Abstract
This paper has been motivated by general considerations on the topic of Risk Measures, which essentially are convex monotone maps defined on spaces of random variables, possibly with the so-called Fatou property.
We show first that the celebrated Namioka-Klee theorem for linear, positive functionals holds also for convex monotone maps π on Frechet lattices.
It is well-known among the specialists that the Fatou property for risk measures on L ∞ enables a simplified dual representation, via probability measures only. The Fatou property in a general framework of lattices is nothing but the lower order semicontinuity property for π. Our second goal is thus to prove that a similar simplified dual representation holds also for order lower semicontinuous, convex and monotone functionals π defined on more general spaces X (locally convex Frechet lattices). To this end, we identify a link between the topology and the order structure in X—the C-property—that enables the simplified representation. One main application of these results leads to the study of convex risk measures defined on Orlicz spaces and of their dual representation.
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References
Aliprantis, C.D., Border, K.C.: Infinite Dimensional Analysis, 3rd edn. Springer, Berlin (2005)
Aliprantis, C.D., Burkinshaw, O.: Positive Operators. Academic Press, San Diego (1985)
Artzner, P., Delbaen, F., Eber, J.M., Heath, D.: Coherent measures of risk. Math. Financ. 4, 203–228 (1999)
Barrieu, P., El Karoui, N.: Pricing, hedging and optimally designing derivatives via minimization of risk measures. In: Carmona, R. (ed.) Indifference Pricing: Theory and Applications. Princeton University Press, Princeton (2008)
Biagini, S.: An Orlicz spaces duality for utility maximization in incomplete markets. In: Proceedings of Ascona 2005. Progress in Probability. Birkhäuser, Basel (2007)
Biagini, S., Frittelli, M.: A unified framework for utility maximization problems: an Orlicz space approach. Ann. Appl. Prob. 18(3), 929–966 (2008)
Biagini, S., Frittelli, M., Grasselli, M.: Indifference price for general semimartingales. Submitted, 2007
Brezis, H.: Analyse fonctionnelle. Masson, Paris (1983)
Cheridito, P., Li, T.: Risk measures on Orlicz hearts. Math. Financ. 19(2), 189–214 (2009)
Delbaen, F.: Coherent risk measures on general probability spaces. In: Essays in Honour of Dieter Sondermann. Springer, Berlin (2000)
Filipovic, D., Svindland, G.: The canonical model space for law-invariant convex risk measures is L 1. Preprint (2008)
Föllmer, H., Schied, A.: Stochastic Finance. An Introduction in Discrete Time, 2nd edn. De Gruyter, Berlin (2004)
Föllmer, H., Schied, A.: Convex measures of risk and trading constraints. Finance Stoch. 6(4), 429–447 (2002)
Frittelli, M., Rosazza Gianin, E.: Putting order in risk measures. J. Bank. Financ. 26(7), 1473–1486 (2002)
Kozek, A.: Convex integral functionals on Orlicz spaces. Ann. Soc. Math. Pol. Ser. 1, Comment. Math. XXI, 109–134 (1979)
Maccheroni, F., Marinacci, M., Rustichini, A.: A variational formula for the relative Gini concentration index. In press
Namioka, I.: Partially Ordered Linear Topological Spaces. Mem. Am. Math. Soc., vol. 24. Princeton University Press, Princeton (1957)
Rao, M.M., Ren, Z.D.: Theory of Orlicz Spaces. Marcel Dekker, New York (1991)
Rüschendorf, L., Kaina, M.: On convex risk measures on Lp-spaces. Preprint (2007)
Ruszczynski, A., Shapiro, A.: Optimization of convex risk measures. Math. Oper. Res. 31(3), 433–452 (2006)
Zaanen, A.C.: Riesz Spaces II. North-Holland Math. Library. North-Holland, Amsterdam (1983)
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Biagini, S., Frittelli, M. (2009). 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. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02608-9_1
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DOI: https://doi.org/10.1007/978-3-642-02608-9_1
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