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Balkema G., Embrechts P. High Risk Scenarios and Extremes: A Geometric Approach

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Balkema G., Embrechts P. High Risk Scenarios and Extremes: A Geometric Approach
European Mathematical Society, 2007. — 389 p. — (Zurich Lectures in Advanced Mathematics). — ISBN 978-3-03719-035-7.
Quantitative Risk Management (QRM) has become a field of research of considerable importance to numerous areas of application, including insurance, banking, energy, medicine, and reliability. Mainly motivated by examples from insurance and finance, the authors develop a theory for handling multivariate extremes. The approach borrows ideas from portfolio theory and aims at an intuitive approach in the spirit of the Peaks over Thresholds method. The point of view is geometric. It leads to a probabilistic description of what in QRM language may be referred to as a high risk scenario: the conditional behaviour of risk factors given that a large move on a linear combination (portfolio, say) has been observed. The theoretical models which describe such conditional extremal behaviour are characterized and their relation to the limit theory for coordinatewise maxima is explained. The first part is an elegant exposition of coordinatewise extreme value theory; the second half develops the more basic geometric theory. Besides a precise mathematical deduction of the main results, the text yields numerous discussions of a more applied nature. A twenty page preview introduces the key concepts; the extensive introduction provides links to financial mathematics and insurance theory. The book is based on a graduate course on point processes and extremes. It could form the basis for an advanced course on multivariate extreme value theory or a course on mathematical issues underlying risk. Students in statistics and finance with a mathematical, quantitative background are the prime audience. Actuaries and risk managers involved in data based risk analysis will find the models discussed in the book stimulating. The text contains many indications for further research.
Point Processes
An intuitive approach
Poisson point processes
The distribution
Convergence
Converging sample clouds
The univariate theory: maxima and exceedances
Componentwise maxima
High Risk Limit Laws
High risk scenarios
The Gauss-exponential domain, rotund sets
The Gauss-exponential domain, unimodal distributions
Flat functions and flat measures
Heavy tails and bounded vectors
The multivariate GPDs
Thresholds
Exceedances over horizontal thresholds
Horizontal thresholds – examples
Heavy tails and elliptic thresholds
Heavy tails – examples
Regular variation and excess measures
Open problems
The stochastic model
The statistical analysis
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