Dual representation of monotone convex functions on 𝐿⁰

Kupper, Michael; Svindland, Gregor

doi:10.1090/S0002-9939-2011-10835-9

Dual representation of monotone convex functions on $L^0$
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by Michael Kupper and Gregor Svindland PDF

Proc. Amer. Math. Soc. 139 (2011), 4073-4086 Request permission

Abstract:

We study monotone convex functions $\psi :{L}^0(\Omega ,\mathcal {F} ,\mathbb {P})\to (-\infty ,\infty ]$ and derive a dual representation as well as conditions that ensure the existence of a $\sigma$-additive subgradient. The results are motivated by applications in economic agents’ choice theory.

References

Charalambos D. Aliprantis and Kim C. Border, Infinite-dimensional analysis, 2nd ed., Springer-Verlag, Berlin, 1999. A hitchhiker’s guide. MR 1717083, DOI 10.1007/978-3-662-03961-8
W. Brannath and W. Schachermayer, A bipolar theorem for $L^0_+(\Omega ,\scr F,\mathbf P)$, Séminaire de Probabilités, XXXIII, Lecture Notes in Math., vol. 1709, Springer, Berlin, 1999, pp. 349–354. MR 1768009, DOI 10.1007/BFb0096525
Patrick Cheridito, Freddy Delbaen, and Michael Kupper, Coherent and convex monetary risk measures for unbounded càdlàg processes, Finance Stoch. 9 (2005), no. 3, 369–387. MR 2211713, DOI 10.1007/s00780-004-0150-7
Cherny, A. and Kupper, M. (2008), Divergence Utilities, preprint.
Freddy Delbaen, Coherent risk measures on general probability spaces, Advances in finance and stochastics, Springer, Berlin, 2002, pp. 1–37. MR 1929369
Drapeau, S. and Kupper, M. (2010), Risk Preferences and Their Robust Representation, preprint.
Nelson Dunford and Jacob T. Schwartz, Linear operators. Part I, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1988. General theory; With the assistance of William G. Bade and Robert G. Bartle; Reprint of the 1958 original; A Wiley-Interscience Publication. MR 1009162
Ivar Ekeland and Roger Témam, Convex analysis and variational problems, Corrected reprint of the 1976 English edition, Classics in Applied Mathematics, vol. 28, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1999. Translated from the French. MR 1727362, DOI 10.1137/1.9781611971088
Damir Filipović, Michael Kupper, and Nicolas Vogelpoth, Separation and duality in locally $L^0$-convex modules, J. Funct. Anal. 256 (2009), no. 12, 3996–4029. MR 2521918, DOI 10.1016/j.jfa.2008.11.015
Hans Föllmer and Alexander Schied, Stochastic finance, Second revised and extended edition, De Gruyter Studies in Mathematics, vol. 27, Walter de Gruyter & Co., Berlin, 2004. An introduction in discrete time. MR 2169807, DOI 10.1515/9783110212075
Tiexin Guo, Relations between some basic results derived from two kinds of topologies for a random locally convex module, J. Funct. Anal. 258 (2010), no. 9, 3024–3047. MR 2595733, DOI 10.1016/j.jfa.2010.02.002
Guo, T. (2010), Recent progress in random metric theory and its applications to conditional risk measures, arXiv:1006.0697v7.
Paul R. Halmos and L. J. Savage, Application of the Radon-Nikodym theorem to the theory of sufficient statistics, Ann. Math. Statistics 20 (1949), 225–241. MR 30730, DOI 10.1214/aoms/1177730032
Kardaras, C. and Žitković, G. (2010), Forward-convex convergence of sequences in $L^0_+$, preprint.
Kardaras, C. (2010), Strictly positive support points of convex sets in $L^0_+$, preprint.
M. Krein and V. Šmulian, On regularly convex sets in the space conjugate to a Banach space, Ann. of Math. (2) 41 (1940), 556–583. MR 2009, DOI 10.2307/1968735
Gregor Svindland, Subgradients of law-invariant convex risk measures on $L^1$, Statist. Decisions 27 (2009), no. 2, 169–199. MR 2662721, DOI 10.1524/stnd.2009.1040