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**High risk scenarios and extremes. A geometric approach.**
*(English)*
Zbl 1121.91055

Zurich Lectures in Advanced Mathematics. Zürich: European Mathematical Society (EMS) (ISBN 978-3-03719-035-7/pbk). xiii, 375 p. (2007).

The monograph consists of five chapters, which require a substantial mathematical background for a reader’s understanding. Anyway the mathematical approach is accessible to a wide audience.

The general treatment involves two main themes: the first one deals with the coordinatewise extreme value theory and the second one with the geometric theory. The exhaustive and elegant theoretical development provides suitable tools to financial mathematics and insurance theory, within the quantitative risk management context.

Chapter I deals with basic topics on point processes within the standard theory framework.

Chapter II is devoted to the theory of maxima from the univariate case up to the theory of coordinatewise maxima.

Chapter III, after an introductory overview, treats the high risk limit laws, particularly deepening a one-parameter family of limit laws, the multivariate Generalized Pareto Distributions (GPDs).

In Chapter IV exceedances over horizontal and elliptic thresholds are investigated; within this context, a relevant space is dedicated to specific examples and to the relation to the limit theory for coordinatewise extremes. Moreover heavy tailed distributions normalized by linear contractions are investigated; the Chapter ends with a sum-up of the theory of multivariate regular variation and the spectral decomposition theorem.

Finally Chapter V points out a list of open problems, also highlighting the technical difficulties in applying the theory to concrete data sets.

A comprehensive bibliograpry and an index complete the book.

The volume can be used in advanced courses on multivariate extreme value theory as well as on mathematical risk theory, and as a reference text for actuaries and risk managers interested to quantitative models for data based risk analysis.

The general treatment involves two main themes: the first one deals with the coordinatewise extreme value theory and the second one with the geometric theory. The exhaustive and elegant theoretical development provides suitable tools to financial mathematics and insurance theory, within the quantitative risk management context.

Chapter I deals with basic topics on point processes within the standard theory framework.

Chapter II is devoted to the theory of maxima from the univariate case up to the theory of coordinatewise maxima.

Chapter III, after an introductory overview, treats the high risk limit laws, particularly deepening a one-parameter family of limit laws, the multivariate Generalized Pareto Distributions (GPDs).

In Chapter IV exceedances over horizontal and elliptic thresholds are investigated; within this context, a relevant space is dedicated to specific examples and to the relation to the limit theory for coordinatewise extremes. Moreover heavy tailed distributions normalized by linear contractions are investigated; the Chapter ends with a sum-up of the theory of multivariate regular variation and the spectral decomposition theorem.

Finally Chapter V points out a list of open problems, also highlighting the technical difficulties in applying the theory to concrete data sets.

A comprehensive bibliograpry and an index complete the book.

The volume can be used in advanced courses on multivariate extreme value theory as well as on mathematical risk theory, and as a reference text for actuaries and risk managers interested to quantitative models for data based risk analysis.

Reviewer: Emilia Di Lorenzo (Napoli)