Monotone and cash-invariant convex functions and hulls. (English) Zbl 1119.91051

Summary: This paper provides some useful results for convex risk measures. In fact, we consider convex functions on a locally convex vector space E which are monotone with respect to the preference relation implied by some convex cone and invariant with respect to some numeraire (‘cash’). As a main result, for any function \(f\), we find the greatest closed convex monotone and cash-invariant function majorized by \(f\). We then apply our results to some well-known risk measures and problems arising in connection with insurance regulation.


91B30 Risk theory, insurance (MSC2010)
91B84 Economic time series analysis
Full Text: DOI


[1] Aliprantis, C.D.; Border, K.C., Infinite dimensional analysis, (1999), Springer · Zbl 0938.46001
[2] Artzner, P.; Delbaen, F.; Eber, J.M.; Heath, D., Coherent measures of risk, Mathematical finance, 9, 3, 203-228, (1999) · Zbl 0980.91042
[3] Barrieu, P.; El Karoui, N., Optimal derivatives design under dynamic risk measures, Mathematics of finance, contemporary mathematics, 351, 13-25, (2004) · Zbl 1070.91019
[4] Barrieu, P.; El Karoui, N., Inf-convolution of risk measures and optimal risk transfer, Finance and stochastics, 9, 269-298, (2005) · Zbl 1088.60037
[5] Bühlmann, H.; Jewell, W.S., Optimal risk exchanges, Astin bulletin, 10, 243-262, (1979) · Zbl 0679.62090
[6] Bühlmann, H., The general economic premium principle, Astin bulletin, 14, 13-21, (1984)
[7] Burgert, Ch, Rüschendorf, L., 2005. Allocation of risks and equilibrium in markets with finitely many traders. Preprint
[8] Delbaen, F., 2000. Coherent Risk Measures, Cattedra Galileiana. Scuola Normale Superiore di Pisa
[9] Deprez, O.; Gerber, U., On convex principles of premium calculation, Insurance: mathematics and economics, 4, 179-189, (1985) · Zbl 0579.62090
[10] Ekeland, I.; Témam, R., Convex analysis and variational problems, (1999), SIAM · Zbl 0939.49002
[11] Filipović, D., Kupper, M., 2005. Equilibrium and optimality for monetary utility functions under constraints. Preprint
[12] Filipović, D., Kupper, M., 2006. Optimal capital and risk transfers for group diversification, Mathematical Finance (in press)
[13] Fischer, T., 2001. Examples of coherent risk measures depending on one-sided moments. Preprint
[14] Föllmer, H.; Kabanov, Y.M., Optimal decomposition and Lagrange multipliers, Finance and stochastics, 2, 69-81, (1998) · Zbl 0894.90016
[15] Föllmer, H.; Schied, A., Convex measures of risk and trading constraints, Finance and stochastics, 6, 4, 429-447, (2002) · Zbl 1041.91039
[16] Föllmer, H., Schied, A., 2002b. Stochastic finance, an introduction in discretetime, de Gruyter Studies in Mathematics 27
[17] Frittelli, M.; Rosazza Gianin, E., Putting order in risk measures, Journal of banking and finance, 26, 7, 1473-1486, (2002)
[18] Goovaerts, M.J.; De Vylder, F.; Haezendonck, J., Insurance premiums, (1984), North-Holland Amsterdam · Zbl 0532.62082
[19] Heath, D.; Ku, H., Pareto equilibria with coherent measures of risk, Mathematical finance, 14, 163-172, (2004) · Zbl 1090.91033
[20] Jobert, A., Rogers, L.C.G., 2005. Pricing operators and dynamic convex risk measures. Preprint · Zbl 1138.91501
[21] Jouini, E., Schachermayer, W., Touzi, N., 2005. Optimal risk sharing for law invariant monetary utility functions. Preprint · Zbl 1133.91360
[22] Kaas, R.; Goovaerts, M.; Dhaene, J.; Denuit, M., Modern actuarial risk theory, (2001), Kluwer Academic Publishers
[23] Maccheroni, F., Marinacci, M., Rustichini, A., Taboga, M., 2005a. Portfolio selection with monotone mean-variance preferences. Preprint · Zbl 1168.91396
[24] Maccheroni, F., Marinacci, M., Rustichini, A., Taboga, M., 2005b. A variational formula for the relative gini concentration index. Preprint
[25] Rockafellar, R.T., Convex analysis, (1997), Princeton University Press · Zbl 0897.49014
[26] Rockafellar, R.T.; Uryasev, S., Conditional value-at-risk for general loss distributions, Journal of banking and finance, 26, 1443-1471, (2002)
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