On the Construction of Semiquadratic Copulas

We introduce several classes of semiquadratic copulas (i.e. copulas that are quadratic in at least one coordinate of any point of the unit square) of which the diagonal section or the opposite diagonal section are given functions. These copulas are constructed by quadratic interpolation on segments connecting the diagonal (resp. opposite diagonal) of the unit square to the boundaries of the unit square. We provide for each class the necessary and sufficient conditions on a diagonal (resp. opposite diagonal) function and two auxiliary real functions f and g to obtain a copula which has this diagonal (resp. opposite diagonal) function as diagonal (resp. opposite diagonal) section.