar, Stochastic Orders and Their Applications. Probability and Mathematical Statistics (Academic, Boston, 1994)
V. Strassen, The existence of probability measures with given marginals. Ann. Math. Stat. 36, 423–439 (1965)
M. Whitt, Bivariate distributions with given marginals. Ann. Stat. 4, 1280–1289 (1976)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Rüschendorf, L. (2013). Convex Order, Excess of Loss, and Comonotonicity. In: Mathematical Risk Analysis. Springer Series in Operations Research and Financial Engineering. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33590-7_3
Download citation
DOI: https://doi.org/10.1007/978-3-642-33590-7_3
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-33589-1
Online ISBN: 978-3-642-33590-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)