Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

(2002). Modelling Extremal Events for Insurance and Finance. Journal of the American Statistical Association: Vol. 97, No. 457, pp. 360-360.

Modelling Extremal Events for Insurance and Finance

Stable non-Gaussian random processes , by G. Samorodnitsky and M. S. Taqqu. Pp. 632. £49.50. 1994. ISBN 0-412-05171-0 (Chapman and Hall).

Extremal events play an increasingly important role in stochastic modelling in insurance and finance. Over many years, probabilists and statisticians have developed techniques for the description, analysis and prediction of such events. In the present paper, we review the relevant theory which may also be used in the wider context of Operation Research. Various applications from the field of insurance and finance are discussed. Via an extensive list of references, the reader is guided towards further material related to the above problem areas.

https://link.springer.com/content/pdf/10.1007/BF01440733.pdf

Modelling of extremal events in insurance and finance

We study the asymptotic properties of estimators of the tail of a distribution based on the excesses over a threshold. A key idea is the use of Pickands' generalised Pareto distribution and its fitting, in most cases, by the method of maximum likelihood. The results cover all three limiting types of extreme value theory. We propose a new estimator for an index of regular variation and show that it often performs better than Hill's estimator. We give new results for estimating the endpoint of a distribution, extending earlier work by Hall and by Smith and Weissman. Finally, we give detailed results for the domain of attraction of $\exp(-e^{-x})$ and show that, in most cases, our proposed estimator is more efficient than two others, one based on the exponential distribution and the other due to Davis and Resnick. We also touch briefly on the problem of large deviations from a statistical point of view. The results make extensive use of existing work on rates of convergence.

/pdf/estimating-tails-of-probability-distributions-jfr7ti0wc2.pdf

Estimating Tails of Probability Distributions

/pdf/regular-variation-extensions-and-tauberian-theorems-52861nt18x.pdf

Regular variation, extensions and Tauberian theorems

Second-order regular variation, convolution and the central limit theorem

An extension of karamata's tauberian theorem and its connection with complementary convex functions

textThe theory of stable probability distributions and their domains of attraction is derived in a direct way (avoiding the usual route via infinitely divisible distributions) using Fourier transforms. Regularly varying functions play an important role in the exposition.

/pdf/stable-probability-distributions-and-their-domains-of-5axl13to3m.pdf

Stable Probability Distributions and their Domains of Attraction

textThe stability of the financial system at higher loss levels is either characterized by asymptotic dependence or asymptotic independence. If asymptotically independent, the dependency, when present, eventually dies out completely at the more extreme quantiles, as in case of the multivariate normal distribution. Given that financial service firms' equity returns depend linearly on the risk drivers, we show that the marginals' distributions maximum domain of attraction determines the type of systemic (in-)stability. A scale for the amount of dependency at high loss lovels is designed. This permits a characterization of systemic risk inherent to different financial network structures. The theory also suggests the functional form of the economically relevant limit copulas.

https://www.econstor.eu/bitstream/10419/86512/1/07-023.pdf

Jaap Geluk

Papers

Regular variation, extensions and Tauberian theorems

Second-order regular variation, convolution and the central limit theorem

An extension of karamata's tauberian theorem and its connection with complementary convex functions

Stable Probability Distributions and their Domains of Attraction

Weak & Strong Financial Fragility