A Duality Theory for Set-Valued Functions I: Fenchel Conjugation Theory

Hamel, Andreas H.

doi:10.1007/s11228-009-0109-0

Cited by 67 publications

(136 citation statements)

References 44 publications

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“…This can be done in the framework of conlinear spaces (see [11]). However, it turns out that convexity in such a sense is equivalent to graph convexity as used in the present paper, and also to…”

Section: Definition Of Risk Measures Bymentioning

confidence: 99%

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Duality for Set-Valued Measures of Risk

Hamel¹,

Heyde²

2010

SIAM J. Finan. Math.

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125

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Abstract. Extending the approach of Jouini, Meddeb, and Touzi [Finance Stoch., 8 (2004) 1. Introduction. The concept of set-valued coherent measures of risk has been introduced recently by Jouini, Meddeb, and Touzi [16]. The basic question is how to evaluate the (financial) risk of a multivariate random outcome in terms of more than one reference instrument, for example if the regulator accepts deposits in more than one currency. This is of particular importance if transaction costs have to be paid for each transaction between assets including the reference instruments.This question became unexpectedly topical as the European Central Bank decreed [7] that temporarily, until the end of 2009, "the list of assets eligible as collateral in Eurosystem credit operations will be expanded" by "marketable debt instruments denominated in other currencies than the euro, namely the US dollar, the British pound and the Japanese yen, and issued in the euro area. These instruments will be subject to a uniform haircut add-on of 8%." See also [8].The following exposition is based on the model used in [16]; in particular, we assume the presence of proportional transaction costs modeled via a closed convex cone which goes back to [17]. A special feature of this model is that the risk of a d-dimensional random variable is evaluated in terms of m reference instruments with 1 ≤ m ≤ d. Usually m d will be true, but the case m = d is not excluded (and also, of course, neither m = 1, d > 1, nor m = d = 1).We shall make a few generalizations compared to [16]. First, we do not assume a "substitutability condition"; i.e., we do not assume that everything could be transferred into one distinguished currency. Second, we consider a general subspace M of R d as the collection

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Section: Definition Of Risk Measures Bymentioning

confidence: 99%

“…It should be mentioned that duality theories for set-valued convex functions in terms of Fenchel conjugates are a very recent achievement. The interested reader may consult [11] for related results and references.…”

Section: Interpretation Since the Elements V ∈ Kmentioning

confidence: 99%

Duality for Set-Valued Measures of Risk

Hamel¹,

Heyde²

2010

SIAM J. Finan. Math.

Self Cite

125

142

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“…Let Z : = (Z, ≤) be a preordered vector space with corresponding convex cone K. The preorder ≤ can be extended from Z to P(Z) in at least two canonical ways, see for instance Hamel [6] and the references therein. On the one hand, A B if, and only if, A + K ⊇ B and on the other hand A B if, and only if, A ⊆ B − K for A, B ∈ P(Z).…”

Section: Complete Lattices Of Monotone Setsmentioning

confidence: 99%

“…Finally, for F : X → G(Z, K) we define −G : X * × K • → G(Z, K) as −G(x * , z * ) = inf x∈X {F (x) + S(z * , − x * , x )} . This functional is the set-valued Fenchel-Moreau conjugate introduced in Hamel [6] and can be seen as an analogue to the negative Fenchel-Moreau convex conjugate in the scalar case. However, unlike the Fenchel-Moreau conjugate, it is not an automorphism.…”

Section: Theorem 45 Any Lower Level-closed and Quasiconvex Functionmentioning

confidence: 99%

Complete Duality for Quasiconvex and Convex Set-Valued Functions

Drapeau

Hamel

Kupper

2015

Set-Valued Var. Anal

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“…For the set-valued multivariate coherent and convex risk measures, see Jouini et al [14], 2 Discrete Dynamics in Nature and Society Hamel et al [15], and Ararat et al [16]. For more study about risk measure and set-valued risk measure in recent years, see [3,[17][18][19][20] and the references cited therein.…”

Section: Introductionmentioning

confidence: 99%

Set-Valued Haezendonck-Goovaerts Risk Measure and Its Properties

Dong

Liu

2017

Discrete Dynamics in Nature and Society

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We propose a new set-valued risk measure, which is called set-valued Haezendonck-Goovaerts risk measure. First, we construct the set-valued Haezendonck-Goovaerts risk measure and then provide an equivalent representation. The properties of the set-valued Haezendonck-Goovaerts risk measure are investigated, which show that the set-valued Haezendonck-Goovaerts risk measure is coherent. Finally, an example of set-valued Haezendonck-Goovaerts risk measure is given, which exhibits the fact that the set-valued average value at risk is a particular case of the set-valued Haezendonck-Goovaerts risk measures.

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A Duality Theory for Set-Valued Functions I: Fenchel Conjugation Theory

Cited by 67 publications

References 44 publications

Duality for Set-Valued Measures of Risk

Duality for Set-Valued Measures of Risk

Complete Duality for Quasiconvex and Convex Set-Valued Functions

Set-Valued Haezendonck-Goovaerts Risk Measure and Its Properties

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