We analyze copulas with a nontrivial singular component by using their Markov kernel representation. In particular, we provide existence results for copulas with a prescribed singular component. The constructions not only help to deal with problems related to multivariate stochastic systems of lifetimes when joint defaults can occur with a nonzero probability, but even provide a copula maximizing the probability of joint default.
"On the singular components of a copula." J. Appl. Probab. 52 (4) 1175 - 1182, December 2015. https://doi.org/10.1239/jap/1450802760