We apply the Malliavin calculus to study several non-degeneracy conditions on the coefficients of a stochastic differential equation on the plane, in order to deduce the existence and smoothness of density for the law of the solution.

We consider copulas with a given diagonal section and compute the explicit density of the unique optimal copula which maximizes the entropy. In this sense, this copula is the least informative among the copulas with a given diagonal section. We give an explicit criterion on the diagonal section...