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# 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.

The concept of comonotonicity has applications in financial risk management and actuarial science. In particular, the sum of the components X1 + X2 + ... + Xd is the riskiest if the joint probability distribution of the random vector (X1,X2,...,Xd) is comonotonic. 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.

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

## Definitions

### Comonotonicity of subsets of Rd

A subset S of Rd is called comonotonic if, for all (x1,x2,...,xd) and (y1,y2,...,yd) in S with xi < yi for some i ∈ {1,2,...,d}, it follows that xj ≤ yj for all j ∈ {1,2,...,d}.

This means that S is a totally ordered set.

### Comonotonicity of probability measures on Rd

Let μ be a probability measure on the d-dimensional Euclidean space Rd 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,...,Fd 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 Rd-valued random vectors

An Rd-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 Rd-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 Rd-valued random vector. Then, for every i ∈ {1,2,...,d} and xi ∈ 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 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.

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