Coherence and avoidance of sure loss for standardized functions and semicopulas

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,…,1). We characterize the existence of a k-increasing n-variate function C fulfilling A⩽C⩽B for standardized n-variate functions A,B and discuss methods for constructing such functions. 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⩽C⩽B.