Coherence and avoidance of sure loss for standardized functions and semicopulas

TL;DR

This paper investigates the conditions under which semicopulas and standardized functions are coherent and avoid sure loss, providing characterizations and construction methods for such functions within the framework of multivariate functions.

Contribution

It offers new characterizations and construction procedures for $k$-increasing functions bounded by given standardized functions, extending to functions on infinite meshes.

Findings

Characterization of the existence of $k$-increasing functions between bounds.

Methods for extending functions from countable meshes to the unit box.

Conditions under which bounds coincide with infimum or supremum of feasible functions.

Abstract

We discuss avoidance of sure loss and coherence results for semicopulas and standardized functions, i.e., for grounded, 1-increasing functions with value $1$ at $(1,1,\ldots, 1)$. We characterize the existence of a $k$-increasing $n$-variate function $C$ fulfilling $A\leq C\leq B$ for standardized $n$-variate functions $A,B$ and discuss the method for constructing this function. Our proofs also include procedures for extending functions on some countably infinite mesh to functions on the unit box. We provide a characterization when $A$ respectively $B$ coincides with the pointwise infimum respectively supremum of the set of all $k$-increasing $n$-variate functions $C$ fulfilling $A\leq C\leq B$.