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

Andreas H. Hamel and Frank Heyde
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Andreas H. Hamel: Faculty of Economics and Management, Free University of Bozen-Bolzano, I-39031 Bruneck-Brunico, Italy
Frank Heyde: Faculty of Mathematics and Computer Science, Freiberg University of Mining and Technology, 09596 Freiberg, Germany

Mathematics, 2021, vol. 9, issue 18, 1-33

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.

Keywords: T -translative functions; set-valued functions; complete lattice; risk measures; dual representation; random sets (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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