Hostname: page-component-7b7567c8c-7pwfn Total loading time: 0 Render date: 2026-11-14T18:41:55.045Z Has data issue: false hasContentIssue false
  • English
  • Français

Characterizations of Conditional Comonotonicity

Part of:Distribution theory - ProbabilityMultivariate analysis

Published online by Cambridge University Press:  14 July 2016

Ka Chun Cheung
Ka Chun Cheung*
Affiliation:
University of Calgary
*
∗ Postal address: Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta T3A 2E2, Canada. Email address: kccheung@math.ucalgary.ca
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

The notion of conditional comonotonicity was first used implicitly by Kaas, Dhaene, and Goovaerts (2000) and was formally introduced by Jouini and Napp (2004) as a generalization of the classical concept of comonotonicity. The objective of the present paper is to further investigate this relatively new concept. The main result is that a random vector is comonotonic conditional to a certain σ-field if and only if it is almost surely comonotonic locally on each atom of the conditioning σ-field. We also provide a new proof of a distributional representation and an almost sure representation of a conditionally comonotonic random vector.

Keywords

Comonotonicityconditional comonotonicitylocal comonotonicityregular conditional distributionprojection theoremmeasurable graph theorem

MSC classification

Secondary:60E99: None of the above, but in this section62H20: Measures of association (correlation, canonical correlation, etc.)

Information

Type
Research Article
Information
Journal of Applied Probability , Volume 44 , Issue 3 , September 2007, pp. 607 - 617
Copyright
Copyright © Applied Probability Trust 2007 

References

[1] Cheung, K. C. (2007). Improved convex upper bound via conditional comonotonicity. To appear in Insurance Math. Econom.Google Scholar
[2] Christensen, J. P. R. (1974). Topology and Borel structure. North-Holland, Amsterdam.Google Scholar
[3] Dellacherie, C. and Meyer, P. A. (1978). Probabilities and Potential (North-Holland Math. Studies 29). North-Holland, Amsterdam.Google Scholar
[4] Dhaene, J. et al. (2002). The concept of comonotonicity in actuarial science and finance: theory. Insurance Math. Econom. 31, 3–33.Google Scholar
[5] Dhaene, J. et al. (2002). The concept of comonotonicity in actuarial science and finance: applications. Insurance Math. Econom. 31, 133–161.Google Scholar
[6] Hoffmann-Jørgensen, J. (1971). Existence of conditional probabilities. Math. Scand. 28, 257–264.Google Scholar
[7] Jouini, E. and Napp, C. (2004). Conditional comonotonicity. Decis. Econom. Finance 27, 153–166.Google Scholar
[8] Kaas, R., Dhaene, J. and Goovaerts, M. J. (2000). Upper and lower bounds for sums of random variables. Insurance Math. Econom. 27, 151–168.Google Scholar
[9] Kaas, R. et al. (2002). A simple geometric proof that comonotonic risks have the convex-largest sum. Astin Bull. 32, 71–80.Google Scholar
[10] Kallenberg, O. (2002). Foundations of Modern Probability, 2nd edn. Springer, New York.CrossRefGoogle Scholar
[11] Loève, M. (1978). Probability Theory. II, 4th edn. Springer, New York.Google Scholar
[12] Schmeidler, D. (1986). Integral representation without additivity. Proc. Amer. Math. Soc. 97, 255–261.CrossRefGoogle Scholar
[13] Yaari, M. E. (1987). The dual theory of choice under risk. Econometrica 55, 95–115.CrossRefGoogle Scholar