Unbounded order convergence and application to martingales without probability

Gao, Niushan; Xanthos, Foivos

doi:10.1016/j.jmaa.2014.01.078

Cited by 84 publications

(121 citation statements)

References 15 publications

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“…Indeed, if C is a convex set in X and

x \in {\bar{C}}^{w}

, then by Mazur's theorem,

x \in {\bar{C}}^{w} = {\bar{C}}^{∥ \cdot ∥}

, so that there exists a sequence

(x_{n})

in C such that

false∥ x_{n} - x false∥ \to 0

. Now Gao and Xanthos (, lemma 3.11) or Biagini and Frittelli (, lemma 4) yields a subsequence

(x_{n_{k}})

(x_{n})

such that

x_{n_{k}} \overset{o}{\to} x

in X .…”

Section: Resultsmentioning

confidence: 98%

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On the C‐property and ‐representations of risk measures

Gao

Xanthos

2017

Mathematical Finance

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Abstract. We identify a large class of Orlicz spaces L Φ (µ) for which the topology. We also establish a variant of the C-property and use it to prove a w * -representation theorem for proper convex increasing functionals, satisfying a suitable version of Delbaen's Fatou property, on Orlicz spaces L Φ (µ) with lim t→∞ Φ(t) t = ∞. Our results apply, in particular, to risk measures on all Orlicz spaces L Φ (P) other than L 1 (P).

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“…Indeed, if C is a convex set in X and

x \in {\bar{C}}^{w}

, then by Mazur's theorem,

x \in {\bar{C}}^{w} = {\bar{C}}^{∥ \cdot ∥}

, so that there exists a sequence

(x_{n})

in C such that

false∥ x_{n} - x false∥ \to 0

. Now Gao and Xanthos (, lemma 3.11) or Biagini and Frittelli (, lemma 4) yields a subsequence

(x_{n_{k}})

(x_{n})

such that

x_{n_{k}} \overset{o}{\to} x

in X .…”

Section: Resultsmentioning

confidence: 98%

“…Finally, we remark that most of our results hold in the general framework of Banach lattices; here we need to replace a.e. convergence with the notion of unbounded order convergence, which was recently developed in Gao (2014), Gao and Xanthos (2014), and Gao, Troitsky, and Xanthos (2017).…”

Section: Resultsmentioning

confidence: 99%

On the C‐property and ‐representations of risk measures

Gao

Xanthos

2017

Mathematical Finance

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“…As standard references for basic notions on vector lattices we adopt the books [1,10,14] and on unbounded order convergence the paper [8]. In this article, all vector lattices are assumed to be Archimedean.…”

Section: Theorem 11 (Brezis-lieb Lemma [4 Theorem 2])mentioning

confidence: 99%

The Brezis-Lieb lemma in convergence vector lattices

2018

Turk J Math

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“…The section starts with a counterexample to a question posed in [LC], and culminates in a proof that a Banach lattice is (sequentially) boundedly uo-complete iff it is (sequentially) monotonically complete. This gives the final solution to a problem that has been investigated in [Gao14], [GX14], [GTX17], and [GLX].…”

Section: Introductionmentioning

confidence: 99%

Completeness of unbounded convergences

Taylor

2018

Proc. Amer. Math. Soc.

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Abstract. As a generalization of almost everywhere convergence to vector lattices, unbounded order convergence has garnered much attention. The concept of boundedly uo-complete Banach lattices was introduced by N. Gao and F. Xanthos, and has been studied in recent papers by D. Leung, V.G. Troitsky, and the aforementioned authors. We will prove that a Banach lattice is boundedly uo-complete iff it is monotonically complete. Afterwards, we study completeness-type properties of minimal topologies; minimal topologies are exactly the Hausdorff locally solid topologies in which uo-convergence implies topological convergence.

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Unbounded order convergence and application to martingales without probability

Cited by 84 publications

References 15 publications

On the C‐property and ‐representations of risk measures

On the C‐property and ‐representations of risk measures

The Brezis-Lieb lemma in convergence vector lattices

Completeness of unbounded convergences

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