High risk scenarios and extremes a geometric approach

ZLAM_Balkema_titelei.qxd 5.7.2007 9:46 Uhr Seite M M S E M E S E S E M E S M S ZLAM_Balkema_titelei.qxd 5.7.2007 9:46 Uhr Seite Zurich Lectures in Advanced Mathematics Edited by Erwin Bolthausen (Managing Editor), Freddy Delbaen, Thomas Kappeler (Managing Editor), Christoph Schwab, Michael Struwe, Gisbert Wüstholz Mathematics in Zurich has a long and distinguished tradition, in which the writing of lecture notes volumes and research monographs play a prominent part The Zurich Lectures in Advanced Mathematics series aims to make some of these publications better known to a wider audience The series has three main constituents: lecture notes on advanced topics given by internationally renowned experts, graduate text books designed for the joint graduate program in Mathematics of the ETH and the University of Zurich, as well as contributions from researchers in residence at the mathematics research institute, FIM-ETH Moderately priced, concise and lively in style, the volumes of this series will appeal to researchers and students alike, who seek an informed introduction to important areas of current research Previously published in this series: Yakov B Pesin, Lectures on partial hyperbolicity and stable ergodicity Sun-Yung Alice Chang, Non-linear Elliptic Equations in Conformal Geometry Sergei B Kuksin, Randomly forced nonlinear PDEs and statistical hydrodynamics in space dimensions Pavel Etingof, Calogero-Moser systems and representation theory Published with the support of the Huber-Kudlich-Stiftung, Zürich ZLAM_Balkema_titelei.qxd 5.7.2007 9:46 Uhr Seite Guus Balkema Paul Embrechts High Risk Scenarios and Extremes S E A geometric approach M M E S M S E M E S European Mathematical Society ZLAM_Balkema_titelei.qxd 5.7.2007 9:46 Uhr A A Balkema Department of Mathematics University of Amsterdam Plantage Muidergracht 24 1018 TV Amsterdam Netherlands guus@science.uva.nl Seite P Embrechts Department of Mathematics ETH Zurich 8092 Zurich Switzerland embrechts@math.ethz.ch The cover shows part of the edge and of the convex hull of a realization of the Gauss-exponential point process This point process may be used to model extremes in, for instance, a bivariate Gaussian or hyperbolic distribution The underlying theory is treated in Chapter III 2000 Mathematics Subject Classification 60G70, 60F99, 91B30, 91B70, 62G32, 60G55 ISBN 978-3-03719-035-7 The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography, and the detailed bibliographic data are available on the Internet at http://www.helveticat.ch This work is subject to copyright All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks For any kind of use permission of the copyright owner must be obtained © 2007 European Mathematical Society Contact address: European Mathematical Society Publishing House Seminar for Applied Mathematics ETH-Zentrum FLI C4 CH-8092 Zürich Switzerland Phone: +41 (0)44 632 34 36 Email: info@ems-ph.org Homepage: www.ems-ph.org Typeset using the authors’ TEX files: I Zimmermann, Freiburg Printed on acid-free paper produced from chlorine-free pulp TCF °° Printed in Germany 987654321 Annemarie and my daughters have looked on my labour with a mixture of indulgence and respect Thank you for your patience Guus For Gerda, Krispijn, Eline and Frederik Thank you ever so much for the wonderful love and support over the many years Paul Foreword These lecture notes describe a way of looking at extremes in a multivariate setting We shall introduce a continuous one-parameter family of multivariate generalized Pareto distributions that describe the asymptotic behaviour of exceedances over linear thresholds The one-dimensional theory has proved to be important in insurance, finance and risk management It has also been applied in quality control and meteorology The multivariate limit theory presented here is developed with similar applications in mind Apart from looking at the asymptotics of the conditional distributions given the exceedance over a linear threshold – the so-called high risk scenarios – one may look at the behaviour of the sample cloud in the given direction The theory then presents a geometric description of the multivariate extremes in terms of limiting Poisson point processes Our terminology distinguishes between extreme value theory and the limit theory for coordinatewise maxima Not all extreme values are coordinatewise extremes! In the univariate theory there is a simple relation between the asymptotics of extremes and of exceedances One of the aims of this book is to elucidate the relation between maxima and exceedances in the multivariate setting Both exceedances over linear and elliptic thresholds will be treated A complete classification of the limit laws is given, and in certain instances a full description of the domains of attraction Our approach will be geometrical Symmetry will play an important role The charm of the limit theory for coordinatewise maxima is its close relationship with multivariate distribution functions The univariate marginals allow a quick check to see whether a multivariate limit is feasible and what its marginals will look like Linear and even non-linear monotone transformations of the coordinates are easily accommodated in the theory Multivariate distribution functions provide a simple characterization of the max-stable limit distributions and of their domains of attraction Weak convergence to the max-stable distribution function has almost magical consequences In the case of greatest practical interest, positive vectors with heavy tailed marginal distribution functions, it entails convergence of the normalized sample clouds and their convex hulls Distribution functions are absent in our approach They are so closely linked to coordinatewise maxima that they not accommodate any other interpretation of extremes Moreover, distribution functions obscure an issue which is of paramount importance in the analysis of samples, the convergence of the normalized sample cloud to a limiting Poisson point process Probability measures and their densities on Rd provide an alternative approach which is fruitful both in developing the theory and in handling applications The theory presented here may be regarded as a useful complement to the multivariate theory of coordinatewise maxima viii Foreword These notes contain the text of the handouts, substantially revised, for a Nachdiplom course on point processes and extremes given at the ETH Zurich in the spring semester of 2005, with the twenty sections of the book roughly corresponding to weekly two-hour lectures Acknowledgements Thanks to Matthias Degen, Andrea Höing and Silja Kinnebrock for taking care of the figures, to Marcel Visser for the figures on the AEX, and to Hicham Zmarrou for the figures on the DAX We thank Johanna Nešlehová for her assistance with technical problems We also thank her for her close reading of the extremal sections of the manuscript and her valuable comments A special word of thanks to Nick Bingham for his encouraging words, his extensive commentary on an earlier version of the text, and his advice on matters of style and punctuation The following persons helped in the important final stages of proofreading: Daniel Alai, Matthias Degen, Dominik Lambrigger, Natalia Lysenko, Parthanil Roy and Johanna Ziegel Dietmar Salamon helped us to understand why discontinuities in the normalization are unavoidable in certain dimensions We would also like to thank Erwin Bolthausen and Thomas Kappeler, who as editors of the series gave us useful input early on in the project Thomas Hintermann, Manfred Karbe and Irene Zimmermann did an excellent job transforming the MS into a book Guus Balkema would like to thank the Forschungsinstitut für Mathematik (FIM) of the ETH Zurich for financial support, and the Department of Mathematics of the ETH Zurich for its hospitality He would also like to express his gratitude to the Korteweg–de Vries Instituut of the University of Amsterdam for the pleasant working conditions and the liberal use of their facilities Contents Foreword vii Introduction Preview A recipe Contents Notation I 13 13 31 36 Point Processes 41 An intuitive approach 1.1 A brief shower 1.2 Sample cloud mixtures 1.3 Random sets and random measures 1.4 The mean measure 1.5* Enumerating the points 1.6 Definitions Poisson point processes 2.1 Poisson mixtures of sample clouds 2.2 The distribution of a point process 2.3 Definition of the Poisson point process 2.4 Variance and covariance 2.5* The bivariate mean measure 2.6 Lévy processes 2.7 Superpositions of zero-one point processes 2.8 Mappings 2.9* Inverse maps 2.10* Marked point processes The distribution 3.1 Introduction 3.2* The Laplace transform 3.3 The distribution 3.4* The distribution of simple point processes Convergence 4.1 Introduction 4.2 The state space 4.3 Weak convergence of probability measures on metric spaces 41 41 43 44 45 46 47 48 48 49 50 51 52 54 56 58 58 62 63 63 64 65 67 69 69 70 72 Starred sections may be skipped on a first reading x Contents 4.4 Radon measures and vague convergence 4.5 Convergence of point processes Converging sample clouds 5.1 Introduction 5.2 Convergence of convex hulls, an example 5.3 Halfspaces, convex sets and cones 5.4 The intrusion cone 5.5 The convergence cone 5.6* The support function 5.7 Almost-sure convergence of the convex hulls 5.8 Convergence to the mean measure 76 78 81 81 83 84 87 89 92 93 96 II Maxima 100 The univariate theory: maxima and exceedances 6.1 Maxima 6.2 Exceedances 6.3 The domain of the exponential law 6.4 The Poisson point process associated with the limit law 6.5* Monotone transformations 6.6* The von Mises condition 6.7* Self-neglecting functions Componentwise maxima 7.1 Max-id vectors 7.2 Max-stable vectors, the stability relations 7.3 Max-stable vectors, dependence 7.4 Max-stable distributions with exponential marginals on 1; 0/ 7.5* Max-stable distributions under monotone transformations 7.6 Componentwise maxima and copulas 100 100 101 101 102 104 105 108 110 111 112 114 117 119 121 III High Risk Limit Laws 123 High risk scenarios 8.1 Introduction 8.2 The limit relation 8.3 The multivariate Gaussian distribution 8.4 The uniform distribution on a ball 8.5 Heavy tails, returns and volatility in the DAX 8.6 Some basic theory The Gauss-exponential domain, rotund sets 9.1 Introduction 123 123 125 126 128 130 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84 c.E/, convex hull, 38 cl.E/, closure, 38 D C , 37, 101, 121 D , 37, 233, 346 D h , 37, 185, 192, 346 D OS , 37 D " , 37, 120, D _ , 37, 113, 346 D ^ , 222 D.0/, 135 D /, 176, 186 D.W /, 126 , 88 @ W H ! @, 132 @ W zn ! @, 132 F , 84, 92 €, 90 GL.d /, 15 H , 84 H ƒ , 88 H , 82 H /, 88 H E , 216 int.E/, interior, 38 M " , 104 MSE, 117 N , 67 o.E t /: x t D, 227 P , 72 R, 76 RE, 33, 135, 136, U, 148 U0 , 148 adapted, 236, 251, 256, 268, 356 admissible, 199 affine expansion, 285 function, 84, 85 transformation, 15 Ansatz, 14, 24, 125, 231 asymptotic affine transformations, 16 convex sets, 23 functions, 132 measures, 209 tail continuity, 104 atom of a measure, 44 atom of a partition, 46, 250 balance, 230, 265, 272, 275, 359 theorem, 265 Basel guidelines, basic inequality, 159 beta density, spherical, 135, 178 Beurling slowly varying, 108 binomial distribution, 56 bivariate mean measure, 52 bland, 13, 124 box, coordinate, 273 CAT, 28, 36, 112, 199 Cauchy distribution, 17, 30 Cauchy equation, 297 chain, 334, 335 compact group, 294, 340, 341 370 Index compact support, 76 compact, relatively, 76 compensated Poisson point process, 97 completely steady, 218, 219 component of the identity, 343 componentwise maxima, 110, 121, 264 346, see also coordinatewise extremes condition on the boundary, 125, 186 cone, 84, 218, 222 convergence, 32, 89, 90, 94, 185, 200, 312, 314 intrusion, 32, 88, 94, 200, 312, 313, 320 proper, 84 consistency theorem, 96 contamination, 224, 360 convergence convex sets, 85 direction, 92 in distribution, 69, 79 of densities, 152 of point process, 69, 78 of types theorem, 14, 355 convex, 84 hull, 38, 82, 83, 93, 118, 225, 243 hulls, convergence of, 83, 94 set, adapted, 236 sets, convergence of, 85 support, 132, 220 coordinate affine transformations, 28, 36, 112, see also CAT coordinate box, 273 coordinatewise extremes, 110, 111, 224, 273, 346, see also componentwise maxima copula, 115, 116, 218 convergence theorem, 122 Poisson, 117, 359 sample, 359 counting measure, 76 covariance, 51, 55, 130, 137, 198, 207, 298, 353, 359 CTT, 14 DAX, 130 decomposition spectral, 22, 38, 258, 275, 333 theorem, 118 decreasing, 87 degenerate distribution, 14 Delta Project, density power family, 134 Weibull, 33, 135 dependency, 122 dependogram, 116 diffuse, 44 direction, 85 of convergence, 92 discrete skeleton, 223, 299, 350 distribution Cauchy, 17 degenerate, 14 Euclidean Pareto, 131, 171 exponential, 101, 117 extreme value, 101 Fréchet, 346 Gauss, 29, 126, 217, 350 Gauss-exponential, 127 generalized Pareto, see GPD Gumbel, 29, 101, 103, 114, 115 high risk, 124 hyperbolic, 161 max-stable, 101 non-degenerate, 14 parabolic power, 135 Pareto, 198 Pascal, 188 point process, 49, 50, 63, 65, 66, 67 stable, 282 Student, 5, 29, 171 Index uniform, see uniform distribution divergence of a sequence, 132 domain, 21 attraction, 21, 113 elliptic attraction, 190, 233, 240 exceedance, 25, 101, 211 high risk, 126 horizontal, 184 max-stable affine norming, 113 monotone norming, 120 double-exponential, see Gumbel egg-shaped, 136 elliptic attraction, 233, 240 distribution, 178 threshold, 25, 37, 38, 234, 285 elliptical models, enumeration of the points, 46 Esscher transform, 211 Euclidean Pareto, 23, 24, 131, 135, 171, 172, 176, 235, 255, 267, 351 exceedance, 101 domain, 21 elliptic threshold, 25, 38, 231, 234, 285 horizontal threshold, 17, 21, 183 univariate, 101 excess function, 166 excess measure, 19, 102, 179, 195, 198, 204, 235, 294, 295, 301, 318, 323, 344 elementary, 318 for expansions, 238 exchangeable, 49 expansion, 235, 236, 236 affine, 285 group, 236 linear, 233, 285 371 scalar, 221, 222, 266, 268, 277, 278, 345, exponent measure, 103, 111, 180, 218, 224, 346 exponential family: natural, 211 law, 101 marginals, 117 polar coordinates, 238 extension, 326 extension lemma, 312 extension theorem, 192, 294 extreme value, 101 extreme value theory, 2, 19 fiber bundle, 142 fixed points, 59 flat, 106, 159, 167, 171, 254 flimsy, 185 Fréchet distribution, 346 Galambos, theorem, 122 gauge function, 138, 265 Gauss distribution, 29, 126, 217, 350 Gauss-exponential, 23, 83, 135, 146, 349–351, 353–355 Gaussian space, 97 Gaussian field, 97 generalized inverse, 115 generator of a Lie group, 196, 199, 202, 297, 342 geometric extreme value theory, 27 geometry, 161, 253 Riemann, 163 global behaviour, 146, 233 GPD, 17, 101, 176, 186, 196, 205, 333, 349–351, 354, 358 Greenspan Uncertainty Principle, Grigelionis, 57, 81 group compact, 294, 340, 341 Lie, 315, 336, 337 372 Index one-parameter, 19, 297, 336 orthogonal, 341 Gumbel distribution, 29, 101, 103, 114, 115 marginals, 115, 121 Haan, de Haan–Resnick theorem, 113 Haar measure, 340, 341 halfspace, 84 halfspace horizontal, 17, 36 heavy tails, 28, 130, 170, 222, 230, 263, 275,347 high risk distribution, 124 limit distribution, 126 limit vector, 126 scenario, 17, 31, 35, 38, 123, 124, 142, 183, 220, 231, 263 homogeneous-elliptic, 216 horizontal, 183 halfspace, 17, 36 threshold, 18, 21, 25, 37, 183, 204, 211, 341, 355 hull, convex, 38, 83, 86, 93, 118, 225, 243 hyperbolic distribution, 161 increasing partitions, 46 independence, 26, 51, 118, 180 initial position, 140 initial transformation, 141 instability, 291 integral, stochastic, 45 interpolation, 256, 299 intrusion cone, 32, 88, 94, 200, 312, 313, 320 invariant, 195, 316 Jordan spectral decomposition, 334 Keesten, 267 Laplace transform, 63, 64, 201, 296, 319 lcscH space, 70 Lebesgue measure, 21, 83, 88, 112, 117, 129, 167, 174, 175, 179, 193, 218–220, 222, 315, 316, 341, 349, 350, 358 level set, 15, 24, 134, 136, 235, 278, 351, 353 Lévy process, 54, 282 max-, 112 Lie algebra, 315, 330, 331, 336, 337 Lie bracket, 337 Lie group, 315, 336, 337 light upper tail, 28 linear expansion, 233, 285 group, 233, 240, 278 local limit law, 175 local scale, 253 local symmetry, 329 Lorentz group, 176 loss, 123 loss integral, 97, 99 lower endpoint, 111 marked point process, 44, 62, 146, 185 master sequence, 108 max-id, 100, 111 max-Lévy process, 112 max-stable, 101, 112, 113, 114–119, 222, 284 maximum, componentwise, 110, 121 mean measure, 18, 20, 45, 96 bivariate, 52 Meerschaert spectral decomposition, 305 Mises, von Mises condition, 105, 136 Janossi density, 53 Jordan form, 21, 196, 198, 203, 204, 228, Mises, von Mises function, 105, 206, 217, 302 235, 237, 262, 268, 278, 297, mixture, 43 304, 318, 320, 333 Index 373 marked, 44, 62, 146, 185 simple, 67 Poisson copula, 117, 359 Poisson mixture, 48 Poisson point process, 48, 50, 102 compensated, 52 inverse map, 58 map, 58 polar coordinates, 172, 234, 235, 238 portfolio, 124 needle, 83 positive-homogeneous, 92, 117, 161, 265, non-degenerate, 126 266 distribution, 14 power family, 134 non-life insurance, 230 power laws, 101 Prohorov theorem, 73 one-parameter group, 19, 21, 297, 324, projection, 198 336, 340 theorem, 179, 180 operational risk, proper cone, 84 operator stable, 34, 282 orbit, 200, 316, 320 quantile, 124 order statistics, 103 quantitative risk management, orthogonal group, 341 quasiconvex, 134 quotients, convergence of, 143, 247 parabolic power laws, 175 Radon measure, 19, 43, 76 parabolic cap, 129, 350 rain shower, 41 parabolic power laws, 135 random counting/ measure, 44 paraboloid, 21, 83, 129, 174 random set, 44 Pareto, 101 regular variation, 22, 101, 295, 296, 300, Euclidean, 171 312 generalized, 101 relatively compact, 76 distribution, see GPD representation, 115 multivariate GPD, 176 Skorohod, 73, 74, 93 parameter, 37, 176, 186, 300 theorem, 196, 293 Pareto distribution, 198 residual life times, 101 partition, 46 risk, 13, 110, 183 separating points, 46 high, 123 Pascal distribution, 188 level, 123 Peaks Over Thresholds, 19 theory, 123 point process, 41, 44, 47 rotund, 33, 138, 174, 353 convergence, 69, 78 rotund-exponential, 33, 135, 136, 353 distribution, 49, 50, 63, 65, 66, 67 Möbius band, 142 Möbius transform, 118 model function, 108 monotone transformation, 104 multivariate, 119 multivariate GPD, 176 monotone transformations, 119 slow variation, 157 374 Index symmetry, 126, 180, 216, 352 roughening of Lebesgue measure, 107, 168, 247, 252, 270, 349 stability relation, 113 stable distribution, 282 sample cloud, 43, 44, 81, 146 stable process, 55, 282 mixture, 44 standard distribution, 353 sample copula, 359 state space, 70 scalar normalization, 226, 264, point process, 47 see also expansion, scalar steady, 95, 185, 200 scalar symmetry, 268 completely, 218, 219 scale function, 105 stress testing, 30 scenario analysis, 30 structure theorem, 178 scenario, high risk, 17, 31, 35, 38, 123, Student distribution, 5, 29, 171 124, 142, 220, 263 sturdy, 185, 200 SDT, 34, 199, 258, 304, 305 completely, 201 self-neglecting, 108 measure, 88 self-similar, 188, 355 subspace theorem, 75 separation theorem, 86 superposition, 56 sequence, divergence of, 132 support, 92, 140 shape of distribution, 14 compact, 76 shear, 140, 237, 257, 311, 318, 319, 354 function, 92 sign-invariant, 353 process, 92 simple point process, 44, 68 symmetry, 127, 271, 314, 333 skeleton, discrete, 223, 299, 350 excess measure, 179, 193 Skorohod representation theorem, 73, 74, group, 179, 301, 341 93 local, 329 slow variation, 157 maximal, 278 small set property, 46 paraboloid, 129 spectral scalar, 268 decomposition, 22, 38, 258, 275, 333 spherical, 180 theorem, see SDT tail property, 19, 27, 349 theorem, affine, 309 tail self-similar, 188, 355 theorem, discrete, 262 theorem distribution, 196 asymptotic invariance, 133 measure, 20, 26, 38, 99, 196, 203, 238, 357 balance, 265 probability measure, 34 basic inequality, 159 stability, 203 consistency, 96 spherical continuity, 75 beta density, 135, 178 continuous mapping, 73 distribution, 178 convergence of convex hulls, 94 convergence of types, 14, 355 Student, 29, 135, 350 Index copula convergence, 122 decomposition, 118 extension, 192, 294 extension lemma, 312 Galambos, 122 geometric invariance, 131 Grigelionis, 57 Haan, de Haan–Resnick, 113 high risk probabilities, 134 local section lemma, 325 Meerschaert spectral decomposition, 305 polar representation, 242 power families, 134 Prohorov, 73 projection, 179, 180, 199 representation, 115, 196, 293 separation, 86, 87 Skorohod’s representation, 73, 74, 93 spectral decomposition, 22, 258, 277, 305 affine, 309 discrete, 262 structure, 178 subspace, 75 uniqueness, 328 weak convergence, 77 thinned point process, 62 threshold elliptic, 25, 37, 38, 234, 285 horizontal, 18, 37, 183 tight, 69, 73, 78, 79, 90 time change, 247, 254, 298 transformation affine, 15 coordinatewise affine, see CAT monotone, 104 multivariate monotone, 119 twisting, 256, 269, 270 type of distribution, 14 type, convergence theorem, 14, 355 375 typical, 206 density, 23, 235, 245, 252, 254, 255, 278, 280, 355 distribution, 205, 206, 209, 243 uniform distribution, 14, 42, 114, 117, 119, 129, 174, 206, 212, 213, 215, 279, 292, 340, 353 spherical, 128 unimodal, 15, 24, 38, 145–147, 157, 278, 349, 350, 353, 359 distribution, 134, 137, 147, 354 function, 134 uniqueness theorem, 328 univariate, 100, 301 upper endpoint, 111 upper halfspace, 18, 125 vague convergence, 76, 83, 225, 244, 247 Value-at-Risk, VaR, 123 variance, see covariance variation, regular, 22, 101, 295, 296, 300, 312 vary like, 22, 207, 297 regularly, 24, 38, 234, 296 at infinity, 234 with exponent, 297 slowly, 157, 223 vertex, 29, 82, 175, 176, 218, 221 vertical component, 36, 185 vertical translation, 209, 211 volatility, 130 weak convergence, 72, 74, 77, 147, 192, 208 weakly, 73, 74, 89 Weibull density, 33, 135 XS-measure, 316, 349 zero-one point process, 44, 56 ... ZLAM_Balkema_titelei.qxd 5.7.2007 9:46 Uhr Seite Guus Balkema Paul Embrechts High Risk Scenarios and Extremes S E A geometric approach M M E S M S E M E S European Mathematical Society ZLAM_Balkema_titelei.qxd... in the important final stages of proofreading: Daniel Alai, Matthias Degen, Dominik Lambrigger, Natalia Lysenko, Parthanil Roy and Johanna Ziegel Dietmar Salamon helped us to understand why discontinuities... exists a linear functional Ô and a real constant c such that Z D c a. s For instance, all Gaussian densities have the same shape Shape (or type) is a geometric concept Given a non-degenerate Gaussian