Implicit max-stable extremal integrals

Recently, the notion of implicit extreme value distributions has been established, which is based on a given loss function f ≥ 0. From an application point of view, one is rather interested in extreme loss events that occur relative to f than in the corresponding extreme values itself. In this context, so-called f -implicit α -Fréchet max-stable distributions arise and have been used to construct independently scattered sup-measures that possess such margins. In this paper we solve an open problem in Goldbach ( 2016 ) by developing a stochastic integral of a deterministic function g ≥ 0 with respect to implicit max-stable sup-measures. The resulting theory covers the construction of max-stable extremal integrals (see Stoev and Taqqu Extremes 8 , 237–266 ( 2005 )) and, at the same time, reveals striking parallels.