© 1997

Modelling Extremal Events

for Insurance and Finance

Part of the Applications of Mathematics book series (SMAP, volume 33)

Table of contents

  1. Front Matter
    Pages I-XV
  2. Paul Emberchts, Claudia Klüppelberg, Thomas Mikosch
    Pages 1-19
  3. Paul Emberchts, Claudia Klüppelberg, Thomas Mikosch
    Pages 21-57
  4. Paul Emberchts, Claudia Klüppelberg, Thomas Mikosch
    Pages 59-111
  5. Paul Emberchts, Claudia Klüppelberg, Thomas Mikosch
    Pages 113-179
  6. Paul Emberchts, Claudia Klüppelberg, Thomas Mikosch
    Pages 181-218
  7. Paul Emberchts, Claudia Klüppelberg, Thomas Mikosch
    Pages 219-282
  8. Paul Emberchts, Claudia Klüppelberg, Thomas Mikosch
    Pages 283-370
  9. Paul Emberchts, Claudia Klüppelberg, Thomas Mikosch
    Pages 371-412
  10. Paul Emberchts, Claudia Klüppelberg, Thomas Mikosch
    Pages 413-550
  11. Back Matter
    Pages 551-648

About this book


Both in insurance and in finance applications, questions involving extremal events (such as large insurance claims, large fluctuations, in financial data, stock-market shocks, risk management, ...) play an increasingly important role. This much awaited book presents a comprehensive development of extreme value methodology for random walk models, time series, certain types of continuous-time stochastic processes and compound Poisson processes, all models which standardly occur in applications in insurance mathematics and mathematical finance. Both probabilistic and statistical methods are discussed in detail, with such topics as ruin theory for large claim models, fluctuation theory of sums and extremes of iid sequences, extremes in time series models, point process methods, statistical estimation of tail probabilities. Besides summarising and bringing together known results, the book also features topics that appear for the first time in textbook form, including the theory of subexponential distributions and the spectral theory of heavy-tailed time series. A typical chapter will introduce the new methodology in a rather intuitive (tough always mathematically correct) way, stressing the understanding of new techniques rather than following the usual "theorem-proof" format. Many examples, mainly from applications in insurance and finance, help to convey the usefulness of the new material. A final chapter on more extensive applications and/or related fields broadens the scope further. The book can serve either as a text for a graduate course on stochastics, insurance or mathematical finance, or as a basic reference source. Its reference quality is enhanced by a very extensive bibliography, annotated by various comments sections making the book broadly and easily accessible.


Analysis Statistical Methods extreme value theory insurance risk mathematical finance modeling sets tail estimation time series analysis

Authors and affiliations

  1. 1.Department of MathematicsETH ZurichZurichSwitzerland
  2. 2.Center for Mathematical SciencesMunich University of TechnologyGarchingGermany
  3. 3.Laboratory of Actuarial MathematicsUniversity of CopenhagenCopenhagenDenmark

Bibliographic information

Industry Sectors
Finance, Business & Banking


From the reviews:


"…excellent, comprehensive treatise on the subject of extremal events modeling. The authors have responded well to the demands of extreme value practitioners for such a text. 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…An understanding of modes of convergence, specifically weak convergence, is essential to fully appreciate the text, but the authors’ intuitive writing style makes most of the basic ideas accessible even to the uninitiated…The authors do an excellent job of organizing these topics and also provide a very useful 20-page ‘Reader Guidelines’ section…[the book] makes an excellent contribution to unifying important concepts in extreme value theory and modeling of extremal events. Aside from its obvious use as a reference for practitioners and theoreticians alike, 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…As the authors point out this may not be the kind of book that you want to tackle form cover to cover initially, but it is my bet you will eventually discover that you have done just that as you repeatedly reference this hefty volume throughout the years."


"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, to plot sample paths of various processes 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 with the range of applications of the subject. While there are a number of books available which cover most of the topics herein, I know of none which presents such a range of theory and applications of extremal processes in one volume, at a level easily understood by users of the methodology. I highly recommend the book to all who work in the area, or in related areas. (...) The combination of skills and expertise of the three authors of this book is impressive. Their reading covers not only the traditional and classical works in the area but a great deal of the modern development, too. (They give 646 references to books and articles in the literature.) Their book concludes with copious appendices setting out the basic probability theory and some of the regular variation theory required for understanding the rest of the development. In summary, this is a worthwhile book in an extremely important area."


"(...) This book impresses me as being exceptionally well written, scholarly beyond question, more than a little daunting, and likely to become a classic in its field."


"The book is the first in the area that strikes a proper balance between mathematical rigor and scope (...) and the statistically-oriented applications for the practitioner."


"(...) the indispensable starting point for anyone interested in contemporary applications and extensions of classical EVT."


"This is an encyclopedic handbook of theory and statistical praxis, of great value to actuaries and statisticians in the fields concerned, which gives an up to date picture of this fast developing field, and at the same time a useful and well motivated text book for those who need a guide for entering the area without getting lost either in pure theory or messy practice."


"Given the nature of the subject (...) the book is easy to read.(...) The narrative style is marvellous, invariably connecting theoretical concepts to the real world objects they are supposed to describe, (...)."


"There are a number of texts available on Extreme Value Theory (EVT). This is the essential one to read. It is authoritative and extremely well written…A nice feature of Embrechts et al is an opening 20-page ‘reader guideline’ that gives an overview of the material before the start of the main text."