Risk measures in ordered normed linear spaces with non-empty cone-interior. (English) Zbl 1233.91149

Summary: We use tools from the theory of partially ordered normed linear spaces, especially the bases of cones. This work extends the well-known results for convex and coherent risk measures. Its linchpin consists in the replacement of the riskless bond by some interior point in the cone of the space of risks, which stands as the alternative numeraire.


91B30 Risk theory, insurance (MSC2010)
Full Text: DOI


[1] Aliprantis, C. D.; Border, K. C., Infinite Dimensional Analysis, A Hitchhiker’s Guide, (1999), Springer · Zbl 0938.46001
[2] Aliprantis, C. D.; Burkinshaw, O., Positive Operators, (1985), Academic Press: Academic Press San Diego, New York · Zbl 0567.47037
[3] Aliprantis, C. D.; Burkinshaw, O., Principles of Real Analysis, (1998), Academic Press · Zbl 0436.46009
[4] Artzner, P.; Delbaen, F.; Eber, J. M.; Heath, D., Coherent measures of risk, Mathematical Finance, 9, 203-228, (1999) · Zbl 0980.91042
[5] Biagini, S.; Fritelli, M., On the extension of the Namioka–Klee theorem and on the Fatou property for risk measures, (Optimality and Risk: Modern Trends in Mathematical Finance, (2009), Springer: Springer Berlin, Heidelberg), 1-28 · Zbl 1188.91085
[6] Borwein, J. M., Automatic continuity of convex relations, Proceedings of the American Mathematical Society, 99, 49-55, (1987) · Zbl 0615.46004
[7] Cheridito, P.; Li, T., Risk measures on Orlicz hearts, Mathematical Finance, 19, 189-214, (2009) · Zbl 1168.91409
[8] Delbaen, F., Coherent risk measures on general probability spaces, (Advances in Finance and Stochastics: Essays in Honour of Dieter Sondermann, (2002), Springer-Verlag: Springer-Verlag Berlin, New York), 1-38 · Zbl 1020.91032
[9] Föllmer, H.; Schied, A., Convex measures of risk and trading constraints, Finance and Stochastics, 6, 429-447, (2002) · Zbl 1041.91039
[10] Fritelli, M.; Gianin, E., Putting order in risk measures, Journal of Banking & Finance, 26, 1473-1486, (2002)
[11] Hamel, A.H., 2006. Translative sets and functions and their applications to risk measure theory and nonlinear separation. IMPA. Preprint D 21/2006.
[12] Herings, P. J.-J.; Polemarchakis, H., Pareto improving price regulation when the asset market is incomplete, Economic Theory, 25, 135-154, (2005) · Zbl 1114.91040
[13] Jameson, G., (Ordered Linear Spaces. Ordered Linear Spaces, Lecture Notes in Mathematics, vol. 141, (1970), Springer-Verlag) · Zbl 0196.13401
[14] Jaschke, S.; Küchler, U., Coherent risk measures and good-deal bounds, Finance and Stochastics, 5, 181-200, (2001) · Zbl 0993.91023
[15] Kaina, M.; Rüschendorf, L., On convex risk measures on \(L^p\)-spaces, Mathematical Methods of Operations Research, 69, 475-495, (2009) · Zbl 1168.91019
[16] Kreps, D. M., Arbitrage and equilibrium in economies with infinitely many commodities, Journal of Mathematical Economics, 8, 15-35, (1981) · Zbl 0454.90010
[17] Polyrakis, I. A., Demand functions and reflexivity, Journal of Mathematical Analysis and Applications, 338, 695-704, (2008) · Zbl 1222.91034
[18] Stoica, G., Relevant coherent measures of risk, Journal of Mathematical Economics, 42, 794-806, (2006) · Zbl 1142.91045
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.