Abstract
This paper presents relations between several types of closedness of a law-invariant convex set in a rearrangement invariant space \({\mathcal {X}}\). In particular, we show that order closedness, \(\sigma ({\mathcal {X}},{\mathcal {X}}_n^\sim )\)-closedness, and \(\sigma ({\mathcal {X}},L^\infty )\)-closedness of a law-invariant convex set in \({\mathcal {X}}\) are equivalent, where \({\mathcal {X}}_n^\sim \) is the order continuous dual of \({\mathcal {X}}\). We also provide some application to proper quasiconvex law-invariant functionals with the Fatou property.
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Notes
In [1], all Banach function spaces \({\mathcal {X}}\) are assumed to satisfy the following condition: \(f\in {\mathcal {X}}\) and \(\Vert f\Vert _{{\mathcal {X}}}=\sup _n\Vert f_n\Vert _{{\mathcal {X}}}\) whenever \(\{f_n\}\) is an increasing norm bounded sequence in \({\mathcal {X}}_+\) and f is the pointwise limit of \(\{f_n\}\). The results we cite from [1] remain true without assuming this extra condition.
References
Bennett, C., Sharpley, R.: Interpolation of Operators. Academic Press Inc., Boston (1988)
Chen, S., Gao, N., Xanthos, F.: The strong Fatou property of risk measures. Depend. Model. 6, 183–196 (2018)
Delbaen, F.: Coherent risk measures on general probability spaces. In: Sandmann, Klaus, Schönbucher, Philip J. (eds.) Advances in Finance and Stochastics, pp. 1–37. Springer, Berlin (2002)
Gao, N., Leung, D., Munari, C., Xanthos, F.: Fatou property, representation, and extension of law-invariant risk measures on general Orlicz spaces. Finance Stoch. 22, 395–415 (2018)
Gao, N., Leung, D., Xanthos, F.: Closedness of convex sets in Orlicz spaces with applications to dual representation of risk measures. Studia Math. 249, 329–347 (2019)
Gao, N., Leung, D., Xanthos, F.: Duality for unbounded order convergence and applications. Positivity 22, 711–725 (2018)
Gao, N., Xanthos, F.: On the C-property and \(w^*\)-representations of risk measures. Math. Finance 28, 748–754 (2018)
Jouini, E., Schachermayer, W., Touzi, N.: Law invariant risk measures have the Fatou property. Adv. Math. Econ. 9, 49–71 (2006)
Krein, S.G., Petunin, Y.J., Semenov, E.M.: Interpolation of Linear Operators. Translations of Mathematical Monographs, vol. 54. American Mathematical Society, Providence (1982)
Meyer-Nieberg, P.: Banach Lattices. Universitext. Springer, Berlin (1991)
Owari, K.: Maximum Lebesgue extension of monotone convex functions. J. Func. Anal. 266, 3572–3611 (2014)
Schaefer, H.H.: Banach Lattices and Positive Operators. Springer, Berlin (1974)
Svindland, G.: Continuity properties of law-invariant (quasi-)convex risk functions on \(L^\infty \). Math. Finance Econ. 3, 39–43 (2010)
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Made Tantrawan is supported by NUS Research Scholarship. Denny H. Leung is partially supported by AcRF Grant R-146-000-242-114.
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Tantrawan, M., Leung, D.H. On closedness of law-invariant convex sets in rearrangement invariant spaces. Arch. Math. 114, 175–183 (2020). https://doi.org/10.1007/s00013-019-01398-3
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DOI: https://doi.org/10.1007/s00013-019-01398-3