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**Modelling extremal events for insurance and finance.**
*(English)*
Zbl 0873.62116

Applications of Mathematics. 33. Berlin: Springer. xv, 645 p. (1997).

This well-written book is a modern text on the modelling of extremal events with special emphasis on applications to insurance and finance. It is intended as a consulting book for all those working in the broader financial faced with questions concerning extremal or rare events. Most of the chapter can be used very constructively in teaching a special-topics course in insurance or mathematical finance. The material is presented in a very clear, well-organized fashion. The authors provide examples to illustrate the theoretical concepts discussed. The book consists of 8 chapters. Its organization is as follows.

Chapter 1: Risk theory. In this chapter the authors study the classical model for insurance risk. They mainly deal with the mathematical analysis of ruin estimation in the case of heavy-tailed claim sizes. Chapter 2: Fluctuations of sums. The chapter surveys a general asymptotic theory for sums of iid random variables. This includes classical results such as the central limit theorem, the law of large numbers, the law of the iterated logarithm, the functional central limit theorem and their ramifications and refinements. The fluctuations of random sums, which are of particular interest in insurance, are also studied. Chapter 3: Fluctuations of maxima. Chapter 4: Fluctuations of upper order statistics. In Chapter 3 and 4 classical extreme value theory is presented. In particular, the authors describe and study maxima, upper order statistics, records and excesses over thresholds. Extensions to randomly indexed samples and to stationary sequences are also given.

Chapter 5: An approach to extremes via point processes. In this chapter, the authors deal with point process techniques which occur in modern extreme value theory. Attention is concentrated on the point process of exceedances of a given threshold by sequences of random variables. Chapter 6: Statistical methods for extremal events. This chapter contains a collection of methods for statistical inference based on extreme values in a sample. Chapter 7: Time series analysis for heavy-tailed processes. The chapter presents some recent research of time series with large fluctuations, relevant for many financial time series. The authors pay particular attention to linear processes with heavy-tailed innovations.

Chapter 8: Special topics. In this chapter the authors show that the methodology of extremal event modelling applies to a wide variety of problems. In particular, they consider the extremal index, a large claim index, perpetuities, ARCH processes and reinsurance treaties. The last part of the book is Appendix, which contains the basic results on modes of convergence, weak convergence in metric space, regular variation, subexponentiality and presents some renewal theory. The bibliography contains 646 items.

Chapter 1: Risk theory. In this chapter the authors study the classical model for insurance risk. They mainly deal with the mathematical analysis of ruin estimation in the case of heavy-tailed claim sizes. Chapter 2: Fluctuations of sums. The chapter surveys a general asymptotic theory for sums of iid random variables. This includes classical results such as the central limit theorem, the law of large numbers, the law of the iterated logarithm, the functional central limit theorem and their ramifications and refinements. The fluctuations of random sums, which are of particular interest in insurance, are also studied. Chapter 3: Fluctuations of maxima. Chapter 4: Fluctuations of upper order statistics. In Chapter 3 and 4 classical extreme value theory is presented. In particular, the authors describe and study maxima, upper order statistics, records and excesses over thresholds. Extensions to randomly indexed samples and to stationary sequences are also given.

Chapter 5: An approach to extremes via point processes. In this chapter, the authors deal with point process techniques which occur in modern extreme value theory. Attention is concentrated on the point process of exceedances of a given threshold by sequences of random variables. Chapter 6: Statistical methods for extremal events. This chapter contains a collection of methods for statistical inference based on extreme values in a sample. Chapter 7: Time series analysis for heavy-tailed processes. The chapter presents some recent research of time series with large fluctuations, relevant for many financial time series. The authors pay particular attention to linear processes with heavy-tailed innovations.

Chapter 8: Special topics. In this chapter the authors show that the methodology of extremal event modelling applies to a wide variety of problems. In particular, they consider the extremal index, a large claim index, perpetuities, ARCH processes and reinsurance treaties. The last part of the book is Appendix, which contains the basic results on modes of convergence, weak convergence in metric space, regular variation, subexponentiality and presents some renewal theory. The bibliography contains 646 items.

Reviewer: W.Dziubdziela (Czȩstochowa)

### MSC:

62P05 | Applications of statistics to actuarial sciences and financial mathematics |

60G70 | Extreme value theory; extremal stochastic processes |

60-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to probability theory |

62-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to statistics |

60F05 | Central limit and other weak theorems |

60F17 | Functional limit theorems; invariance principles |