Abstract
The paper introduces a new notion of vector-valued risk function, a crucial notion in Actuarial and Financial Mathematics. Both deviations and expectation bounded or coherent risk measures are defined and analyzed. The relationships with both scalar and vector risk functions of previous literature are discussed, and it is pointed out that this new approach seems to appropriately integrate several preceding points of view. The framework of the study is the general setting of Banach lattices and Bochner integrable vector-valued random variables. Sub-gradient linked representation theorems and practical examples are provided.
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References
Artzner P., Delbaen F., Eber J.M., Heath D.: Coherent measures of risk. Mathematical Finance 9, 203–228 (1999)
Balbás A., Balbás B., Balbás R.: Minimizing measures of risk by saddle point conditions. Journal of Computational and Applied Mathematics 234, 2924–2931 (2010)
Balbás A., Balbás B., Balbás R.: CAPM and APT-like models with risk measures. Journal of Banking & Finance 34, 1166–1174 (2010)
Balbás A., Balbás B., Heras A.: Optimal reinsurance with general risk measures. Insurance: Mathematics and Economics 44, 374–384 (2009)
Balbás A., Jiménez-Guerra P.: Set-valued vector risk functions. The Journal of Financial Decision Making 6, 41–49 (2010)
Benati S.: The optimal portfolio problem with coherent risk measure constraints. European Journal of Operational Research 150, 572–584 (2003)
Cascos I., Molchanov I.: Multivariate risks and depth trimmed regions. Finance & Stochastics 11, 373–397 (2007)
Cheridito P., Delbaen F., Kupper M.: Coherent and convex monetary risk measures for unbounded cádlág processes. Finance & Stochastics 9, 369–387 (2005)
Diestel J., Uhl J.J.: Vector measures. American Mathematical Society, Providence Rhode, Island (1977)
Föllmer H., Schied A.: Convex measures of risk and trading constraints. Finance & Stochastics 6, 429–447 (2002)
M. Frittelli and E. Rosazza Gianin, Dynamic convex risk measures. In: Szegö, G. (Ed.), Risk Measures for the 21st Century. Wiley, New York, (2004) pp. 227-248.
E.A. Galperin, Pareto analysis vis-à-vis balance apace approach in multiobjective global optimization. Journal of Optimization Theory and Applications 93 (1997), 3, 533-545.
Goovaerts M., Kaas R., Dhaene J., Tang Q.: A new classes of consistent risk measures. Insurance: Mathematics and Economics 34, 505–516 (2004)
Hamel A., Heyde F.: Duality for set-valued measures of risk. SIAM Journal of Financial Mathematics 1, 66–95 (2010)
Jouini E., Meddeb M., Touci N.: Vector-valued coherent risk measures. Finance & Stochastics 8, 531–552 (2004)
Konno H., Akishino K., Yamamoto R.: Optimization of a long-short portfolio under non-convex transaction costs. Computational Optimization and Applications 32, 115–132 (2005)
Meyer-Nieberg P.: Banach lattices. Springer-Verlag, New York (1991)
Nakano Y.: Efficient hedging with coherent risk measure. Journal of Mathematical Analysis and Applications 293, 345–354 (2004)
Ogryczak W., Ruszczynski A.: From stochastic dominance to mean risk models: Semideviations and risk measures. European Journal of Operational Research 116, 33–50 (1999)
Roorda B., Schumacher J.M.: The strictest common relaxation of a family of risk measures. Insurance: Mathematics and Economics 48, 29–34 (2011)
Rockafellar R.T., Uryasev S.: Zabarankin M., Optimality conditions in portfolio analysis with general deviations measures. Mathematical Programming, Ser. B 108, 515–540 (2006)
Rockafellar R.T., Uryasev S., Zabarankin M.: Generalized deviations in risk analysis. Finance & Stochastics 10, 51–74 (2006)
M. Schweizer, Variance-optimal hedging in discrete time. Mathematics of Operations Research 20 (1995), 1, 1-32.
C. Zalinescu, Convex analysis in general vector spaces. World Scientific Publishing Co,(2002).
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The authors thank Professor C. Zalinescu and the anonymous reviewer for their useful comments and suggestions. This research was partially supported by “Comunidad Autónoma de Madrid” (Spain), Grant S2009ESP − 1685, and “Ministerio de Ciencia en Innovación” (Spain), Grant ECO2009 − 14457 − C04. The usual caveat applies.
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Balbás, A., Balbás, R. & Jiménez-Guerra, P. Vector Risk Functions. Mediterr. J. Math. 9, 563–574 (2012). https://doi.org/10.1007/s00009-011-0153-5
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DOI: https://doi.org/10.1007/s00009-011-0153-5