Embrechts P., Klüppelberg C., Mikosch T. Modelling Extremal Events: for Insurance and Finance

Springer, 9th Corrected printing, 2012. — 650 p. — ISBN: 978-3540609315.A reader's first impression on leafing through this book is of the large number of graphs and diagrams, used to illustrate shapes of distributions and to show real data examples in various ways. A closer reading reveals a nice mix of theory and applications, with the copious graphical illustrations alluded to. Such a mixture is of course dear to the heart of the applied probabilist/statistician, and should impress even the most ardent theorists.
The book is comprehensive treatise on the subject of extremal events modeling. Although it was clearly and admittedly motivated by practical questions of workers in finance, insurance, and reinsurance, the book contains the mathematical rigor and generality that will interest the extreme value theoretician. This text may be used to teach a graduate-level course in mathematical finance or a special topics course in stochastic processes with or without a financial emphasis.Reader Guidelines.
Risk Theory.
The Ruin Problem
The Cramér–Lundberg Estimate.
Ruin Theory for Heavy–Tailed Distributions.
Cramér–Lundberg Theory for Large Claims: a Discussion.
Fluctuations of Sums.
The Laws of Large Numbers.
The Central Limit Problem.
Refinements of the CLT.
The Functional CLT: Brownian Motion Appears.
Random Sums.
Fluctuations of MAXIMA.
Limit Probabilities for MAXIMA.
Weak Convergence of MAXIMA Under Affine Transformations.
Maximum Domains of Attraction and Norming Constants.
The Generalised Extreme Value Distribution and the Generalised Pareto Distribution.
Almost Sure Behaviour of MAXIMA.
Fluctuations of Upper Order Statistics.
Order Statistics.
The Limit Distribution of Upper Order Statistics.
The Limit Distribution of Randomly Indexed Upper Order Statistics.
Some Extreme Value Theory for Stationary Sequences.
An Approach to Extremes via Point Processes.
Basic Facts About Point Processes.
Weak Convergence of Point Processes.
Point Processes of Exceedances.
Applications of Point Process Methods to IID Sequences.
Some Extreme Value Theory for Linear Processes.
Statistical Methods for Extremal Events.Exploratory Data Analysis for Extremes.
Parameter Estimation for the Generalised Extreme Value Distribution.
Estimating Under Maximum Domain of Attraction Conditions.
Fitting Excesses Over a Threshold.
Time Series Analysis for Heavy–Tailed Processes.
A Short Introduction to Classical Time Series Analysis.
Heavy–Tailed Time Series.
Estimation of the Autocorrelation Function.
Estimation of the Power Transfer Function.
Parameter Estimation for ARMA Processes.
Some Remarks About Non–Linear Heavy–Tailed Models.
Special Topics.
The Extremal Index.
A Large Claim Index.
When and How Ruin Occurs.
Perpetuities and ARCH Processes.
On the Longest Success–Run.
Some Results on Large Deviations.
Reinsurance Treaties.
Stable Processes.
Self–Similarity.
Appendix.
A1 Modes of Convergence.
A2 Weak Convergence in Metric Spaces.
A3 Regular Variation and Subexponentiality.
A4 Some Renewal Theory.
List of Abbreviations and Symbols.