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. In two dimensions it is also possible to consider perfect negative dependence, which 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, see e.g. Template:Harvtxt and Template:Harvtxt. In particular, the sum of the components Template:Math is the riskiest if the joint probability distribution of the random vector Template:Math is comonotonic. Furthermore, the Template:Math-quantile of the sum equals the sum of the Template:Math-quantiles of its components, hence comonotonic random variables are quantile-additive. In practical risk management terms it means that there is minimal (or eventually no) variance reduction from diversification.

For extensions of comonotonicity, see Template:Harvtxt and Template:Harvtxt.

Definitions

Comonotonicity of subsets of Template:Math

A subset Template:Math of Template:Math is called comonotonic (sometimes also nondecreasing) if, for all Template:Math and Template:Math in Template:Math with Template:Math for some Template:Math}, it follows that Template:Math for all Template:Math}.

This means that Template:Math is a totally ordered set.

Comonotonicity of probability measures on Template:Math

Let Template:Math be a probability measure on the Template:Math-dimensional Euclidean space Template:Math and let Template:Math denote its multivariate cumulative distribution function, that is

$F(x_1,\dots ,x_n):=\mu (\{(y_1,\dots ,y_n)\in {ℝ}^n\mid y_1\le x_1,\dots ,y_n\le x_n\}),(x_1,\dots ,x_n)\in {ℝ}^n.$

Furthermore, let Template:Math denote the cumulative distribution functions of the Template:Math one-dimensional marginal distributions of Template:Math, that means

$F_i(x):=\mu (\{(y_1,\dots ,y_n)\in {ℝ}^n\mid y_i\le x\}),x\in ℝ$

for every Template:Math}. Then Template:Math is called comonotonic, if

$F(x_1,\dots ,x_n)=\underset{i\in \{1,\dots ,n\}}{\min }{F}_i(x_i),(x_1,\dots ,x_n)\in {ℝ}^n.$

Note that the probability measure Template:Math is comonotonic if and only if its support Template:Math is comonotonic according to the above definition.

Comonotonicity of Template:Math-valued random vectors

An Template:Math-valued random vector Template:Math is called comonotonic, if its multivariate distribution (the pushforward measure) is comonotonic, this means

$\mathrm{Pr}(X_1\le x_1,\dots ,X_n\le x_n)=\underset{i\in \{1,\dots ,n\}}{\min }\mathrm{Pr}(X_i\le x_i),(x_1,\dots ,x_n)\in {ℝ}^n.$

Properties

An Template:Math-valued random vector Template:Math is comonotonic if and only if it can be represented as

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

where Template:Math stands for equality in distribution, on the right-hand side are the left-continuous generalized inverses of the cumulative distribution functions Template:Math, and Template:Math is a uniformly distributed random variable on the unit interval. More generally, a random vector is comonotonic if and only if it agrees in distribution with a random vector where all components are non-decreasing functions (or all are non-increasing functions) of the same random variable.

Upper bounds

Upper Fréchet–Hoeffding bound for cumulative distribution functions

Let Template:Math be an Template:Math-valued random vector. Then, for every Template:Math},

$\mathrm{Pr}(X_1\le x_1,\dots ,X_n\le x_n)\le \mathrm{Pr}(X_i\le x_i),(x_1,\dots ,x_n)\in {ℝ}^n,$

hence

$\mathrm{Pr}(X_1\le x_1,\dots ,X_n\le x_n)\le \underset{i\in \{1,\dots ,n\}}{\min }\mathrm{Pr}(X_i\le x_i),(x_1,\dots ,x_n)\in {ℝ}^n,$

with equality everywhere if and only if Template:Math is comonotonic.

Upper bound for the covariance

Let Template:Math be a bivariate random vector such that the expected values of Template:Math, Template:Math and the product Template:Math exist. Let Template:Math be a comonotonic bivariate random vector with the same one-dimensional marginal distributions as Template:Math. Then it follows from Höffding's formula for the covariance and the upper Fréchet–Hoeffding bound that

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

and, correspondingly,

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

with equality if and only if Template:Math is comonotonic.

Note that this result generalizes the rearrangement inequality and Chebyshev's sum inequality.

See also

• Copula (probability theory)

References

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