Modelling dependence in finance using copulas - Thierry Roncalli's ...

2.3 Copulas in a nutshell
A copula function C is a multivariate probability distribution with
uniform [0, 1] margins.
C (F 1 (x 1 ) , . . . , F N (x N )) defines a multivariate cdf F with margins
F 1 , . . . , F N ⇒ F is a probability distribution with given marginals.
The copula function of the random variables (X 1 , . . . , X N ) is
invariant under strictly increasing transformations (∂ x h n (x) > 0):
C 〈X 1 , . . . , X N 〉 = C 〈h 1 (X 1 ) , . . . , h N (X N )〉
... the copula is invariant while the margins may be changed at will,
it follows that is precisely the copula which captures those properties
of the joint distribution which are invariant under a.s. strickly
increasing transformations (Schweizer and Wolff [1981]).
⇒ Copula = dependence function of r.v. (Deheuvels [1978]).
Modelling dependence in finance using copulas
Copulas and multivariate financial models 2-4