Abstract
We provide a variety of results for quasiconvex, law-invariant functionals defined on a general Orlicz space, which extend well-known results from the setting of bounded random variables. First, we show that Delbaen’s representation of convex functionals with the Fatou property, which fails in a general Orlicz space, can always be achieved under the assumption of law-invariance. Second, we identify the class of Orlicz spaces where the characterization of the Fatou property in terms of norm-lower semicontinuity by Jouini, Schachermayer and Touzi continues to hold. Third, we extend Kusuoka’s representation to a general Orlicz space. Finally, we prove a version of the extension result by Filipović and Svindland by replacing norm-lower semicontinuity with the (generally non-equivalent) Fatou property. Our results have natural applications to the theory of risk measures.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aliprantis, Ch.D., Border, K.C.: Infinite Dimensional Analysis: A Hitchhiker’s Guide, 3rd edn. Springer, Berlin (2006)
Artzner, Ph., Delbaen, F., Eber, J.-M., Heath, D.: Coherent measures of risk. Math. Finance 9, 203–228 (1999)
Belomestny, D., Krätschmer, V.: Central limit theorems for law-invariant coherent risk measures. J. Appl. Probab. 49, 1–21 (2012)
Belomestny, D., Krätschmer, V.: Optimal stopping under probability distortions. Math. Oper. Res. 42, 806–833 (2017)
Biagini, S., Frittelli, M.: On the extension of the Namioka–Klee theorem and on the Fatou property for risk measures. In: Delbaen, F., et al. (eds.) Optimality and Risk—Modern Trends in Mathematical Finance. The Kabanov Festschrift, pp. 1–28. Springer, Berlin (2009)
Cheridito, P., Li, T.: Risk measures on Orlicz hearts. Math. Finance 19, 189–214 (2009)
Delbaen, F.: Coherent risk measures on general probability spaces. In: Sandmann, K., Schönbucher, P.J. (eds.) Advances in Finance and Stochastics: Essays in Honour of Dieter Sondermann, pp. 1–37. Springer, Berlin (2002)
Delbaen, F.: Risk measures for non-integrable random variables. Math. Finance 19, 329–333 (2009)
Delbaen, F., Owari, K.: On convex functions on the duals of \(\Delta_{2}\)-Orlicz spaces. Preprint (2016). Available online at https://arxiv.org/abs/1611.06218
Drapeau, S., Kupper, M.: Risk preferences and their robust representations. Math. Oper. Res. 38, 28–62 (2013)
Edgar, G.A., Sucheston, L.: Stopping Times and Directed Processes. Cambridge University Press, Cambridge (1992)
Farkas, W., Koch-Medina, P., Munari, C.: Beyond cash-additive risk measures: when changing the numeraire fails. Finance Stoch. 18, 145–173 (2014)
Filipović, D., Svindland, G.: The canonical model space for law-invariant convex risk measures is \(L^{1}\). Math. Finance 22, 585–589 (2012)
Föllmer, H., Schied, A.: Stochastic Finance: An Introduction in Discrete Time, 3rd edn. De Gruyter, Berlin (2011)
Frittelli, M., Rosazza Gianin, E.: Putting order in risk measures. J. Bank. Finance 26, 1473–1486 (2002)
Frittelli, M., Rosazza Gianin, E.: Law invariant convex risk measures. Adv. Math. Econ. 7, 33–46 (2005)
Gao, N., Leung, D., Xanthos, F.: Closedness of convex sets in Orlicz spaces with applications to dual representation of risk measures. Preprint (2016). Available online at https://arxiv.org/abs/1610.08806
Gao, N., Xanthos, F.: On the C-property and \(w^{*}\)-representations of risk measures. Math. Finance (2017). https://doi.org/10.1111/mafi.12150. Available online at: http://onlinelibrary.wiley.com/doi/10.1111/mafi.12150/full
Jouini, E., Schachermayer, W., Touzi, N.: Law invariant risk measures have the Fatou property. Adv. Math. Econ. 9, 49–71 (2006)
Kaina, M., Rüschendorf, L.: On convex risk measures on \(L^{p}\)-spaces. Math. Methods Oper. Res. 69, 475–495 (2009)
Koch-Medina, P., Munari, C.: Law-invariant risk measures: extension properties and qualitative robustness. Stat. Risk. Model. 31, 1–22 (2014)
Krätschmer, V., Schied, A., Zähle, H.: Comparative and qualitative robustness for law-invariant risk measures. Finance Stoch. 18, 271–295 (2014)
Kusuoka, S.: On law-invariant coherent risk measures. Adv. Math. Econ. 3, 83–95 (2001)
Munari, C.: Measuring Risk Beyond the Cash-Additive Paradigm. Ph.D. Dissertation, ETH Zurich (2015). Available online at https://www.research-collection.ethz.ch/handle/20.500.11850/102246
Orihuela, J., Ruiz Galán, M.: Lebesgue property for convex risk measures on Orlicz spaces. Math. Financ. Econ. 6, 15–35 (2012)
Owari, K.: Maximal Lebesgue extension of monotone convex functions. J. Funct. Anal. 266, 3572–3611 (2014)
Shapiro, A.: On Kusuoka representation of law invariant risk measures. Math. Oper. Res. 38, 142–152 (2013)
Svindland, G.: Convex Risk Measures Beyond Bounded Risks. Ph.D. Dissertation, LMU Munich (2008). Available online at http://edoc.ub.uni-muenchen.de/9715/
Svindland, G.: Continuity properties of law-invariant (quasi-)convex risk functions on \(L^{\infty}\). Math. Financ. Econ. 3, 39–43 (2010)
Acknowledgements
We are grateful for the comments of an anonymous referee that improved the clarity of some proofs in the paper. The first author is a PIMS Postdoctoral Fellow. The second author is supported by AcRF grant R-146-000-242-114. The fourth author acknowledges the support of an NSERC grant.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Gao, N., Leung, D., Munari, C. et al. Fatou property, representations, and extensions of law-invariant risk measures on general Orlicz spaces. Finance Stoch 22, 395–415 (2018). https://doi.org/10.1007/s00780-018-0357-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00780-018-0357-7