    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:
  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 Berkes, I, Philipp, W:Approximation theorems for independent and weakly dependent random vectors. Ann. Probab. 7(1), 29-54 (1979) 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 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 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 Delbaen, F:Coherent risk measures on general probability spaces. Advances in Finance and Stochastics, pp. 1-37. Springer, Berlin (2002) 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 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 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 Faugeras, OP, Rüschendorf, L:Markov morphisms:a combined copula and mass transportation approach to multivariate quantiles. Math. Applicanda. 45, 3-45 (2017) 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 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] 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 Kellerer, HG:Duality theorems for marginal problems. Z. Wahrsch. Verw. Gebiete. 67(4), 399-432 (1984). https://doi.org/10.1007/BF00532047 Koenker, R:Quantile Regression. Econometric Society Monographs, vol. 38, p. 349. Cambridge University Press, Cambridge (2005). https://doi.org/10.1017/CBO9780511754098 Lehmann, EL:Some concepts of dependence. Ann. Math. Statist. 37, 1137-1153 (1966). https://doi.org/10.1214/aoms/1177699260 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 Müller, A:Integral probability metrics and their generating classes of functions. Adv. in Appl. Probab. 29(2), 429-443 (1997) 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) Nelsen, RB:An Introduction to Copulas. 2nd edn. Springer Series in Statistics, p. 269. Springer, New York (2006) 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) 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) 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 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 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 Rüschendorf, L.:Monotonicity and unbiasedness of tests via a.s. constructions. Statistics. 17(2), 221-230(1986). https://doi.org/10.1080/02331888608801931 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 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 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 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) Strassen, V:The existence of probability measures with given marginals. Ann. Math. Statist. 36, 423-439(1965). https://doi.org/10.1214/aoms/1177700153 Tchen, AH:Inequalities for distributions with given marginals. Ann. Probab. 8(4), 814-827 (1980) Villani, C:Topics in Optimal Transportation. Graduate Studies in Mathematics, vol. 58, p. 370. American Mathematical Society (2003). https://doi.org/10.1007/b12016 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:
  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 Berkes, I, Philipp, W:Approximation theorems for independent and weakly dependent random vectors. Ann. Probab. 7(1), 29-54 (1979) 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 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 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 Delbaen, F:Coherent risk measures on general probability spaces. Advances in Finance and Stochastics, pp. 1-37. Springer, Berlin (2002) 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 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 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 Faugeras, OP, Rüschendorf, L:Markov morphisms:a combined copula and mass transportation approach to multivariate quantiles. Math. Applicanda. 45, 3-45 (2017) 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 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] 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 Kellerer, HG:Duality theorems for marginal problems. Z. Wahrsch. Verw. Gebiete. 67(4), 399-432 (1984). https://doi.org/10.1007/BF00532047 Koenker, R:Quantile Regression. Econometric Society Monographs, vol. 38, p. 349. Cambridge University Press, Cambridge (2005). https://doi.org/10.1017/CBO9780511754098 Lehmann, EL:Some concepts of dependence. Ann. Math. Statist. 37, 1137-1153 (1966). https://doi.org/10.1214/aoms/1177699260 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 Müller, A:Integral probability metrics and their generating classes of functions. Adv. in Appl. Probab. 29(2), 429-443 (1997) 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) Nelsen, RB:An Introduction to Copulas. 2nd edn. Springer Series in Statistics, p. 269. Springer, New York (2006) 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) 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) 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 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 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 Rüschendorf, L.:Monotonicity and unbiasedness of tests via a.s. constructions. Statistics. 17(2), 221-230(1986). https://doi.org/10.1080/02331888608801931 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 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 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 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) Strassen, V:The existence of probability measures with given marginals. Ann. Math. Statist. 36, 423-439(1965). https://doi.org/10.1214/aoms/1177700153 Tchen, AH:Inequalities for distributions with given marginals. Ann. Probab. 8(4), 814-827 (1980) Villani, C:Topics in Optimal Transportation. Graduate Studies in Mathematics, vol. 58, p. 370. American Mathematical Society (2003). https://doi.org/10.1007/b12016 Zolotarev, VM:Modern Theory of Summation of Random Variables. Modern Probability and Statistics, p. 412. VSP, Utrecht (1997). https://doi.org/10.1515/9783110936537 Ugo Bessi. The stochastic value function in metric measure spaces. Discrete and Continuous Dynamical Systems, 2017, 37 (4) : 1819-1839. doi: 10.3934/dcds.2017076  David Kinderlehrer, Adrian Tudorascu. Transport via mass transportation. Discrete and Continuous Dynamical Systems - B, 2006, 6 (2) : 311-338. doi: 10.3934/dcdsb.2006.6.311  Gershon Wolansky. Limit theorems for optimal mass transportation and applications to networks. Discrete and Continuous Dynamical Systems, 2011, 30 (1) : 365-374. doi: 10.3934/dcds.2011.30.365  A. Daducci, A. Marigonda, G. Orlandi, R. Posenato. Neuronal Fiber--tracking via optimal mass transportation. 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