    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

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:
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Exposé fondamental.[Basic exposition], With a foreword by Capéraà, Van Cutsem and Alain Baille Google Scholar  Delbaen, F:Coherent risk measures on general probability spaces. Advances in Finance and Stochastics, pp. 1-37. Springer, Berlin (2002) Google Scholar  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 Google Scholar  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 Google Scholar  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 Google Scholar  Faugeras, OP, Rüschendorf, L:Markov morphisms:a combined copula and mass transportation approach to multivariate quantiles. Math. Applicanda. 45, 3-45 (2017) Google Scholar  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 Google Scholar  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] Google Scholar  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 Google Scholar  Kellerer, HG:Duality theorems for marginal problems. Z. Wahrsch. Verw. Gebiete. 67(4), 399-432 (1984). https://doi.org/10.1007/BF00532047 Google Scholar  Koenker, R:Quantile Regression. Econometric Society Monographs, vol. 38, p. 349. Cambridge University Press, Cambridge (2005). https://doi.org/10.1017/CBO9780511754098 Google Scholar  Lehmann, EL:Some concepts of dependence. Ann. Math. Statist. 37, 1137-1153 (1966). https://doi.org/10.1214/aoms/1177699260 Google Scholar  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 Google Scholar  Müller, A:Integral probability metrics and their generating classes of functions. Adv. in Appl. Probab. 29(2), 429-443 (1997) Google Scholar  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) Google Scholar  Nelsen, RB:An Introduction to Copulas. 2nd edn. Springer Series in Statistics, p. 269. Springer, New York (2006) Google Scholar  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) Google Scholar  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) Google Scholar  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 Google Scholar  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 Google Scholar  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 Google Scholar  Rüschendorf, L.:Monotonicity and unbiasedness of tests via a.s. constructions. Statistics. 17(2), 221-230(1986). https://doi.org/10.1080/02331888608801931 Google Scholar  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 Google Scholar  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 Google Scholar  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 Google Scholar  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) Google Scholar  Strassen, V:The existence of probability measures with given marginals. Ann. Math. Statist. 36, 423-439(1965). https://doi.org/10.1214/aoms/1177700153 Google Scholar  Tchen, AH:Inequalities for distributions with given marginals. Ann. Probab. 8(4), 814-827 (1980) Google Scholar  Villani, C:Topics in Optimal Transportation. Graduate Studies in Mathematics, vol. 58, p. 370. American Mathematical Society (2003). https://doi.org/10.1007/b12016 Google Scholar  Zolotarev, VM:Modern Theory of Summation of Random Variables. Modern Probability and Statistics, p. 412. VSP, Utrecht (1997). https://doi.org/10.1515/9783110936537 Google Scholar

show all references

##### References:
  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 Google Scholar  Berkes, I, Philipp, W:Approximation theorems for independent and weakly dependent random vectors. Ann. Probab. 7(1), 29-54 (1979) Google Scholar  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 Google Scholar  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 Google Scholar  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 Google Scholar  Delbaen, F:Coherent risk measures on general probability spaces. Advances in Finance and Stochastics, pp. 1-37. Springer, Berlin (2002) Google Scholar  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 Google Scholar  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 Google Scholar  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 Google Scholar  Faugeras, OP, Rüschendorf, L:Markov morphisms:a combined copula and mass transportation approach to multivariate quantiles. Math. Applicanda. 45, 3-45 (2017) Google Scholar  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 Google Scholar  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] Google Scholar  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 Google Scholar  Kellerer, HG:Duality theorems for marginal problems. Z. Wahrsch. Verw. Gebiete. 67(4), 399-432 (1984). https://doi.org/10.1007/BF00532047 Google Scholar  Koenker, R:Quantile Regression. Econometric Society Monographs, vol. 38, p. 349. Cambridge University Press, Cambridge (2005). https://doi.org/10.1017/CBO9780511754098 Google Scholar  Lehmann, EL:Some concepts of dependence. Ann. Math. Statist. 37, 1137-1153 (1966). https://doi.org/10.1214/aoms/1177699260 Google Scholar  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 Google Scholar  Müller, A:Integral probability metrics and their generating classes of functions. Adv. in Appl. Probab. 29(2), 429-443 (1997) Google Scholar  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) Google Scholar  Nelsen, RB:An Introduction to Copulas. 2nd edn. Springer Series in Statistics, p. 269. Springer, New York (2006) Google Scholar  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) Google Scholar  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) Google Scholar  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 Google Scholar  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 Google Scholar  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 Google Scholar  Rüschendorf, L.:Monotonicity and unbiasedness of tests via a.s. constructions. Statistics. 17(2), 221-230(1986). https://doi.org/10.1080/02331888608801931 Google Scholar  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 Google Scholar  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 Google Scholar  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 Google Scholar  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) Google Scholar  Strassen, V:The existence of probability measures with given marginals. Ann. Math. Statist. 36, 423-439(1965). https://doi.org/10.1214/aoms/1177700153 Google Scholar  Tchen, AH:Inequalities for distributions with given marginals. Ann. Probab. 8(4), 814-827 (1980) Google Scholar  Villani, C:Topics in Optimal Transportation. Graduate Studies in Mathematics, vol. 58, p. 370. American Mathematical Society (2003). https://doi.org/10.1007/b12016 Google Scholar  Zolotarev, VM:Modern Theory of Summation of Random Variables. Modern Probability and Statistics, p. 412. VSP, Utrecht (1997). https://doi.org/10.1515/9783110936537 Google Scholar
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