Complete Duality for Quasiconvex and Convex Set-Valued Functions

Drapeau, Samuel; Hamel, Andreas H.; Kupper, Michael

doi:10.1007/s11228-015-0332-9

Cited by 6 publications

(4 citation statements)

References 15 publications

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“…Nonlinear maps restricted to the family L p (R d ) of p-integrable random vectors and sets having the form of a random vector plus a cone have been studied by Cascos and Molchanov [6] and Hamel and Heyde [12,13]; comprehensive duality results have been proved by Drapeau et al [9]. In our terminology, these studies concern the case when the argument of a superlinear expectation is the sum of a random vector and a convex cone, which in Hamel et al [14] is allowed to be random, but is the same for all random vectors involved.…”

Section: Introductionmentioning

confidence: 99%

“…In our terminology, these studies concern the case when the argument of a superlinear expectation is the sum of a random vector and a convex cone, which in Hamel et al [14] is allowed to be random, but is the same for all random vectors involved. However, for general set-valued arguments, it does not seem possible to rely on the approach of [9,12,13], since the known techniques of set-valued optimisation theory (see e.g. Khan and Tammer [21]) do not suffice to handle functions whose arguments belong to a nonlinear space.…”

Section: Introductionmentioning

confidence: 99%

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Nonlinear expectations of random sets

Molchanov

Mühlemann

2020

Finance Stoch

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Sublinear functionals of random variables are known as sublinear expectations; they are convex homogeneous functionals on infinite-dimensional linear spaces. We extend this concept for set-valued functionals defined on measurable set-valued functions (which form a nonlinear space) or, equivalently, on random closed sets. This calls for a separate study of sublinear and superlinear expectations, since a change of sign does not alter the direction of the inclusion in the set-valued setting.We identify the extremal expectations as those arising from the primal and dual representations of nonlinear expectations. Several general construction methods for nonlinear expectations are presented and the corresponding duality representation results are obtained. On the application side, sublinear expectations are naturally related to depth trimming of multivariate samples, while superlinear ones can be used to assess utilities of multiasset portfolios.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Nonlinear expectations of random sets

Molchanov

Mühlemann

2020

Finance Stoch

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“…Nonlinear maps restricted to the family L p (R d ) of p-integrable random vectors have been studied in [4,9], the comprehensive duality results can be found in [7]. In our framework, these studies concern the cases when the argument of a superlinear expectation is the sum of a random vector and a convex cone.…”

Section: Introductionmentioning

confidence: 99%

“…In our framework, these studies concern the cases when the argument of a superlinear expectation is the sum of a random vector and a convex cone. However, for general set-valued arguments, it is not possible to rely the approach of [9,7], since the known techniques of set-valued optimisation theory (see, e.g., [16]) are not applicable.…”

Section: Introductionmentioning

confidence: 99%

Expectations of Random Sets

Molchanov

2017

Probability Theory and Stochastic Modelling

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Sublinear functionals of random variables are known as sublinear expectations; they are convex homogeneous functionals on infinite-dimensional linear spaces. We extend this concept for set-valued functionals defined on measurable set-valued functions (which form a nonlinear space), equivalently, on random closed sets. This calls for a separate study of sublinear and superlinear expectations, since a change of sign does not convert one to the other in the set-valued setting.We identify the extremal expectations as those arising from the primal and dual representations of them. Several general construction methods for nonlinear expectations are presented and the corresponding duality representation results are obtained. On the application side, sublinear expectations are naturally related to depth trimming of multivariate samples, while superlinear ones can be used to assess utilities of multiasset portfolios.

show abstract