Banach Journal of Mathematical Analysis

On Banach spaces of vector-valued random variables and their duals motivated by risk measures

Thomas Kalmes and Alois Pichler

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We introduce Banach spaces of vector-valued random variables motivated from mathematical finance. So-called risk functionals are defined in a natural way on these Banach spaces, and it is shown that these functionals are Lipschitz continuous. Since the risk functionals cannot be defined on strictly larger spaces of random variables, this creates an area of particular interest with regard to the spaces presented. We elaborate key properties of these Banach spaces and give representations of their dual spaces in terms of vector measures with values in the dual space of the state space.

Article information

Banach J. Math. Anal. (2018), 35 pages.

Received: 29 March 2017
Accepted: 19 June 2017
First available in Project Euclid: 8 September 2017

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Primary: 46E30: Spaces of measurable functions (Lp-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
Secondary: 46E40: Spaces of vector- and operator-valued functions 62P05: Applications to actuarial sciences and financial mathematics

vector-valued random variables Banach spaces of random variables rearrangement invariant spaces dual representation risk measures


Kalmes, Thomas; Pichler, Alois. On Banach spaces of vector-valued random variables and their duals motivated by risk measures. Banach J. Math. Anal., advance publication, 8 September 2017. doi:10.1215/17358787-2017-0026.

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  • [1] C. Acerbi, Spectral measures of risk: A coherent representation of subjective risk aversion, J. Banking Finance 26 (2002), no. 7, 1505–1518.
  • [2] A. Ahmadi-Javid and A. Pichler, Norms and Banach spaces induced by the entropic value-at-risk, to appear in Math. Financ. Econ.
  • [3] P. Artzner, F. Delbaen, J.-M. Eber, and D. Heath, Coherent measures of risk, Math. Finance 9 (1999), no. 3, 203–228.
  • [4] F. Bellini and E. Rosazza Gianin, Haezendonck–Goovaerts risk measures and Orlicz quantiles, Insurance Math. Econom. 51 (2012), no. 1, 107–114.
  • [5] C. Burgert and L. Rüschendorf, Consistent risk measures for portfolio vectors, Insurance Math. Econom. 38 (2006), no. 2, 289–297.
  • [6] P. Cheridito and T. Li, Risk measures on Orlicz hearts, Math. Finance 19 (2009), no. 2, 189–214.
  • [7] D. Dentcheva and A. Ruszczyński, Optimization with stochastic dominance constraints, SIAM J. Optim. 14 (2003), no. 2, 548–566.
  • [8] D. Dentcheva and A. Ruszczyński, Convexification of stochastic ordering, C. R. Acad. Bulgare Sci. 57 (2004), no. 4, 11–16.
  • [9] D. Dentcheva and A. Ruszczyński, Portfolio optimization with stochastic dominance constraints, J. Banking Finance 30 (2006), no. 2, 433–451.
  • [10] J. Diestel, Geometry of Banach Spaces—Selected Topics, Lecture Notes in Math. 485, Springer, Berlin, 1975.
  • [11] J. Diestel and J. J. Uhl, Jr., Vector Measures, Math. Surveys 15, Amer. Math. Soc., Providence, 1977.
  • [12] I. Ekeland, A. Galichon, and M. Henry, Comonotonic measures of multivariate risks, Math. Finance 22 (2012), no. 1, 109–132.
  • [13] I. Ekeland and W. Schachermayer, Law invariant risk measures on ${L}^{\infty}({\mathbb{R}}^{d})$, Stat. Risk Model. 28 (2001), no. 3, 195–225.
  • [14] D. Filipović and G. Svindland, The canonical model space for law-invariant convex risk measures is ${L}^{1}$, Math. Finance 22 (2012), no. 3, 585–589.
  • [15] I. S. Gradshteyn and J. M. Ryzhik, Table of Integrals, Series, and Products, 6th ed., Academic Press, San Diego, 2000.
  • [16] I. Halperin, Function spaces, Canad. J. Math. 5 (1953), 273–288.
  • [17] G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge Math. Lib., Cambridge Univ. Press, Cambridge, 1988.
  • [18] M. Kupper and G. Svindland, Dual representation of monotone convex functions on ${L}^{0}$, Proc. Amer. Math. Soc. 139 (2011), no. 11, 4073–4086.
  • [19] S. Kusuoka, “On law invariant coherent risk measure” in Advances in Mathematical Economics, Vol. 3, Math. Econ. 3, Springer, Tokyo, 2001, 83–95.
  • [20] G. G. Lorentz, On the theory of spaces $\Lambda$, Pacific J. Math. 1 (1951), no. 3, 411–429.
  • [21] G. G. Lorentz, Bernstein Polynomials, 2nd ed., Chelsea, New York, 1986.
  • [22] A. J. McNeil, R. Frey, and P. Embrechts, Quantitative Risk Management: Concepts, Techniques and Tools, Princet. Ser. Finance, Princeton Univ. Press, Princeton, 2005.
  • [23] W. Ogryczak and A. Ruszczyński, Dual stochastic dominance and related mean-risk models, SIAM J. Optim. 13 (2002), no. 1, 60–78.
  • [24] G. Ch. Pflug, “Some remarks on the value-at-risk and the conditional value-at-risk” in Probabilistic Constrained Optimization, Nonconvex Optim. Appl. 49, Kluwer, Dordrecht, 2000, 272–281.
  • [25] G. Ch. Pflug and W. Römisch, Modeling, Measuring and Managing Risk, World Scientific, Hackensack, NJ, 2007.
  • [26] A. Pichler, The natural Banach space for version independent risk measures, Insurance Math. Econom. 53 (2013), no. 2, 405–415.
  • [27] A. Pichler, Insurance pricing under ambiguity, Eur. Actuar. J. 4 (2014), no. 2, 335–364.
  • [28] A. Pichler, A quantitative comparison of risk measures, Ann. Oper. Res. 254 (2017), no. 1–2, 251–275.
  • [29] L. Rüschendorf, Law invariant convex risk measures for portfolio vectors, Statist. Decisions 24 (2006), no. 1, 97–108.
  • [30] G. Svindland, Subgradients of law-invariant convex risk measures on ${L}^{1}$, Statist. Decisions 27 (2009), no. 2, 169–199.
  • [31] D. Williams, Probability with Martingales, Cambridge Univ. Press, Cambridge, 1991.