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 + ... + Xd is the riskiest if the joint probability distribution of the random vector (X1,X2,...,Xd) 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 Rd

A subset S of Rd is called comonotonic[5] 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[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

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