Abstract
A theory for set-valued functions is developed, which are translative with respect to a linear operator. It is shown that such functions cover a wide range of applications, from projections in Hilbert spaces, set-valued quantiles for vector-valued random variables, to scalar or set-valued risk measures in finance with defaultable or nondefaultable securities. Primal, dual, and scalar representation results are given, among them an infimal convolution representation, which is not so well known even in the scalar case. Along the way, new concepts of set-valued lower/upper expectations are introduced and dual representation results are formulated using such expectations. An extension to random sets is discussed at the end. The principal methodology consisted of applying the complete lattice framework of set optimization.
Highlights
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Set-valued risk measures for multivariate financial positions [1,2,3,4] make sense if there is more than one asset eligible for risk-compensating deposits or as an accounting unit
Typical situation is Kabanov’s model of a multicurrency market with frictions [5]. Objects such as superhedging prices turn into sets of superhedging portfolios since the operation of taking the infimum or supremum can no longer be performed in a meaningful way if applied to sets in IRd with respect to the order relations generated, e.g., by solvency cones
Summary
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. A typical situation is Kabanov’s model of a multicurrency market with frictions [5] In such market models, objects such as superhedging prices turn into sets of superhedging portfolios since the operation of taking the infimum or supremum can no longer be performed in a meaningful way if applied to sets in IRd with respect to the order relations generated, e.g., by solvency cones. A general framework for set-valued translative functions and their representation by scalar families is provided, and it is shown that it does cover set-valued risk measures, and set-valued lower and upper expectations, projections, aggregation mappings, and set-valued quantiles for multivariate positions.
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