Multi-utility representations of incomplete preferences induced by set-valued risk measures

Abstract

We establish a variety of numerical representations of preference relations induced by set-valued risk measures. Because of the general incompleteness of such preferences, we have to deal with multi-utility representations. We look for representations that are both parsimonious (the family of representing functionals is indexed by a tractable set of parameters) and well behaved (the representing functionals satisfy nice regularity properties with respect to the structure of the underlying space of alternatives). The key to our results is a general dual representation of set-valued risk measures that unifies the existing dual representations in the literature and highlights their link with duality results for scalar risk measures.

Set-Valued T-Translative Functions and Their Applications in Finance

“…First, a type of extended real-valued function is introduced, which will provide scalarizing functions for set-valued T-translative functions. Special cases of such functions were considered in [2,39], but the first reference in a finance context seems to be ( [17], Formula (CR), p. 590).…”

“…3.4), this property for T = id Z and Z a linear subspace of X was called translation invariance with respect to (Z, −z * ). In ( [39], Prop. 4.2), it was called "translative along M", where M ⊆ X corresponds to TZ in Definition 7.…”

“…A set-valued convex analysis based on such representations via families of scalar functions can be found in [41] and a more general approach in [42]. The properties of such representations within a finance context were also discussed in [39].…”