Abstract

We extend to the framework of locally L0L^0-convex modules some results from the classical convex analysis. Namely, randomized versions of the Mazur lemma and the Krein-Šmulian theorem under mild stability properties are provided.

Author information

Contact details are reproduced from the original publication and may be historical.

José M. Zapata

Dpto. Matemáticas, Universidad de Murcia, 30100 Espinardo–Murcia, Spain

jmzg1@um.es

Keywords

  • Locally L0-convex module
  • stability properties
  • Mazur lemma
  • Krein-Šmulian theorem

Mathematics Subject Classification

46H25, 47H40, 52A41, 91B30, 91B70

References

  1. P. Artzner, F. Delbaen, J.-M. Eber, D. Heath: Coherent risk measures, Math. Finance 9 (1999) 203–228.https://doi.org/10.1111/1467-9965.00068Google Scholar
  2. J. Bion-Nadal: Conditional risk measure and robust representation of convex conditional risk measures, CMAP Preprint 557 (2004).Google Scholar
  3. F. Delbaen: Monetary Utility Functions, University of Osaka, Osaka (2011).Google Scholar
  4. F. Delbaen, W. Schachermayer: The Mathematics of Arbitrage, Springer-Verlag, Berlin (2006).Google Scholar
  5. K. Detlefsen, G. Scandolo: Conditional and dynamic convex risk measures, Finance Stoch. 9 (2005) 539–561.https://doi.org/10.1007/s00780-005-0159-6Google Scholar
  6. S. Drapeau, A. Jamneshan, M. Karliczek, M. Kupper: The algebra of conditional sets and the concepts of conditional topology and compactness, J. Math. Anal. Appl. 437(1) (2016) 561–589.https://doi.org/10.1016/j.jmaa.2015.11.057Google Scholar
  7. K.-T. Eisele, S. Taieb: Weak topologies for modules over rings of bounded random variables, J. Math. Anal. Appl. 421(2) (2015) 1334–1357.https://doi.org/10.1016/j.jmaa.2014.07.062Google Scholar
  8. D. Filipovic, M. Kupper, N. Vogelpoth: Separation and duality in locally L0 - convex module, J. Funct. Anal. 256 (2009) 3996–4029.https://doi.org/10.1016/j.jfa.2008.11.015Google Scholar
  9. D. Filipovic, M. Kupper, N. Vogelpoth: Approches to conditional risk, SIAM J. Financial Math. (2012) 402–432.https://doi.org/10.1137/090773076Google Scholar
  10. H. Föllmer, A. Schied: Convex measures of risk and trading constraints, Finance Stoch. 6(4) (2002) 429–447.https://doi.org/10.1007/s007800200072Google Scholar
  11. H. Föllmer, A. Schied: Stochastic Finance: an Introduction in Discrete Time, Walter de Gruyter, Berlin (2011).https://doi.org/10.1515/9783110218053Google Scholar
  12. M. Fritelli, E. Gianin: Putting order in risk measures, J. Banking Finance 26 (2002) 1473–1486.https://doi.org/10.1016/S0378-4266(02)00270-4Google Scholar
  13. T. Guo: The relation of Banach–Alaoglu Theorem and Banach–Bourbaki–Kakutani–Šmulian Theorem in complete random normed modules to stratification structure, Sci. China Ser. A 51 (2008) 1651–1663.https://doi.org/10.1007/s11425-008-0047-6Google Scholar
  14. T. Guo: Relations between some basic results derived from two kids of topologies for a random locally convex module, J. Funct. Anal. 258 (2010) 3024–-3047.https://doi.org/10.1016/j.jfa.2010.02.002Google Scholar
  15. T. Guo, X. Chen: Random duality, Science in China Series A: Mathematics 52(10) (2009) 2084–2098.https://doi.org/10.1007/s11425-009-0149-9Google Scholar
  16. T. Guo, G. Shi: The algebraic structure of finitely generated L0 (F, K)-modules and the Helly theorem in random normed modules, J. Math. Anal. Appl. 381(2) (2011) 833–842.https://doi.org/10.1016/j.jmaa.2011.03.069Google Scholar
  17. T. Guo, H. Xiao, X. Chen: A basic strict separation theorem in random locally convex modules, Nonlinear Anal.: Theory, Methods Appl. 71(9) (2009) 3794–3804.https://doi.org/10.1016/j.na.2009.02.038Google Scholar
  18. T. Guo, S. Zhao, X. Zeng: On random convex analysis – the analytic foundation of the module approach to conditional risk measures, arXiv preprint 1210.1848 (2012).Google Scholar
  19. T. Guo, S. Zhao, X. Zeng: The relations among the three kinds of conditional risk measures, Sci. China Math. 57(8) (2014) 1753–1764.https://doi.org/10.1007/s11425-014-4840-0Google Scholar
  20. T. Guo, S. Zhao, X. Zeng: Random convex analysis (I): separation and Fenchel-Moreau duality in random locally convex modules, arXiv preprint 1503.08695 (2015).Google Scholar
  21. T. Guo, S. Zhao, X. Zeng: Random convex analysis (II): continuity and subdifferentiability theorems in L0 -pre-barreled random locally convex modules, arXiv preprint 1503.08637 (2015).Google Scholar
  22. R. Haydon, M. Levy, Y. Raynaud: Randomly Normed Spaces, Hermann, Paris (1991).Google Scholar
  23. J. Orihuela, J. M. Zapata: Stability in locally L0 -convex modules and a conditional version of James’ compactness theorem, J. Math. Anal. Appl. 452(2) (2017) 1101–1127.https://doi.org/10.1016/j.jmaa.2017.03.048Google Scholar
  24. M. Wu, T. Guo: A counterexample shows that not every locally L0 -convex topology is necessarily induced by a family of L0 -seminorms, arXiv preprint 1501.04400 (2015).Google Scholar
  25. J. M. Zapata: On the characterization of locally L0 -convex topologies induced by a family of L0 -seminorms, J. Convex Analysis 24(2) (2017) 383–391.https://doi.org/10.68381/jca24026Google Scholar

Suggested citation

J. M. Zapata. “Randomized Versions of the Mazur Lemma and the Krein–Šmulian Theorem.” Journal of Convex Analysis 25 (2018), No. 3, 939–956. https://doi.org/10.68381/jca25061

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Published by Heldermann Verlag, 2018. Rights now held by Banach Press LLC.