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 X₁ + X₂ + ... + X_d is the riskiest if the joint probability distribution of the random vector (X₁,X₂,...,X_d) is comonotonic.^[2] Furthermore, the α-quantile of the sum equals of the sum of the α-quantiles of its components, hence comonotonic random variables are quantile-additive.^[3][4]

For extensions of comonotonicity, see Jouini & Napp (2004) and Puccetti & Scarsini (2010).

Definitions

Comonotonicity of subsets of R^d

A subset S of R^d is called comonotonic^[5] if, for all (x₁,x₂,...,x_d) and (y₁,y₂,...,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 d-dimensional Euclidean space R^d and let F denote its multivariate cumulative distribution function, that is

$F(x_1,\ldots,x_d):=\mu\bigl(\{(y_1,\ldots,y_d)\in{\mathbb R}^d\mid y_1\le x_1,\ldots,y_d\le x_d\}\bigr),\qquad (x_1,\ldots,x_d)\in{\mathbb R}^d.$

Furthermore, let F₁,...,F_d denote the cumulative distribution functions of the d one-dimensional marginal distributions of μ, that means

$F_i(x):=\mu\bigl(\{(y_1,\ldots,y_d)\in{\mathbb R}^d\mid y_i\le x\}\bigr),\qquad x\in{\mathbb R}$

for every i ∈ {1,2,...,d}. Then μ is called comonotonic, if

$F(x_1,\ldots,x_d)=\min_{i\in\{1,\ldots,d\}}F_i(x_i),\qquad (x_1,\ldots,x_d)\in{\mathbb R}^d.$

Comonotonicity of R^d-valued random vectors

An R^d-valued random vector $X=(X_1,\ldots,X_d)$ is called comonotonic, if its multivariate distribution (the pushforward measure) is comonotonic, this means

${\mathbb P}(X_1\le x_1,\ldots,X_d\le x_d)=\min_{i\in\{1,\ldots,d\}}{\mathbb P}(X_i\le x_i),\qquad (x_1,\ldots,x_d)\in{\mathbb R}^d.$

Properties

A comonotonic R^d-valued random vector $X=(X_1,\ldots,X_d)$ can be represented as

$(X_1,\ldots,X_d)=_\text{d}(F_{X_1}^{-1}(U),\ldots,F_{X_d}^{-1}(U)), \,$

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 non-decreasing functions (or all are non-increasing functions) of the same random variable.^{[citation needed]}

Upper bounds

Fréchet upper bound for cumulative distribution functions

Let $X=(X_1,\ldots,X_d)$ be an R^d-valued random vector. Then, for every i ∈ {1,2,...,d} and x_i ∈ R,

${\mathbb P}(X_1\le x_1,\ldots,X_d\le x_d)\le{\mathbb P}(X_i\le x_i),$

hence

${\mathbb P}(X_1\le x_1,\ldots,X_d\le x_d)\le\min_{i\in\{1,\ldots,d\}}{\mathbb P}(X_i\le x_i),\qquad (x_1,\ldots,x_d)\in{\mathbb R}^d,$

with equality everywhere if and only if $(X_1,\ldots,X_d)$ 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 one-dimensional marginal distributions as (X,Y).^{[clarification needed]} Then it follows from Höffding's formula for the covariance^[6] and the Fréchet upper bound that

$\text{Cov}(X,Y)\le\text{Cov}(X^*,Y^*)$

and, correspondingly,

${\mathbb E}[XY]\le{\mathbb E}[X^*Y^*]$

with equality if and only if (X,Y) is comonotonic.^[7]

Citations

1. ^ (Sriboonchitta et al. 2010, pp. 149–152)
2. ^ (Kaas et al. 2002, Theorem 6)
3. ^ (Kaas et al. 2002, Theorem 7)
4. ^ (McNeil, Frey & Embrechts 2005, Proposition 6.15)
5. ^ (Kaas et al. 2002, Definition 1)
6. ^ (McNeil, Frey & Embrechts 2005, Lemma 5.24)
7. ^ (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 1593-8883, 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 convex-largest 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 0-691-12255-5, 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 0047-259X, MR2557634, Zbl 1184.62081, http://www.parisschoolofeconomics.eu/IMG/pdf/MED090320-Scarsini.pdf 
• Sriboonchitta, Songsak; Wong, Wing-Keung; Dhompongsa, Sompong; Nguyen, Hung T. (2010), Stochastic Dominance and Applications to Finance, Risk and Economics, Boca Raton, FL: Chapman & Hall/CRC Press, ISBN 978-1-4200-8266-1, MR2590381, Zbl 1180.91010, http://books.google.com/books?id=omxatN4lVCkC