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On the Lebesgue property of monotone convex functions

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Abstract

The Lebesgue property (order-continuity) of a monotone convex function on a solid vector space of measurable functions is characterized in terms of (1) the weak inf-compactness of the conjugate function on the order-continuous dual space, (2) the attainment of the supremum in the dual representation by order-continuous linear functionals. This generalizes and unifies several recent results obtained in the context of convex risk measures.

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References

  1. Aliprantis, C.D., Border, K.C.: Infinite Dimensional Analysis: a Hitchhiker’s Guide, 3rd edn. Springer, Berlin (2006)

    Google Scholar 

  2. Aliprantis, C.D., Burkinshaw, O.: Locally solid Riesz spaces with applications to economics. In: Mathematical Surveys and Monographs, 2nd ed, vol. 105, pp. xii+344. American Mathematical Society, Providence (2003)

  3. Biagini, S., Frittelli, M.: On the extension of the Namioka-Klee theorem and on the Fatou property for risk measures. In: Optimality and Risk–Modern Trends in Mathematical Finance, pp. 1–28. Springer, Berlin (2009)

  4. Bourbaki, N.: General topology. Chapters 1–4. In: Elements of Mathematics (Berlin). Springer, Berlin. Translated from the French, Reprint of the 1966 edition (1989)

  5. Cheridito, P., Li, T.: Risk measures on Orlicz hearts. Math. Financ. 19, 189–214 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  6. Delbaen, F.: Coherent Risk Measures. Cattedra Galileiana. [Galileo Chair]. Scuola Normale Superiore, Classe di Scienze, Pisa (2000)

    Google Scholar 

  7. Delbaen, F.: Differentiability properties of utility functions. In: Optimality and Risk–Modern Trends in Mathematical Finance, pp. 39–48. Springer, Berlin (2009)

  8. Delbaen, F.: Risk measures for non-integrable random variables. Math. Financ. 19, 329–333 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  9. Föllmer, H., Schied, A.: Stochastic Finance. An Introduction in Discrete Time, 3rd edn. Walter de Gruyter & Co., Berlin (2011)

    Book  Google Scholar 

  10. Jouini, E., Schachermayer, W., Touzi, N., et al.: Law invariant risk measures have the fatou property. Adv. Math. Econ. 9, 49–71 (2006)

    Article  MathSciNet  Google Scholar 

  11. Kaina, M., Rüschendorf, L.: On convex risk measures on \(L^p\)-spaces. Math. Methods Oper. Res. 69, 475–495 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  12. Krätschmer, V. : On \(\sigma \)-additive robust representation of convex risk measures for unbounded financial positions in the presence of uncertainty about the market model. SFB 649 Discussion Paper 2007–010, Humboldt-Universität zu Berlin (2007)

  13. Moreau, J.-J.: Sur la fonction polaire d’une fonction semi-continue supérieurement. C. R. Acad. Sci. Paris 258, 1128–1130 (1964)

    MATH  MathSciNet  Google Scholar 

  14. Orihuela, J., Ruiz Galán, M.: A coercive James’s weak compactness theorem and nonlinear variational problems. Nonlinear Anal. 75, 598–611 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  15. Orihuela, J., Ruiz Galán, M.: Lebesgue property of convex risk measures on Orlicz spaces. Math. Financ. Econ. 6, 15–35 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  16. Owari, K.: Maximum Lebesgue extension of monotone convex functions. Preprint, arXiv:1304.7934 (2013)

    Google Scholar 

  17. Rao, M.M., Ren, Z.D.: Theory of Orlicz Spaces. Marcel Dekker Inc., New York (1991)

    MATH  Google Scholar 

  18. Ruszczyński, A., Shapiro, A.: Optimization of convex risk functions. Math. Oper. Res. 31, 433–452 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  19. Zaanen, A.C.: Riesz spaces. II, North-Holland Mathematical Library, vol. 30. North-Holland Publishing Co., Amsterdam (1983)

    Google Scholar 

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Acknowledgments

The author thanks Sara Biagini for helpful comments on an earlier version of the paper. He also thanks two anonymous referees for careful reading. The financial support of the Center for Advanced Research in Finance (CARF) at the Graduate School of Economics of the University of Tokyo is gratefully acknowledged.

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  1. Graduate School of Economics, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo , 113-0033, Japan

    Keita Owari

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  1. Keita Owari
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Owari, K. On the Lebesgue property of monotone convex functions. Math Finan Econ 8, 159–167 (2014). https://doi.org/10.1007/s11579-013-0111-z

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