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About this book
Both in insurance and in finance applications, questions involving extremal events (such as large insurance claims, large fluctuations in financial data, stock market shocks, risk management, ...) play an increasingly important role. This book sets out to bridge the gap between the existing theory and practical applications both from a probabilistic as well as from a statistical point of view. Whatever new theory is presented is always motivated by relevant real-life examples. The numerous illustrations and examples, and the extensive bibliography make this book an ideal reference text for students, teachers and users in the industry of extremal event methodology.
Written for:
actuarial trainees, actuaries, other actuarial mathematics trainees; research and development departments of insurance companies; actuarial students at universities, applied probabilists, statisticians
Keywords:
extreme value theory
insurance risk
mathematical finance
tail estimation
time series analysis

ISBN: 3-540-60931-8
TITLE: Modelling Extremal Events for Insurance and Finance
AUTHOR: Embrechts, Paul; Klüppelberg, Claudia; Mikosch, Thomas
TOC:

Reader Guidelines 1
1 Risk Theory 21
1.1 The Ruin Problem 22
1.2 The Cramér Lundberg Estimate 28
1.3 Ruin Theory for Heavy Tailed Distributions 36
1.3.1 Some Preliminary Results 37
1.3.2 Cramér Lundberg Theory for Subexponential
Distributions 39
1.3.3 The Total Claim Amount in the Subexponential Case 44
1.4 Cramér Lundberg Theory for Large Claims: a Discussion 49
1.4.1 Some Related Classes of Heavy Tailed Distributions 49
1.4.2 The Heavy Tailed Cramér Lundberg Case Revisited 53
2 Fluctuations of Sums 59
2.1 The Laws of Large Numbers 60
2.2 The Central Limit Problem 70
2.3 Refinements of the CLT 82
2.4 The Functional CLT: Brownian Motion Appears 88
2.5 Random Sums 96
2.5.1 General Randomly Indexed Sequences 96
2.5.2 Renewal Counting Processes 103
2.5.3 Random Sums Driven by Renewal Counting Processes 106
3 Fluctuations of Maxima 113
3.1 Limit Probabilities for Maxima 114
3.2 Weak Convergence of Maxima Under Affine Transformations 120
3.3 Maximum Domains of Attraction and Norming Constants 128
3.3.1 The Maximum Domain of Attraction of the Fréchet
Distribution Phi_alpha(x) = exp{-x^-alpha} 130
3.3.2 The Maximum Domain of Attraction of the Weibull.
Distribution Psi_alpha(x) = exp{-(-x)^alpha} 134
3.3.3 The Maximum Domain of Attraction of the Gumbel
Distribution Lambda(x) = exp{-exp{-x}} 138
3.4 The Generalised Extreme Value Distribution and the
Generalised Pareto Distribution 152
3.5 Almost Sure Behaviour of Maxima 168
4 Fluctuations of Upper Order Statistics 181
4.1 Order Statistics 182
4.2 The Limit Distribution of Upper Order Statistics 196
4.3 The Limit Distribution of Randomly Indexed
Upper Order Statistics 204
4.4 Some Extreme Value Theory for Stationary Sequences 209
5 An Approach to Extremes via Point Processes 219
5.1 Basic Facts About Point Processes 220
5.1.1 Definition and Examples 220
5.1.2 Distribution and Laplace Functional 225
5.1.3 Poisson Random Measures 226
5.2 Weak Convergence of Point Processes 232
5.3 Point Processes of Exceedances 237
5.3.1 The IID Case 238
5.3.2 The Stationary Case 242
5.4 Applications of Point Process Methods to IID Sequences 247
5.4.1 Records and Record Times 248
5.4.2 Embedding Maxima in Extremal Processes 250
5.4.3 The Frequency of Records and the Growth
of Record Times 254
5.4.4 Invariance Principle for Maxima 260
5.5 Some Extreme Value Theory for Linear Processes 263
5.5.1 Noise in the Maximum Domain of Attraction
of the Fréchet Distribution Phi_alpha 264
5.5.2 Subexponential Noise in the Maximum Domain
of Attraction of the Gumbel Distribution Lambda 277
6 Statistical Methods for Extremal Events 283
6.1 Introduction 283
6.2 Exploratory Data Analysis for Extremes 290
6.2.1 Probability and Quantile Plots 290
6.2.2 The Mean Excess Function 294
6.2.3 Gumbel's Method of Exceedances 303
6.2.4 The Return Period 305
6.2.5 Records as an Exploratory Tool 307
6.2.6 The Ratio of Maximum and Sum 309
6.3 Paxameter Estimation for the Generalised Extreme
Value Distribution 316
6.3.1 Maximum Likelihood Estimation 317
6.3.2 Method of Probability Weighted Moments 321
6.3.3 Tail and Quantile Estimation, a First Go 323
6.4 Estimating Under Maximum Domain
of Attraction Conditions 325
6.4.1 Introduction 325
6.4.2 Estimating the Shape Parameter xi 327
6.4.3 Estimating the Norming Constants 345
6.4.4 Tail and Quantile Estimation 348
6.5 Fitting Excesses Over a Threshold 352
6.5.1 Fitting the GPD 352
6.5.2 An Application to Real Data 358
7 Time Series Analysis for Heavy Tailed Processes 371
7.1 A Short Introduction to Classical Time Series Analysis 372
7.2 Heavy Tailed Time Series 378
7.3 Estimation of the Autocorrelation Function 381
7.4 Estimation of the Power Transfer Function 386
7.5 Parameter Estimation for ARMA Processes 393
7.6 Some Remarks About Non Linear Heavy Tailed Models 403
8 Special Topics 413
8.1 The Extremal Index 413
8.1.1 Definition and Elementary Properties 413
8.1.2 Interpretation and Estimation of the Extremal Index 418
8.1.3 Estimating the Extremal Index from Data 424
8.2 A Large Claim Index 430
8.2.1 The Problem 430
8.2.2 The Index 431
8.2.3 Some Examples 433
8.2.4 On Sums and Extremes 436
8.3 When and How Ruin Occurs 439
8.3.1 Introduction 439
8.3.2 The Cramér Lundberg Case 444
8.3.3 The Large Claim Case 449
8.4 Perpetuities and ARCH Processes 454
8.4.1 Stochastic Recurrence Equations and Perpetuities 455
8.4.2 Basic Properties of ARCH Processes 461
8.4.3 Extremes of ARCH Processes 473
8.5 On the Longest Success Run 481
8.5.1 The Total Variation Distance to a
Poisson Distribution 483
8.5.2 The Almost Sure Behaviour 486
8.5.3 The Distributional Behaviour 493
8.6 Some Results on Large Deviations 498
8.7 Reinsurance Treaties 503
8.7.1 Introduction 503
8.7.2 Probabilistic Analysis 507
8.8 Stable Processes 521
8.8.1 Stable Random Vectors 522
8.8.2 Symmetric Stable Processes 526
8.8.3 Stable Integrals 527
8.8.4 Examples 532
8.9 Self Similarity 541
Appendix 551
A1 Modes of Convergence 551
A1.1 Convergence in Distribution 551
A1.2 Convergence in Probability 552
A1.3 Almost Sure Convergence 553
A1.4 L^p Convergence 553
A1.5 Convergence to Types 554
A1.6 Convergence of Generalised Inverse Functions 554
A2 Weak Convergence in Metric Spaces 555
A2.1 Preliminaries about Stochastic Processes 555
A2.2 The Spaces C[0,1] and D[0,1] 557
A2.3 The Skorokhod Space D(0,infinity) 559
A2.4 Weak Convergence 559
A2.5 The Continuous Mapping Theorem 561
A2.6 Weak Convergence of Point Processes 562
A3 Regular Variation and Subexponentiality 564
A3.1 Basic Results on Regular Variation 564
A3.2 Properties of Subexponential Distributions 571
A3.3 The Tail Behaviour of Weighted Sums
of Heavy Tailed Random Variables 583
A4 Some Renewal Theory 587
References 591
Index 626
List of Abbreviations and Symbols 641
END

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