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Copula Theory and Its Applications - Proceedings of the Workshop Held in Warsaw, 25-26 September 2009  
Copula Theory and Its Applications - Proceedings of the Workshop Held in Warsaw, 25-26 September 2009
von: Piotr Jaworski, Fabrizio Durante, Wolfgang Karl Härdle, Tomasz Rychlik
Springer-Verlag, 2010
ISBN: 9783642124655
327 Seiten, Download: 3790 KB
 
Format:  PDF
geeignet für: Apple iPad, Android Tablet PC's Online-Lesen PC, MAC, Laptop

Typ: B (paralleler Zugriff)

 

 
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Inhaltsverzeichnis

  Foreword 5  
  Preface 6  
  Contents 7  
  Contributors 13  
  Part I Surveys 17  
     1 Copula Theory: An Introduction 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  
        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  
        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  
        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  
        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  
        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  
        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  
        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  
        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  
        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  
        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  
        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  
        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  
        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  
        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  
        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  
        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  


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