# Implicit max-stable extremal integrals

## TL;DR

This paper develops a stochastic integral framework for implicit max-stable extremal integrals, extending existing max-stable extremal integral theory and addressing an open problem in the field.

## Contribution

It introduces a stochastic integral with respect to implicit max-stable sup-measures, bridging the gap between implicit extreme value distributions and extremal integrals.

## Key findings

- Constructed a stochastic integral for implicit max-stable sup-measures.
- Revealed parallels between implicit and classical max-stable extremal integrals.
- Solved an open problem in the theory of max-stable extremal integrals.

## Abstract

Recently, the notion of implicit extreme value distributions has been established, which is based on a given loss function $f \ge 0$. 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 $\alpha$-Fr\'{e}chet 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 [7] by developing a stochastic integral of a deterministic function $g\ge 0$ with respect to implicit max-stable sup-measures. The resulting theory covers the construction of max-stable extremal integrals (see [14]) and, at the same time, reveals striking parallels.