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. and . In particular, the sum of the components X1 + X2 + · · · + Xn is the riskiest if the joint probability distribution of the random vector (X1, X2, . . . , Xn) 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. In practical risk management terms it means that there is minimal (or eventually no) variance reduction from diversification. For extensions of comonotonicity, see and . A subset S of Rn is called comonotonic (sometimes also nondecreasing) if, for all (x1, x2, . . . , xn) and (y1, y2, . . . , yn) in S with xi < yi for some i ∈ {1, 2, . . . , n}, it follows that xj ≤ yj for all j ∈ {1, 2, . . . , n}. This means that S is a totally ordered set. Let μ be a probability measure on the n-dimensional Euclidean space Rn and let F denote its multivariate cumulative distribution function, that is Furthermore, let F1, . . . , Fn denote the cumulative distribution functions of the n one-dimensional marginal distributions of μ, that means for every i ∈ {1, 2, . . . , n}. Then μ is called comonotonic, if Note that the probability measure μ is comonotonic if and only if its support S is comonotonic according to the above definition. An Rn-valued random vector X = (X1, . . . , Xn) is called comonotonic, if its multivariate distribution (the pushforward measure) is comonotonic, this means An Rn-valued random vector X = (X1, . . .