The Kreps‐Yan theorem for L∞

Rokhlin, Dmitry B.

doi:10.1155/ijmms.2005.2749

Cited by 20 publications

(15 citation statements)

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“…Lemma 3 in the case X = L ∞ (P) is proved in [7, Lemma 2.5]. We should note that this claim is true also for a cone C that is not closed.…”

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confidence: 86%

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The Kreps-Yan theorem for Banach ideal spaces

Rokhlin

2009

Sib Math J

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Consider a closed convex cone C in a Banach ideal space X on some measure space with σ-finite measure. We prove that the fulfilment of the conditions C ∩X + = {0} and C ⊃ −X + guarantees the existence of a strictly positive continuous functional on X whose restriction to C is nonpositive.Given a complete measure space (Ω, F ) with (a countably additive function) measure μ : F → [0, ∞], consider the vector space L 0 (μ) = L 0 (Ω, F , μ) of the classes of μ-equivalent almost everywhere finite Fmeasurable functions. This space is a vector lattice (Riesz space) with respect to the natural order generated by the cone L 0 + (μ) of nonnegative elements [1,2]. Take some solid subspace (or an ideal) X of L 0 (μ), which means that X is a linear subset of L 0 (μ) such that x ∈ X and |y| ≤ |x| implies that y ∈ X. Suppose that X is equipped with a norm satisfying x ≤ y whenever |x| ≤ |y| for x, y ∈ X (a monotone norm) and that X is complete with respect to this norm. In this case (X, · ) is called [1, 3] a Banach ideal space on (Ω, F , μ).Take a Banach ideal space X with the cone X + = {x ∈ X : x ≥ 0} of nonnegative elements. An element g of the topologically dual space X is called strictly positive whenever x, g := g(x) > 0 for x ∈ X + \{0}. Somewhat modifying the terminology of [4], we will say that X possesses the Kreps-Yan property whenever given a closed convex cone C ⊂ X such thatthere is a strictly positive element g ∈ X satisfying x, g ≤ 0 for x ∈ C.The problem of the existence of g of this type can be posed in a broader context of locally convex spaces with a cone. Some results in this direction, as well as further references and commentaries to the articles of Kreps [5] and Yan [6] can be found in [4].Say that a topological space (X, τ ) possesses the Lindelöf property whenever each τ -open covering of X has a countable subcovering. It is known that if the weak topology σ(X, X ) on X possesses the Lindelöf property (for brevity, X is called weakly Lindelöf) and the set of strictly positive functionals g ∈ X is nonempty then X possesses the Kreps-Yan property [7, Theorem 1.

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“…Lemma 3 in the case X = L ∞ (P) is proved in [7, Lemma 2.5]. We should note that this claim is true also for a cone C that is not closed.…”

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confidence: 86%

“…The general strategy of the proof follows [7] whose arguments are adjusted to L ∞ . We prove first the existence of a strictly positive element f ∈ X bounded above on {x ∈ C : x − ≤ 1} (Lemma 1).…”

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confidence: 99%

The Kreps-Yan theorem for Banach ideal spaces

Rokhlin

2009

Sib Math J

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“…The role of the existence of an unconditional basic sequence in a Banach space is also quoted in this section independently from the results provided in [8], as an important condition for the extraction of results concerning FTAP. This condition is not irrelevant to ( [9], Th. 1.1), about Lindelöf Properties of weak topology, but here it mainly concerns the construction of a Strictly Positive Projection Operator.…”

Section: Some Remarks On Previous Work About the Fundamental Asset Prmentioning

confidence: 97%

On the Order Form of the Fundamental Theorems of Asset Pricing

Kountzakis¹

2014

JMF

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In this article, we provide an order-form of the First and the Second Fundamental Theorem of Asset Pricing both in the one-period market model for a finite and infinite state-space and in the case of multi-period model for a finite state-space and a finite time-horizon. The space of the financial positions is supposed to be a Banach lattice. We also prove relevant results in the case where the space of the financial positions is not ordered by a lattice cone. Keywords Strictly Positive Extension, Positive Projection, Sublattice, Complete Market, Incomplete Market1 J − ∆ of J  (note that J µ ≥ ). Suppose that the first J vectors of this set are linearly independent. If we suppose that the vectors , 1, 2, , J i y i J

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“…Thus, to look for finitely additive EMM's seems to make sense; see Berti et al (2013Berti et al ( , 2014a, Cassese (2007), Rokhlin (2005) and Rokhlin and Schachermayer (2006).…”

Section: Introductionmentioning

confidence: 99%