Comonotonicity

In probability theory, comonotonicity mainly refers to the perfect positive dependence between the components of a random vector, essentially saying that they can be represented as increasing functions of a single random variable. Perfect negative dependence is called countermonotonicity.
Comonotonicity is also related to the comonotonic additivity of the Choquet integral.^{[1]}
The concept of comonotonicity has applications in financial risk management and actuarial science. In particular, the sum of the components X_{1} + X_{2} + ... + X_{d} is the riskiest if the joint probability distribution of the random vector (X_{1},X_{2},...,X_{d}) is comonotonic.^{[2]} Furthermore, the αquantile of the sum equals of the sum of the αquantiles of its components, hence comonotonic random variables are quantileadditive.^{[3]}^{[4]}
For extensions of comonotonicity, see Jouini & Napp (2004) and Puccetti & Scarsini (2010).
Contents
Definitions
Comonotonicity of subsets of R^{d}
A subset S of R^{d} is called comonotonic^{[5]} if, for all (x_{1},x_{2},...,x_{d}) and (y_{1},y_{2},...,y_{d}) in S with x_{i} < y_{i} for some i ∈ {1,2,...,d}, it follows that x_{j} ≤ y_{j} for all j ∈ {1,2,...,d}.
This means that S is a totally ordered set.
Comonotonicity of probability measures on R^{d}
Let μ be a probability measure on the ddimensional Euclidean space R^{d} and let F denote its multivariate cumulative distribution function, that is
Furthermore, let F_{1},...,F_{d} denote the cumulative distribution functions of the d onedimensional marginal distributions of μ, that means
for every i ∈ {1,2,...,d}. Then μ is called comonotonic, if
Comonotonicity of R^{d}valued random vectors
An R^{d}valued random vector is called comonotonic, if its multivariate distribution (the pushforward measure) is comonotonic, this means
Properties
A comonotonic R^{d}valued random vector can be represented as
where =_{d} stands for equality in distribution and U is a uniformly distributed random variable on the unit interval. It can be proved that a random vector is comonotonic if and only if all marginals are nondecreasing functions (or all are nonincreasing functions) of the same random variable.^{[citation needed]}
Upper bounds
Fréchet upper bound for cumulative distribution functions
Let be an R^{d}valued random vector. Then, for every i ∈ {1,2,...,d} and x_{i} ∈ R,
hence
with equality everywhere if and only if is comonotonic.
Upper bound for the covariance
Let (X,Y) be a bivariate random vector such that the expected values of X, Y and the product XY exist. Let (X * ,Y * ) be a comonotonic bivariate random vector with the same onedimensional marginal distributions as (X,Y).^{[clarification needed]} Then it follows from Höffding's formula for the covariance^{[6]} and the Fréchet upper bound that
and, correspondingly,
with equality if and only if (X,Y) is comonotonic.^{[7]}
Citations
 ^ (Sriboonchitta et al. 2010, pp. 149–152)
 ^ (Kaas et al. 2002, Theorem 6)
 ^ (Kaas et al. 2002, Theorem 7)
 ^ (McNeil, Frey & Embrechts 2005, Proposition 6.15)
 ^ (Kaas et al. 2002, Definition 1)
 ^ (McNeil, Frey & Embrechts 2005, Lemma 5.24)
 ^ (McNeil, Frey & Embrechts 2005, Theorem 5.25(2))
References
 Jouini, Elyès; Napp, Clotilde (2004), "Conditional comonotonicity", Decisions in Economics and Finance 27 (2): 153–166, ISSN 15938883, MR2104639, Zbl 1063.60002, http://www.ceremade.dauphine.fr/~jouini/DEF180RR.pdf
 Kaas, Rob; Dhaene, Jan; Vyncke, David; Goovaerts, Marc J.; Denuit, Michel (2002), "A simple geometric proof that comonotonic risks have the convexlargest sum", ASTIN Bulletin 32 (1): 71–80, MR1928014, Zbl 1061.62511, http://www.casact.org/library/astin/vol32no1/71.pdf
 McNeil, Alexander J.; Frey, Rüdiger; Embrechts, Paul (2005), Quantitative Risk Management. Concepts, Techniques and Tools, Princeton Series in Finance, Princeton, NJ: Princeton University Press, ISBN 0691122555, MR2175089, Zbl 1089.91037, http://books.google.com/books?id=f5J_OZPeq50C
 Puccetti, Giovanni; Scarsini, Marco (2010), "Multivariate comonotonicity", Journal of Multivariate Analysis 101 (1): 291–304, ISSN 0047259X, MR2557634, Zbl 1184.62081, http://www.parisschoolofeconomics.eu/IMG/pdf/MED090320Scarsini.pdf
 Sriboonchitta, Songsak; Wong, WingKeung; Dhompongsa, Sompong; Nguyen, Hung T. (2010), Stochastic Dominance and Applications to Finance, Risk and Economics, Boca Raton, FL: Chapman & Hall/CRC Press, ISBN 9781420082661, MR2590381, Zbl 1180.91010, http://books.google.com/books?id=omxatN4lVCkC
Categories: Theory of probability distributions
 Statistical dependence
 Covariance and correlation
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