Several algorithms for constructing copulas via ⁎-product decompositions

For two given measure-preserving functions defined on the unit interval , the function given by is a copula. Although the theoretical problem for constructing this copula is completely solved, in practice it is a rather difficult task. The principal problem is the reverse implication (that is, to prove that and are measure-preserving when is a copula). We provide new proof of this fact with a technique that is far from the previous ones already known in the literature. Indeed, finding two measure-preserving functions and , such that , for a given , is equivalent to a suitable decomposition of such copula in the form ( ⁎), where id denotes the identity function. We also provide explicit algorithms which solve this problem in various contexts such as the measure preserving functions and are monotonic, as well as the copula is a diagonal copula, an extreme copula, an extremal biconic copula, an Archimedean copula, a conic copula, a copula invariant under truncations, or an -migrative copula.