Modelling Extremal Events for Insurance and Finance

From the back cover of the book: Both in insurance and finance applications, questions involving extremal events (such as large insurance claims, large fluctuations in financial data, stock market shocks, risk management, reinsurance products ...) play an increasingly important role. This much awaited book presents a comprehensive treatment of extreme value methodology for random walk models, time series, certain types of continuous-time stochastic processes and compound Poisson processes, all standard models which occur in applications in insurance mathematics and mathematical finance. Both probabilistic and statistical methods are discussed in detail, with such topics as ruin theory for large claim models, fluctuation theory of sums and extremes, extremes in time series models, point process methods, statistical estimation of quantiles and tail probabilities. Besides summarising and bringing together known results, the book also features topics which appear for the first time in textbook form, including the theory of subexponential distributions and the spectral theory of heavy-tailed time series.

A typical chapter will introduce the new methodology in a rather intuitive (though always mathematically correct) way, stressing the understanding of new techniques rather than following the usual «theorem-proof» format. Various examples, mainly from the realm of insurance and finance, help to convey the usefulness of the new material. A final chapter on more extensive applications and/or related fields broadens the scope further.

The book can serve either as a text for a graduate course on applied probability and statistics, insurance and/or finance, or as a basic reference source. Its reference quality is enhanced by a very extensive bibliography, annotated by various notes and comments sections making the book broadly and easily accessible.
 

Table of Contents

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 Definitions 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 Functions 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 Parameter 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,\infty) 559
  A2.4 Weak Convergence 559
  A2.5 The Continuous Mapping Theorem 561
  A2.6 Weak Convergence of Point Processes 562
A3 Regular Results 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


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