Deviation Measures on Banach Spaces and Applications

Abstract

In this article we generalize the notion of the deviation measure, which were initially defined on spaces of squarely integrable random variables, as an extension of the notion of standard deviation. We extend them both under a frame which requires some elements from the theory of partially ordered linear spaces and also under a frame which refers to some closed subspace, whose elements are supposed to have zero deviation. This subspace denotes in general a set of risk-less assets, since in finance deviation measures may replace standard deviation as a measure of risk. In the last sections of the article we treat the minimization of deviation measures over a set of financial positions as a zero-sum game between the investor and the nature and we determine the solution of such a minimization problem via min-max theorems.

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Kountzakis, C. (2013). Deviation Measures on Banach Spaces and Applications. Journal of Financial Risk Management, 2, 13-28. doi: 10.4236/jfrm.2013.21003.

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1] Acerbi, C., & Tasche, D. (2002). On the coherence of the expected shortfall. Journal of Banking and Finance, 26, 1487-1503. doi:10.1016/S0378-4266(02)00283-2
[2] Aliprantis, C. D., & Border, K. C. (1999). Infinite dimensional analysis, A hitchhiker’s guide (2nd ed.). New York: Springer. doi:10.1007/978-3-662-03961-8
[3] Artzner, P., Delbaen, F., Eber, J. M., & Heath, D. (1999). Coherent measures of risk. Mathematical Finance, 9, 203-228. doi:10.1111/1467-9965.00068
[4] Barbu, V., & Precupanu, T. (1986). Convexity and optimization in banach spaces. D. Riedel Publishing Company, Kluwer Academic Publishers Group, Dordrecht.
[5] Delbaen, F. (2002). Coherent risk measures on general probability spaces. In: Advances in finance and stochastics: essays in honour of Dieter Sondermann (pp. 1-38). Springer-Verlag: Berlin. doi:10.1007/978-3-662-04790-3_1
[6] Denis, L., & Martini, C. (2006). A theoretical framework for the pricing of contingent claims in the presence of model uncertainty. Annals of Applied Probability, 16, 827-852. doi:10.1214/105051606000000169
[7] Dhaene, G., Goovaerts, M. J., Kaas, R., Tang, Q., Vanduffel, S., & Vyncke, D. (2003). Solvency capital, risk measures and comonotonicity: A review. Research Report OR0416, Leuven: Catholic University of Leuven.
[8] Gong, X. H. (1994). Connectedness of efficient solution sets for set valued maps in normed spaces. Journal of Optimization Theory and Applications, 83, 83-96. doi:10.1007/BF02191763
[9] Grechuk, B., Molyboha, A., & Zabarankin, M. (2009). Maximum entropy principle with general deviation measures. Mathematics of Operations Research, 34, 445-467. doi:10.1287/moor.1090.0377
[10] Grechuk, B., Molyboha, A., & Zabarankin, M. (2011). Cooperative games with general deviation measures. Mathematical Finance, Forthcoming.
[11] Jameson, G. (1970). Ordered linear spaces. Lecture notes in mathematics (Vol. 141). Berlin: Springer-Verlag.
[12] Jaschke, S., & Küchler, U. (2001). Coherent risk measures and good deal bounds. Finance and Stochastics, 5, 181-200. doi:10.1007/PL00013530
[13] Kaina, M., & Rüschendorf, L. (2009). On convex risk measures on -spaces. Mathematical Methods of Operations Research, 69, 475-495. doi:10.1007/s00186-008-0248-3
[14] Konstantinides, D. G., & Kountzakis, C. (2011). Risk measures in ordered normed linear spaces with non-empty cone-interior. Insurance: Mathematics and Economics, 48, 111-122. doi:10.1016/j.insmatheco.2010.10.003
[15] Kountzakis, C. E. (2011). On efficient portfolio selection using convex risk measures. Mathematics and Financial Economics, 4, 223-252. doi:10.1007/s11579-011-0043-4
[16] Kountzakis, C. E. (2011). Risk measures on ordered non-reflexive banach spaces. Journal of Mathematical Analysis and Applications, 373, 548-562. doi:10.1016/j.jmaa.2010.08.013
[17] Kroll, Y., Levy, H., & Markowitz, H. M. (1984). Mean-variance versus direct utility maximization. Journal of Finance, 39, 47-61. doi:10.1111/j.1540-6261.1984.tb03859.x
[18] Kusuoka, S. (2001). On law invariant coherent risk measures. In: Advances in mathematical economics (Vol. 3, pp. 83-95). Tokyo: Springer. doi:10.1007/978-4-431-67891-5_4
[19] Luenberger, D. G. (1969). Optimization by vector space methods. New York: John Wiley and Sons Inc.
[20] Markowitz, H. (1952). Portfolio selection. Journal of Finance, 7, 77-91.
[21] Peng, S. (2007). G-expectation, G-brownian motion and related stochastic calulus of It Type. In: stochastic analysis and applications, abel symposium (Vol. 2, pp. 541-567). Berlin: Springer.
[22] Rockafellar, R. T., Uryasev, S., & Zabarankin, M. (2003). Deviation measures in risk analysis and optimization. Research Report 2002-7, University of Florida.
[23] Rockafellar, R. T., Uryasev, S., & Zabarankin, M. (2006a). Generalized deviations in risk analysis. Finance and Stochastics, 10, 51-74. doi:10.1007/s00780-005-0165-8
[24] Rockafellar, R. T., Uryasev, S., & Zabarankin, M. (2006b). Master funds in portfolio analysis with general deviation measures. Journal of Banking and Finance, 30, 743-778. doi:10.1016/j.jbankfin.2005.04.004
[25] Rockafellar, R. T., Uryasev, S., & Zabarankin, M. (2006c). Optimality conditions in portfolio analysis with general deviation measures. Ma thematical Programming, 108, 515-540. doi:10.1007/s10107-006-0721-9
[26] Rockafellar, R. T., Uryasev, S., & Zabarankin, M. (2007). Equilibrium with investors with a diversity of deviation measures Journal of Banking and Finance, 31, 3251-3268. doi:10.1016/j.jbankfin.2007.04.002
[27] Soner, H. M., Touzi, N., & Zhang, J. (2011). A martingale representation theorem for the G-expectation. Stochastic Processes and their Applications, 121, 265-287. doi:10.1016/j.spa.2010.10.006
[28] Tasche, D. (2002). Expected shortfall and beyond. Journal of Banking and Finance, 26, 1519-1533. doi:10.1016/S0378-4266(02)00272-8
[29] Von Neumann, J. (1928). Zur theorie der gesellschaftsspiele. Mathematische Annalen, 100, 295-320. doi:10.1007/BF01448847
[30] Xanthos, F. (2009). Conic characterizations of reflexive banach spaces. Master’s Degree Dissertation, Athens: National Technical University of Athens.

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