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

Thomas Kalmes; Alois Pichler

doi:10.1215/17358787-2017-0026

October 2018 On Banach spaces of vector-valued random variables and their duals motivated by risk measures

Thomas Kalmes, Alois Pichler

Banach J. Math. Anal. 12(4): 773-807 (October 2018). DOI: 10.1215/17358787-2017-0026

Abstract

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.

References

1.

[1] C. Acerbi, Spectral measures of risk: A coherent representation of subjective risk aversion, J. Banking Finance 26 (2002), no. 7, 1505–1518.[1] C. Acerbi, Spectral measures of risk: A coherent representation of subjective risk aversion, J. Banking Finance 26 (2002), no. 7, 1505–1518.

2.

[2] A. Ahmadi-Javid and A. Pichler, Norms and Banach spaces induced by the entropic value-at-risk, to appear in Math. Financ. Econ. MR3709386 06821358 10.1007/s11579-017-0197-9[2] A. Ahmadi-Javid and A. Pichler, Norms and Banach spaces induced by the entropic value-at-risk, to appear in Math. Financ. Econ. MR3709386 06821358 10.1007/s11579-017-0197-9

3.

[3] P. Artzner, F. Delbaen, J.-M. Eber, and D. Heath, Coherent measures of risk, Math. Finance 9 (1999), no. 3, 203–228. 0980.91042 10.1111/1467-9965.00068[3] P. Artzner, F. Delbaen, J.-M. Eber, and D. Heath, Coherent measures of risk, Math. Finance 9 (1999), no. 3, 203–228. 0980.91042 10.1111/1467-9965.00068

4.

[4] F. Bellini and E. Rosazza Gianin, Haezendonck–Goovaerts risk measures and Orlicz quantiles, Insurance Math. Econom. 51 (2012), no. 1, 107–114. 1284.91205 10.1016/j.insmatheco.2012.03.005[4] F. Bellini and E. Rosazza Gianin, Haezendonck–Goovaerts risk measures and Orlicz quantiles, Insurance Math. Econom. 51 (2012), no. 1, 107–114. 1284.91205 10.1016/j.insmatheco.2012.03.005

5.

[5] C. Burgert and L. Rüschendorf, Consistent risk measures for portfolio vectors, Insurance Math. Econom. 38 (2006), no. 2, 289–297. MR2212528 10.1016/j.insmatheco.2005.08.008[5] C. Burgert and L. Rüschendorf, Consistent risk measures for portfolio vectors, Insurance Math. Econom. 38 (2006), no. 2, 289–297. MR2212528 10.1016/j.insmatheco.2005.08.008

6.

[6] P. Cheridito and T. Li, Risk measures on Orlicz hearts, Math. Finance 19 (2009), no. 2, 189–214. 1168.91409 10.1111/j.1467-9965.2009.00364.x[6] P. Cheridito and T. Li, Risk measures on Orlicz hearts, Math. Finance 19 (2009), no. 2, 189–214. 1168.91409 10.1111/j.1467-9965.2009.00364.x

7.

[7] D. Dentcheva and A. Ruszczyński, Optimization with stochastic dominance constraints, SIAM J. Optim. 14 (2003), no. 2, 548–566. 1055.90055 10.1137/S1052623402420528[7] D. Dentcheva and A. Ruszczyński, Optimization with stochastic dominance constraints, SIAM J. Optim. 14 (2003), no. 2, 548–566. 1055.90055 10.1137/S1052623402420528

8.

[8] D. Dentcheva and A. Ruszczyński, Convexification of stochastic ordering, C. R. Acad. Bulgare Sci. 57 (2004), no. 4, 11–16. 1083.90031[8] D. Dentcheva and A. Ruszczyński, Convexification of stochastic ordering, C. R. Acad. Bulgare Sci. 57 (2004), no. 4, 11–16. 1083.90031

9.

[9] D. Dentcheva and A. Ruszczyński, Portfolio optimization with stochastic dominance constraints, J. Banking Finance 30 (2006), no. 2, 433–451.[9] D. Dentcheva and A. Ruszczyński, Portfolio optimization with stochastic dominance constraints, J. Banking Finance 30 (2006), no. 2, 433–451.

10.

[10] J. Diestel, Geometry of Banach Spaces—Selected Topics, Lecture Notes in Math. 485, Springer, Berlin, 1975. 0307.46009[10] J. Diestel, Geometry of Banach Spaces—Selected Topics, Lecture Notes in Math. 485, Springer, Berlin, 1975. 0307.46009

11.

[11] J. Diestel and J. J. Uhl, Jr., Vector Measures, Math. Surveys 15, Amer. Math. Soc., Providence, 1977.[11] J. Diestel and J. J. Uhl, Jr., Vector Measures, Math. Surveys 15, Amer. Math. Soc., Providence, 1977.

12.

[12] I. Ekeland, A. Galichon, and M. Henry, Comonotonic measures of multivariate risks, Math. Finance 22 (2012), no. 1, 109–132. 1278.91085 10.1111/j.1467-9965.2010.00453.x[12] I. Ekeland, A. Galichon, and M. Henry, Comonotonic measures of multivariate risks, Math. Finance 22 (2012), no. 1, 109–132. 1278.91085 10.1111/j.1467-9965.2010.00453.x

13.

[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.[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.

[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.[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.

[15] I. S. Gradshteyn and J. M. Ryzhik, Table of Integrals, Series, and Products, 6th ed., Academic Press, San Diego, 2000. 0981.65001[15] I. S. Gradshteyn and J. M. Ryzhik, Table of Integrals, Series, and Products, 6th ed., Academic Press, San Diego, 2000. 0981.65001

16.

[16] I. Halperin, Function spaces, Canad. J. Math. 5 (1953), 273–288.[16] I. Halperin, Function spaces, Canad. J. Math. 5 (1953), 273–288.

17.

[17] G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge Math. Lib., Cambridge Univ. Press, Cambridge, 1988.[17] G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge Math. Lib., Cambridge Univ. Press, Cambridge, 1988.

18.

[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. 1239.46053 10.1090/S0002-9939-2011-10835-9[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. 1239.46053 10.1090/S0002-9939-2011-10835-9

19.

[19] S. Kusuoka, “On law invariant coherent risk measure” in Advances in Mathematical Economics, Vol. 3, Math. Econ. 3, Springer, Tokyo, 2001, 83–95. 1010.60030 10.1007/978-4-431-67891-5[19] S. Kusuoka, “On law invariant coherent risk measure” in Advances in Mathematical Economics, Vol. 3, Math. Econ. 3, Springer, Tokyo, 2001, 83–95. 1010.60030 10.1007/978-4-431-67891-5

20.

[20] G. G. Lorentz, On the theory of spaces $\Lambda$, Pacific J. Math. 1 (1951), no. 3, 411–429.[20] G. G. Lorentz, On the theory of spaces $\Lambda$, Pacific J. Math. 1 (1951), no. 3, 411–429.

21.

[21] G. G. Lorentz, Bernstein Polynomials, 2nd ed., Chelsea, New York, 1986.[21] G. G. Lorentz, Bernstein Polynomials, 2nd ed., Chelsea, New York, 1986.

22.

[22] A. J. McNeil, R. Frey, and P. Embrechts, Quantitative Risk Management: Concepts, Techniques and Tools, Princet. Ser. Finance, Princeton Univ. Press, Princeton, 2005. 1089.91037[22] A. J. McNeil, R. Frey, and P. Embrechts, Quantitative Risk Management: Concepts, Techniques and Tools, Princet. Ser. Finance, Princeton Univ. Press, Princeton, 2005. 1089.91037

23.

[23] W. Ogryczak and A. Ruszczyński, Dual stochastic dominance and related mean-risk models, SIAM J. Optim. 13 (2002), no. 1, 60–78. 1022.91017 10.1137/S1052623400375075[23] W. Ogryczak and A. Ruszczyński, Dual stochastic dominance and related mean-risk models, SIAM J. Optim. 13 (2002), no. 1, 60–78. 1022.91017 10.1137/S1052623400375075

24.

[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.[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.

[25] G. Ch. Pflug and W. Römisch, Modeling, Measuring and Managing Risk, World Scientific, Hackensack, NJ, 2007.[25] G. Ch. Pflug and W. Römisch, Modeling, Measuring and Managing Risk, World Scientific, Hackensack, NJ, 2007.

26.

[26] A. Pichler, The natural Banach space for version independent risk measures, Insurance Math. Econom. 53 (2013), no. 2, 405–415. 1304.91129 10.1016/j.insmatheco.2013.07.005[26] A. Pichler, The natural Banach space for version independent risk measures, Insurance Math. Econom. 53 (2013), no. 2, 405–415. 1304.91129 10.1016/j.insmatheco.2013.07.005

27.

[27] A. Pichler, Insurance pricing under ambiguity, Eur. Actuar. J. 4 (2014), no. 2, 335–364. 1329.91073 10.1007/s13385-014-0099-7[27] A. Pichler, Insurance pricing under ambiguity, Eur. Actuar. J. 4 (2014), no. 2, 335–364. 1329.91073 10.1007/s13385-014-0099-7

28.

[28] A. Pichler, A quantitative comparison of risk measures, Ann. Oper. Res. 254 (2017), no. 1–2, 251–275. 06764426 10.1007/s10479-017-2397-3[28] A. Pichler, A quantitative comparison of risk measures, Ann. Oper. Res. 254 (2017), no. 1–2, 251–275. 06764426 10.1007/s10479-017-2397-3

29.

[29] L. Rüschendorf, Law invariant convex risk measures for portfolio vectors, Statist. Decisions 24 (2006), no. 1, 97–108. MR2323190[29] L. Rüschendorf, Law invariant convex risk measures for portfolio vectors, Statist. Decisions 24 (2006), no. 1, 97–108. MR2323190

30.

[30] G. Svindland, Subgradients of law-invariant convex risk measures on ${L}^{1}$, Statist. Decisions 27 (2009), no. 2, 169–199.[30] G. Svindland, Subgradients of law-invariant convex risk measures on ${L}^{1}$, Statist. Decisions 27 (2009), no. 2, 169–199.

31.

[31] D. Williams, Probability with Martingales, Cambridge Univ. Press, Cambridge, 1991. 0722.60001[31] D. Williams, Probability with Martingales, Cambridge Univ. Press, Cambridge, 1991. 0722.60001

Citation Download Citation

Thomas Kalmes and Alois Pichler "On Banach spaces of vector-valued random variables and their duals motivated by risk measures," Banach Journal of Mathematical Analysis 12(4), 773-807, (October 2018). https://doi.org/10.1215/17358787-2017-0026

Received: 29 March 2017; Accepted: 19 June 2017; Published: October 2018

Access the abstract

JOURNAL ARTICLE
35 PAGES

DOWNLOAD PDF + SAVE TO MY LIBRARY