Shock models with recovery option via the maxmin copulas

Ever since the historical Sklar's theorem in 1959 copulas have been one of the main tools in modelling dependence of random variables. With the range of applications in applied mathematics expanding and varying from mathematics of finance through system theory to fuzzy set theory, there is a growing need for new types of copulas that could serve as appropriate models in these applications. It is our aim to turn a new page in constructing copulas by setting a counterpart to the famous Marshall copulas (an extension of Marshall–Olkin copulas) that are typically applied to model lifetime of a two-component system. Even a small but essential change of assumptions that the model is applied to such a system with one of the components having a recovery option turns into a substantially different problem on the level of copulas. We give a full study of the augmented case by introducing a new type of copulas, called maxmin, together with concrete applications in server-side web framework, in an economic model and in fuzzy set theory.