Foreword	5
	Preface	6
	Contents	7
	Contributors	13
	Part I Surveys	17
	1 Copula Theory: An Introduction	18
	Fabrizio Durante and Carlo Sempi	18
	1.1 Historical Introduction	18
	1.1.1 Outline	21
	1.2 Preliminaries on Random Variables and Distribution Functions	21
	1.3 Copulas: Definitions and Basic Properties	24
	1.4 Sklar's Theorem	27
	1.5 Copulas and Random Vectors	29
	1.6 Families of Copulas	30
	1.6.1 Elliptical Copulas	31
	1.6.2 Archimedean Copulas	32
	1.6.3 EFGM Copulas	34
	1.7 Constructions of Copulas	35
	1.7.1 Copulas with Given Lower Dimensional Marginals	35
	1.7.2 Copula-to-Copula Transformations	36
	1.7.3 Geometric Constructions of Copulas	37
	1.8 Copula Theory: What's the Future?	38
	References	39
	2 Dynamic Modeling of Dependence in Finance via Copulae Between Stochastic Processes	47
	Tomasz R. Bielecki, Jacek Jakubowski and Mariusz Nieweg?owski	47
	2.1 Introduction	47
	2.2 Lévy Copulae	49
	2.3 Semimartingale Copulae	53
	2.3.1 Copulae for Special Semimartingales	53
	2.3.2 Consistent Semimartingale Copulae	62
	2.4 Markov Copulae	68
	2.4.1 Consistent Markov Processes	69
	2.4.2 Markov Copulae: Generator Approach	71
	2.4.3 Markov Copulae: Symbolic Approach	77
	2.5 Applications in Finance	83
	2.5.1 Pricing Rating-Triggered Step-Up Bonds via Simulation	84
	2.5.2 Model Calibration and Pricing	86
	References	89
	3 Copula Estimation	91
	Barbara Choros, Rustam Ibragimov and Elena Permiakova	91
	3.1 Introduction	91
	3.2 Copula Estimation: Random Samples with Dependent Marginals	92
	3.2.1 Parametric Models: Maximum Likelihood Methods and Inference from Likelihoods for Margins	92
	3.2.2 Semiparametric Estimation	94
	3.2.3 Nonparametric Inference and Empirical Copula Processes	95
	3.3 Copula-Based Time Series and Their Estimation	96
	3.3.1 Copula-Based Characterizations for (Higher-Order) Markov Processes	96
	3.3.2 Parametric and Semiparametric Copula Estimation Methods for Markov Processes	97
	3.3.3 Nonparametric Copula Inference for Time Series	98
	3.3.4 Dependence Properties of Copula-Based Time Series	99
	3.4 Further Copula Inference Methods	100
	3.5 Empirical Applications	101
	References	103
	4 Pair-Copula Constructions of Multivariate Copulas	106
	Claudia Czado	106
	4.1 Introduction	106
	4.2 Pair Copula Constructions of D-Vine, Canonical and Regular Vine Distributions	107
	4.2.1 Pair-Copula Constructions of D-Vine and Canonical Vine Distributions	107
	4.2.2 Regular Vines Distributions and Copulas	109
	4.3 Estimation Methods for Regular Vine Copulas	113
	4.4 Model Selection Among Vine Specifications	116
	4.5 Applications of Vine Distributions	118
	4.6 Summary and Open Problems	119
	References	120
	5 Risk Aggregation	123
	Paul Embrechts and Giovanni Puccetti	123
	5.1 Motivations and Preliminaries	124
	5.1.1 The Mathematical Framework	124
	5.2 Bounds for Functions of Risks: The Coupling-Dual Approach	125
	5.2.1 Application 1: Bounding Value-at-Risk	127
	5.2.2 Application 2: Supermodular Functions	131
	5.3 The Calculation of the Distribution of the Sum of Risks	132
	5.3.1 Open Problems	136
	References	137
	6 Extreme-Value Copulas	139
	Gordon Gudendorf and Johan Segers	139
	6.1 Introduction	139
	6.2 Foundations	140
	6.3 Parametric Models	143
	6.3.1 Logistic Model or Gumbel--Hougaard Copula	144
	6.3.2 Negative Logistic Model or Galambos Copula	144
	6.3.3 Hüsler--Reiss Model	145
	6.3.4 The t-EV Copula	146
	6.4 Dependence Coefficients	146
	6.5 Estimation	148
	6.5.1 Parametric Estimation	149
	6.5.2 Nonparametric Estimation	150
	6.6 Further Reading	152
	References	153
	7 Construction and Sampling of Nested Archimedean Copulas	158
	Marius Hofert	158
	7.1 Introduction	158
	7.2 Nested Archimedean Copulas	160
	7.3 A Sufficient Nesting Condition	162
	7.4 Construction of Nested Archimedean Copulas	164
	7.5 Sampling Nested Archimedean Copulas	166
	7.6 Conclusion	170
	References	170
	8 Tail Behaviour of Copulas	172
	Piotr Jaworski	172
	8.1 Introduction	172
	8.2 Tail Expansions of Copulas	174
	8.2.1 Characterization and Properties of Leading Parts	178
	8.2.2 Relatively Invariant Measures on [0,)n	179
	8.3 Examples of Tail Expansions	180
	8.3.1 Homogeneous Copulas	180
	8.3.2 Diagonal Copulas	180
	8.3.3 Absolutely Continuous Copulas	182
	8.3.4 Archimedean Copulas	183
	8.3.5 Multivariate Extreme Value Copulas	188
	8.4 Applications	189
	8.4.1 Tail Conditional Copulas	189
	8.4.2 Extreme Value Copulas of a Given Copula	191
	8.4.3 Regularly Varying Random Vectors with a Given Copula	192
	8.4.4 Value at Risk	193
	References	196
	9 Copulae in Reliability Theory (Order Statistics, Coherent Systems)	198
	Tomasz Rychlik	198
	9.1 Coherent Systems	198
	9.2 Signatures	200
	9.2.1 Components with i.i.d. Lifetimes	200
	9.2.2 Mixed Systems	201
	9.2.3 Components with Exchangeable Lifetimes	203
	9.3 Bounds for Exchangeable Lifetime Components	205
	9.3.1 Distribution Bounds	205
	9.3.2 Expectation Bounds	207
	9.4 Characterizations of k-Out-of-n System Lifetime Distributions	209
	9.4.1 General Copula Joint Distribution	210
	9.4.2 Absolute Continuous Copula Joint Distribution	211
	9.4.3 Variance Bounds	214
	9.5 Final Remarks	216
	References	217
	10 Copula-Based Measures of Multivariate Association	220
	Friedrich Schmid, Rafael Schmidt, Thomas Blumentritt, Sandra Gaißer and Martin Ruppert	220
	10.1 Introduction and Definitions	220
	10.2 Aspects of Multivariate Association	223
	10.3 Multivariate Generalizations of Spearman's Rho, Kendall's Tau, Blomqvist's Beta, and Gini's Gamma	226
	10.3.1 Spearman's Rho	226
	10.3.2 Kendall's Tau	228
	10.3.3 Blomqvist's Beta	230
	10.3.4 Gini's Gamma	231
	10.4 Information-Based Measures of Multivariate Association	232
	10.5 Measures of Multivariate Association Based on Lp-Distances	235
	10.5.1 2 as a L2-Distance-Based Measure	236
	10.5.2 as a L1-Distance-Based Measure	238
	10.5.3 as a L-Distance-Based Measure	238
	10.6 Multivariate Tail Dependence	239
	References	243
	11 Semi-Copulas and Interpretations of Coincidences Between Stochastic Dependence and Ageing	248
	Fabio Spizzichino	248
	11.1 Introduction	248
	11.2 Univariate Ageing and Dependence Properties of Archimedean Semi-Copulas	251
	11.3 Dependence and Univariate Ageing in Schur-Constant Models	254
	11.4 Level Curves, B functions, Duality, and Interpretation of Coincidence Between Ageing and Dependence	258
	11.5 Summary and Concluding Remarks	262
	References	263
	Part II Contributed Papers	266
	12 A Copula-Based Model for Spatial and Temporal Dependence of Equity Markets	267
	Umberto Cherubini, Fabio Gobbi, Sabrina Mulinacci and Silvia Romagnoli	267
	12.1 Introduction	267
	12.2 A market Model in Discrete Time	268
	12.3 The Martingale Property	269
	12.4 Applications	271
	12.4.1 Multivariate Digital Options	271
	12.4.2 Basket and Spread Options	273
	References	274
	13 Nonparametric and Semiparametric Bivariate Modeling of Petrophysical Porosity-Permeability Dependence from Well Log Data	276
	Arturo Erdely and Martin Diaz-Viera	276
	13.1 Introduction	276
	13.2 Methodology	277
	13.3 Data Analysis	280
	13.4 Final Remarks	285
	References	287
	14 Testing Under the Extended Koziol-Green Model	288
	Auguste Gaddah and Roel Braekers	288
	14.1 Introduction	288
	14.2 Asymptotic Results	291
	14.3 Test Statistics	294
	14.4 Data Example: Survival with Malignant Melanoma	295
	References	297
	15 Parameter Estimation and Application of the Multivariate Skew t-Copula	298
	Tõnu Kollo Gaida Pettere	298
	15.1 Introduction	298
	15.2 Preliminary Notions and Notation	299
	15.3 Construction of a Skew t-Copula	301
	15.4 Parameter Estimation	302
	15.5 Simulation	304
	15.6 Application	305
	References	307
	16 On Analytical Similarities of Archimedean and Exchangeable Marshall-Olkin Copulas	308
	Jan-Frederik Mai and Matthias Scherer	308
	16.1 Introduction	308
	16.2 Complete Monotonicity and d-Monotonicity	310
	16.2.1 Definitions and Examples	310
	16.2.2 Probabilistic Interpretations	311
	16.2.3 d-Monotonicity	312
	16.3 Probabilistic Models and Sampling	314
	16.3.1 The Completely Monotone Case	314
	16.3.2 The Proper d-Monotone Case	315
	References	317
	17 Relationships Between Archimedean Copulas and Morgenstern Utility Functions	319
	Jaap Spreeuw	319
	17.1 Introduction	319
	17.2 Archimedean Copulas	320
	17.3 Utility Functions	320
	17.4 Relationships Between Properties of Utility Functions and Properties of Generators	323
	17.5 Examples	326
	17.5.1 Classical Cases	326
	17.5.2 The HARA Family	327
	17.5.3 The Expo Power Utility	327
	17.5.4 Other Examples of Decreasing Absolute Risk Aversion (DARA) as in Pratt [9]	327
	17.6 Conclusion	329
	References	329
	Index	331