Conditional Interior and Conditional Closure of Random Sets
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Conditional Interior and Conditional Closure of Random Sets Meriam El Mansour1,2 · Emmanuel Lépinette1 Received: 23 January 2020 / Accepted: 9 October 2020 / Published online: 20 October 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In this paper, we introduce two new types of conditional random set taking values in a Banach space: the conditional interior and the conditional closure. The conditional interior is a version of the conditional core, as introduced by A. Truffert and recently developed by Lépinette and Molchanov, and may be seen as a measurable version of the topological interior. The conditional closure is a generalization of the notion of conditional support of a random variable. These concepts are useful for applications in mathematical finance and conditional optimization. Keywords Conditional random set · Conditional optimization · Super-hedging problem · European option · Mathematical finance Mathematics Subject Classification 52-02 · 60D05 · 46N10 · 60-02 · 54-02 · 91-02
1 Introduction The conditional essential supremum and infimum of a real-valued random variable have been introduced in [1]. A generalization is then proposed in [2] for vector-valued random variables with respect to random preference relations. Actually, these two concepts are related to the notion of conditional core as first introduced in [3]1 and developed in [4] for random sets in separable Banach spaces, with respect to a complete 1 E. Lépinette apologizes to A. Truffert for not having quoted her paper [4].
Communicated by Michel Théra.
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Emmanuel Lépinette [email protected] Meriam El Mansour [email protected]
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Paris Dauphine University, PSL Research University, Paris, France
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Gosaef, Faculté des Sciences de Tunis, Tunis, Tunisia
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Journal of Optimization Theory and Applications (2020) 187:356–369
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σ -algebra H. A conditional core of a set-valued mapping Γ (ω), ω ∈ Ω, is defined as the largest H-graph measurable random set Γ (ω) such that Γ (ω) ⊂ Γ (ω). This concept provides a natural conditional risk measure that generalizes the concept of essential infimum for multi-asset portfolios in mathematical finance. Applications are deduced for geometrical market models with transaction costs: see [4,5] and the theory with transaction costs developed in [6]. In this paper, we first introduce the open version of the conditional core as proposed in [3,4]. Precisely, if H is a complete sub-σ -algebra on a probability space, the conditional interior (or open conditional core)of a set-valued mapping Γ (ω), ω ∈ Ω, is defined as the largest H-measurable random open set Γ (ω) such that Γ (ω) ⊂ Γ (ω) P-almost every ω ∈ Ω. It may be seen as a measurable version of the classical interior in topology. One of our main contributions is to show the existence and uniqueness of such a conditional interior for an arbitrary random set in a separable Banach space. Then, the dual concept, the conditional closure, is introduced as a generalization of the co
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