The problem of multivariate extremes and their financial application is of central interest in the field of quantitative risk management and it is the main topic of this monograph. The book is divided into five parts and twenty chapters. After a description of motivations, the first part contains an introduction to the theory of point processes and their applications in extreme value theory. The second part covers the classical univariate and multivariate extreme value theories. The coordinate-wise approach and max-stable distributions are discussed here.

The main part of the book comprises parts III and IV. In the third part a modern geometric (coordinate free) approach to multivariate extremes is broadly studied. Particular interest is paid to heavy tailed distributions and to the generalized Pareto distribution (candidate multivariate extension of the GPD). Extreme value theory is often based on threshold exceedance, which is studied in the fourth part of the book. There is no unique threshold in the multivariate case and the authors study two classes (horizontal and elliptic) of thresholds. In the last part of the book, some open problems in the theory and statistical applications of multivariate extremes are summarized.

The book is written with a broad audience of theoretical and applied mathematicians in mind. Problems are mostly motivated by examples from insurance and finance (portfolio theory) but their solution is purely mathematical. The level of rigour in the book is very high and to understand all the techniques of modern extreme value theory, a solid mathematical background is required. On the other hand, if one just needs to apply the results, the proofs and technical sections may be skipped. The book can also be recommended as the basis for an advanced course on multivariate extreme value theory.

Reviewer:

dh