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Closedness of convex sets in Orlicz spaces with applications to dual representation of risk measures

Author

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  • Niushan Gao
  • Denny H. Leung
  • Foivos Xanthos

Abstract

Let $(\Phi,\Psi)$ be a conjugate pair of Orlicz functions. A set in the Orlicz space $L^\Phi$ is said to be order closed if it is closed with respect to dominated convergence of sequences of functions. A well known problem arising from the theory of risk measures in financial mathematics asks whether order closedness of a convex set in $L^\Phi$ characterizes closedness with respect to the topology $\sigma(L^\Phi,L^\Psi)$. (See [26, p.3585].) In this paper, we show that for a norm bounded convex set in $L^\Phi$, order closedness and $\sigma(L^\Phi,L^\Psi)$-closedness are indeed equivalent. In general, however, coincidence of order closedness and $\sigma(L^\Phi,L^\Psi)$-closedness of convex sets in $L^\Phi$ is equivalent to the validity of the Krein-Smulian Theorem for the topology $\sigma(L^\Phi,L^\Psi)$; that is, a convex set is $\sigma(L^\Phi,L^\Psi)$-closed if and only if it is closed with respect to the bounded-$\sigma(L^\Phi,L^\Psi)$ topology. As a result, we show that order closedness and $\sigma(L^\Phi,L^\Psi)$-closedness of convex sets in $L^\Phi$ are equivalent if and only if either $\Phi$ or $\Psi$ satisfies the $\Delta_2$-condition. Using this, we prove the surprising result that: \emph{If (and only if) $\Phi$ and $\Psi$ both fail the $\Delta_2$-condition, then there exists a coherent risk measure on $L^\Phi$ that has the Fatou property but fails the Fenchel-Moreau dual representation with respect to the dual pair $(L^\Phi, L^\Psi)$}. A similar analysis is carried out for the dual pair of Orlicz hearts $(H^\Phi,H^\Psi)$.

Suggested Citation

  • Niushan Gao & Denny H. Leung & Foivos Xanthos, 2016. "Closedness of convex sets in Orlicz spaces with applications to dual representation of risk measures," Papers 1610.08806, arXiv.org, revised Jun 2017.
  • Handle: RePEc:arx:papers:1610.08806
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    References listed on IDEAS

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    1. Freddy Delbaen, 2009. "Risk Measures For Non‐Integrable Random Variables," Mathematical Finance, Wiley Blackwell, vol. 19(2), pages 329-333, April.
    2. Keita Owari, 2013. "Maximum Lebesgue Extension of Monotone Convex Functions," Papers 1304.7934, arXiv.org, revised Jan 2014.
    3. Elyès Jouini & Walter Schachermayer & Nizar Touzi, 2006. "Law Invariant Risk Measures Have the Fatou Property," Post-Print halshs-00176522, HAL.
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    7. repec:dau:papers:123456789/342 is not listed on IDEAS
    8. Patrick Cheridito & Tianhui Li, 2009. "Risk Measures On Orlicz Hearts," Mathematical Finance, Wiley Blackwell, vol. 19(2), pages 189-214, April.
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    Cited by:

    1. Shengzhong Chen & Niushan Gao & Foivos Xanthos, 2018. "The strong Fatou property of risk measures," Papers 1805.05259, arXiv.org.
    2. Chen Shengzhong & Gao Niushan & Xanthos Foivos, 2018. "The strong Fatou property of risk measures," Dependence Modeling, De Gruyter, vol. 6(1), pages 183-196, October.
    3. Freddy Delbaen & Keita Owari, 2016. "Convex functions on dual Orlicz spaces," Papers 1611.06218, arXiv.org, revised Dec 2017.

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