# American Institute of Mathematical Sciences

January  2018, 3: 6 doi: 10.1186/s41546-018-0032-0

## Risk excess measures induced by hemi-metrics

 1. Toulouse School of Economics-Université Toulouse 1 Capitole, Manufacture des Tabacs, 21 Allée de Brienne, 31000 Toulouse, France; 2. Abteilung für Mathematische Stochastik, Albert-Ludwigs University of Freiburg, Eckerstrasse 1, D-79104 Freiburg, Germany

Received  September 26, 2017 Revised  May 07, 2018 Published  June 2018

The main aim of this paper is to introduce the notion of risk excess measure, to analyze its properties, and to describe some basic construction methods. To compare the risk excess of one distribution Q w.r.t. a given risk distribution P, we apply the concept of hemi-metrics on the space of probability measures. This view of risk comparison has a natural basis in the extension of orderings and hemi-metrics on the underlying space to the level of probability measures. Basic examples of these kind of extensions are induced by mass transportation and by function class induced orderings. Our view towards measuring risk excess adds to the usually considered method to compare risks of Q and P by the values ρ(Q), ρ(P) of a risk measure ρ. We argue that the difference ρ(Q)-ρ(P) neglects relevant aspects of the risk excess which are adequately described by the new notion of risk excess measure. We derive various concrete classes of risk excess measures and discuss corresponding ordering and measure extension properties.
Citation: Olivier P. Faugeras, Ludger Rüschendorf. Risk excess measures induced by hemi-metrics. Probability, Uncertainty and Quantitative Risk, 2018, 3 (0) : 6-. doi: 10.1186/s41546-018-0032-0
##### References:
 [1] Artzner, P, Delbaen, F, Eber, J-M, Heath, D:Coherent measures of risk. Math. Finance. 9(3), 203-228(1999). https://doi.org/10.1111/1467-9965.00068 [2] Berkes, I, Philipp, W:Approximation theorems for independent and weakly dependent random vectors. Ann. Probab. 7(1), 29-54 (1979) [3] Burgert, C, Rüschendorf, L:Consistent risk measures for portfolio vectors. Insurance Math. Econom. 38(2), 289-297 (2006). https://doi.org/10.1016/j.insmatheco.2005.08.008 [4] Cambanis, S, Simons, G, Stout, W:Inequalities for Ek(X, Y) when the marginals are fixed. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete. 36(4), 285-294 (1976). https://doi.org/10.1007/BF00532695 [5] Capéraà, P, Van Cutsem, B:Méthodes et Modèles en Statistique Non Paramétrique, p. 359. Les Presses de l'Université Laval, Sainte-Foy, QC; Dunod, Paris (1988). Exposé fondamental.[Basic exposition], With a foreword by Capéraà, Van Cutsem and Alain Baille [6] Delbaen, F:Coherent risk measures on general probability spaces. Advances in Finance and Stochastics, pp. 1-37. Springer, Berlin (2002) [7] Dudley, RM:Distances of probability measures and random variables. Ann. Math. Statist. 39, 1563-1572(1968). https://doi.org/10.1007/978-1-4419-5821-1_4 [8] Dudley, RM:Probabilities and Metrics, p. 126. Matematisk Institut, Aarhus Universitet, Aarhus (1976). Convergence of laws on metric spaces, with a view to statistical testing, Lecture Notes Series, No. 45 [9] Dudley, RM:Real Analysis and Probability. Cambridge Studies in Advanced Mathematics, vol. 74, p. 555. Cambridge University Press, Cambridge (2002). https://doi.org/10.1017/CBO9780511755347. Revised reprint of the 1989 original [10] Faugeras, OP, Rüschendorf, L:Markov morphisms:a combined copula and mass transportation approach to multivariate quantiles. Math. Applicanda. 45, 3-45 (2017) [11] Föllmer, H, Schied, A:Stochastic Finance. De Gruyter Studies in Mathematics, vol. 27, p. 422. Walter de Gruyter & Co., Berlin (2002). https://doi.org/10.1515/9783110198065. An introduction in discrete time [12] Goubault-Larrecq, J:Non-Hausdorff Topology and Domain Theory. New Mathematical Monographs, vol. 22, p. 491. Cambridge University Press, Cambridge (2013). https://doi.org/10.1017/CBO9781139524438.[On the cover:Selected topics in point-set topology] [13] Jouini, E, Meddeb, M, Touzi, N:Vector-valued coherent risk measures. Finance Stoch. 8(4), 531-552(2004). https://doi.org/10.1007/s00780-004-0127-6 [14] Kellerer, HG:Duality theorems for marginal problems. Z. Wahrsch. Verw. Gebiete. 67(4), 399-432 (1984). https://doi.org/10.1007/BF00532047 [15] Koenker, R:Quantile Regression. Econometric Society Monographs, vol. 38, p. 349. Cambridge University Press, Cambridge (2005). https://doi.org/10.1017/CBO9780511754098 [16] Lehmann, EL:Some concepts of dependence. Ann. Math. Statist. 37, 1137-1153 (1966). https://doi.org/10.1214/aoms/1177699260 [17] Marshall, AW, Olkin, I, Arnold, BC:Inequalities:Theory of Majorization and Its Applications. 2nd edn. Springer Series in Statistics, p. 909. Springer (2011). https://doi.org/10.1007/978-0-387-68276-1 [18] Müller, A:Integral probability metrics and their generating classes of functions. Adv. in Appl. Probab. 29(2), 429-443 (1997) [19] Nachbin, L:Topology and Order. Translated from the Portuguese by Lulu Bechtolsheim. Van Nostrand Mathematical Studies, No. 4, p. 122. D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London(1965) [20] Nelsen, RB:An Introduction to Copulas. 2nd edn. Springer Series in Statistics, p. 269. Springer, New York (2006) [21] Rachev, ST, Rüschendorf, L.:Approximation of sums by compound Poisson distributions with respect to stop-loss distances. Adv. in Appl. Probab. 22(2), 350-374 (1990) [22] Rachev, ST:Probability Metrics and the Stability of Stochastic Models. Wiley Series in Probability and Mathematical Statistics:Applied Probability and Statistics, p. 494. John Wiley & Sons, Ltd., Chichester (1991) [23] Rachev, ST, Rüschendorf, L:Mass Transportation Problems. Vol. I. Probability and its Applications (New York), vol. 1, p. 508. Springer-Verlag, New York (1998). Theory [24] Rachev, ST, Klebanov, LB, Stoyanov, SV, Fabozzi, FJ:The Methods of Distances in the Theory of Probability and Statistics, p. 619. Springer (2013). https://doi.org/10.1007/978-1-4614-4869-3 [25] Rosenberger, J, Gasko, M:Understanding robust and exploratory data analysis. Wiley Classics Library, p. 447. Wiley-Interscience, New York (2000). Chap. Comparing Location Estimators:Trimmed Means, Medians, and Trimean. Revised and updated reprint of the 1983 original [26] Rüschendorf, L.:Monotonicity and unbiasedness of tests via a.s. constructions. Statistics. 17(2), 221-230(1986). https://doi.org/10.1080/02331888608801931 [27] Rüschendorf, L:Fréchet bounds and their applications. In:Dall'Aglio, G, Kotz, S, Salinetti, G (eds.) Advances in Probability Distributions with Given Marginals:Beyond the Copulas, pp. 151-187. Springer, Dordrecht (1991). https://doi.org/10.1007/978-94-011-3466-8 [28] Rüschendorf, L.:On the distributional transform, Sklar's theorem, and the empirical copula process. J. Statist. Plann. Inference. 139(11), 3921-3927 (2009). https://doi.org/10.1016/j.jspi.2009.05.030 [29] Rüschendorf, L.:Mathematical Risk Analysis. Springer Series in Operations Research and Financial Engineering, p. 408. Springer (2013). https://doi.org/10.1007/978-3-642-33590-7. Dependence, risk bounds, optimal allocations and portfolios [30] Sriperumbudur, BK, Fukumizu, K, Gretton, A, Schölkopf, B, Lanckriet, GRG:On the empirical estimation of integral probability metrics. Electron. J. Stat. 6, 1550-1599 (2012) [31] Strassen, V:The existence of probability measures with given marginals. Ann. Math. Statist. 36, 423-439(1965). https://doi.org/10.1214/aoms/1177700153 [32] Tchen, AH:Inequalities for distributions with given marginals. Ann. Probab. 8(4), 814-827 (1980) [33] Villani, C:Topics in Optimal Transportation. Graduate Studies in Mathematics, vol. 58, p. 370. American Mathematical Society (2003). https://doi.org/10.1007/b12016 [34] Zolotarev, VM:Modern Theory of Summation of Random Variables. Modern Probability and Statistics, p. 412. VSP, Utrecht (1997). https://doi.org/10.1515/9783110936537

show all references

##### References:
 [1] Artzner, P, Delbaen, F, Eber, J-M, Heath, D:Coherent measures of risk. Math. Finance. 9(3), 203-228(1999). https://doi.org/10.1111/1467-9965.00068 [2] Berkes, I, Philipp, W:Approximation theorems for independent and weakly dependent random vectors. Ann. Probab. 7(1), 29-54 (1979) [3] Burgert, C, Rüschendorf, L:Consistent risk measures for portfolio vectors. Insurance Math. Econom. 38(2), 289-297 (2006). https://doi.org/10.1016/j.insmatheco.2005.08.008 [4] Cambanis, S, Simons, G, Stout, W:Inequalities for Ek(X, Y) when the marginals are fixed. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete. 36(4), 285-294 (1976). https://doi.org/10.1007/BF00532695 [5] Capéraà, P, Van Cutsem, B:Méthodes et Modèles en Statistique Non Paramétrique, p. 359. Les Presses de l'Université Laval, Sainte-Foy, QC; Dunod, Paris (1988). Exposé fondamental.[Basic exposition], With a foreword by Capéraà, Van Cutsem and Alain Baille [6] Delbaen, F:Coherent risk measures on general probability spaces. Advances in Finance and Stochastics, pp. 1-37. Springer, Berlin (2002) [7] Dudley, RM:Distances of probability measures and random variables. Ann. Math. Statist. 39, 1563-1572(1968). https://doi.org/10.1007/978-1-4419-5821-1_4 [8] Dudley, RM:Probabilities and Metrics, p. 126. Matematisk Institut, Aarhus Universitet, Aarhus (1976). Convergence of laws on metric spaces, with a view to statistical testing, Lecture Notes Series, No. 45 [9] Dudley, RM:Real Analysis and Probability. Cambridge Studies in Advanced Mathematics, vol. 74, p. 555. Cambridge University Press, Cambridge (2002). https://doi.org/10.1017/CBO9780511755347. Revised reprint of the 1989 original [10] Faugeras, OP, Rüschendorf, L:Markov morphisms:a combined copula and mass transportation approach to multivariate quantiles. Math. Applicanda. 45, 3-45 (2017) [11] Föllmer, H, Schied, A:Stochastic Finance. De Gruyter Studies in Mathematics, vol. 27, p. 422. Walter de Gruyter & Co., Berlin (2002). https://doi.org/10.1515/9783110198065. An introduction in discrete time [12] Goubault-Larrecq, J:Non-Hausdorff Topology and Domain Theory. New Mathematical Monographs, vol. 22, p. 491. Cambridge University Press, Cambridge (2013). https://doi.org/10.1017/CBO9781139524438.[On the cover:Selected topics in point-set topology] [13] Jouini, E, Meddeb, M, Touzi, N:Vector-valued coherent risk measures. Finance Stoch. 8(4), 531-552(2004). https://doi.org/10.1007/s00780-004-0127-6 [14] Kellerer, HG:Duality theorems for marginal problems. Z. Wahrsch. Verw. Gebiete. 67(4), 399-432 (1984). https://doi.org/10.1007/BF00532047 [15] Koenker, R:Quantile Regression. Econometric Society Monographs, vol. 38, p. 349. Cambridge University Press, Cambridge (2005). https://doi.org/10.1017/CBO9780511754098 [16] Lehmann, EL:Some concepts of dependence. Ann. Math. Statist. 37, 1137-1153 (1966). https://doi.org/10.1214/aoms/1177699260 [17] Marshall, AW, Olkin, I, Arnold, BC:Inequalities:Theory of Majorization and Its Applications. 2nd edn. Springer Series in Statistics, p. 909. Springer (2011). https://doi.org/10.1007/978-0-387-68276-1 [18] Müller, A:Integral probability metrics and their generating classes of functions. Adv. in Appl. Probab. 29(2), 429-443 (1997) [19] Nachbin, L:Topology and Order. Translated from the Portuguese by Lulu Bechtolsheim. Van Nostrand Mathematical Studies, No. 4, p. 122. D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London(1965) [20] Nelsen, RB:An Introduction to Copulas. 2nd edn. Springer Series in Statistics, p. 269. Springer, New York (2006) [21] Rachev, ST, Rüschendorf, L.:Approximation of sums by compound Poisson distributions with respect to stop-loss distances. Adv. in Appl. Probab. 22(2), 350-374 (1990) [22] Rachev, ST:Probability Metrics and the Stability of Stochastic Models. Wiley Series in Probability and Mathematical Statistics:Applied Probability and Statistics, p. 494. John Wiley & Sons, Ltd., Chichester (1991) [23] Rachev, ST, Rüschendorf, L:Mass Transportation Problems. Vol. I. Probability and its Applications (New York), vol. 1, p. 508. Springer-Verlag, New York (1998). Theory [24] Rachev, ST, Klebanov, LB, Stoyanov, SV, Fabozzi, FJ:The Methods of Distances in the Theory of Probability and Statistics, p. 619. Springer (2013). https://doi.org/10.1007/978-1-4614-4869-3 [25] Rosenberger, J, Gasko, M:Understanding robust and exploratory data analysis. Wiley Classics Library, p. 447. Wiley-Interscience, New York (2000). Chap. Comparing Location Estimators:Trimmed Means, Medians, and Trimean. Revised and updated reprint of the 1983 original [26] Rüschendorf, L.:Monotonicity and unbiasedness of tests via a.s. constructions. Statistics. 17(2), 221-230(1986). https://doi.org/10.1080/02331888608801931 [27] Rüschendorf, L:Fréchet bounds and their applications. In:Dall'Aglio, G, Kotz, S, Salinetti, G (eds.) Advances in Probability Distributions with Given Marginals:Beyond the Copulas, pp. 151-187. Springer, Dordrecht (1991). https://doi.org/10.1007/978-94-011-3466-8 [28] Rüschendorf, L.:On the distributional transform, Sklar's theorem, and the empirical copula process. J. Statist. Plann. Inference. 139(11), 3921-3927 (2009). https://doi.org/10.1016/j.jspi.2009.05.030 [29] Rüschendorf, L.:Mathematical Risk Analysis. Springer Series in Operations Research and Financial Engineering, p. 408. Springer (2013). https://doi.org/10.1007/978-3-642-33590-7. Dependence, risk bounds, optimal allocations and portfolios [30] Sriperumbudur, BK, Fukumizu, K, Gretton, A, Schölkopf, B, Lanckriet, GRG:On the empirical estimation of integral probability metrics. Electron. J. Stat. 6, 1550-1599 (2012) [31] Strassen, V:The existence of probability measures with given marginals. Ann. Math. Statist. 36, 423-439(1965). https://doi.org/10.1214/aoms/1177700153 [32] Tchen, AH:Inequalities for distributions with given marginals. Ann. Probab. 8(4), 814-827 (1980) [33] Villani, C:Topics in Optimal Transportation. Graduate Studies in Mathematics, vol. 58, p. 370. American Mathematical Society (2003). https://doi.org/10.1007/b12016 [34] Zolotarev, VM:Modern Theory of Summation of Random Variables. Modern Probability and Statistics, p. 412. VSP, Utrecht (1997). https://doi.org/10.1515/9783110936537
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