(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

Non-Gaussian processes of Ornstein–Uhlenbeck (OU) type offer the possibility of capturing important distributional deviations from Gaussianity and for flexible modelling of dependence structures. This paper develops this potential, drawing on and extending powerful results from probability theory for applications in statistical analysis. Their power is illustrated by a sustained application of OU processes within the context of finance and econometrics. We construct continuous time stochastic volatility models for financial assets where the volatility processes are superpositions of positive OU processes, and we study these models in relation to financial data and theory.

Non-Gaussian Ornstein–Uhlenbeck-based models and some of their uses in financial economics

This celebrated book has been prepared with readers' needs in mind, remaining a systematic treatment of the subject whilst retaining its vitality. The second volume follows on from the first, concentrating on stochastic integrals, stochastic differential equations, excursion theory and the general theory of processes. Much effort has gone into making these subjects as accessible as possible by providing many concrete examples that illustrate techniques of calculation, and by treating all topics from the ground up, starting from simple cases. Many of the examples and proofs are new; some important calculational techniques appeared for the first time in this book. Together with its companion volume, this book helps equip graduate students for research into a subject of great intrinsic interest and wide application in physics, biology, engineering, finance and computer science.

Diffusions, Markov processes, and martingales

Levy processes are the natural continuous-time analogue of random walks and form a rich class of stochastic processes around which a robust mathematical theory exists. Their mathematical significance is justified by their application in many areas of classical and modern stochastic models.

This textbook forms the basis of a graduate course on the theory and applications of Levy processes, from the perspective of their path fluctuations. Central to the presentation are decompositions of the paths of Levy processes in terms of their local maxima and an understanding of their short- and long-term behaviour.

The book aims to be mathematically rigorous while still providing an intuitive feel for underlying principles. The results and applications often focus on the case of Levy processes with jumps in only one direction, for which recent theoretical advances have yielded a higher degree of mathematical transparency and explicitness. Each chapter has a comprehensive set of exercises with complete solutions.

Introductory Lectures on Fluctuations of Lévy Processes with Applications

SUMMARY A new three-parameter family of distributions on the positive numbers is proposed. It includes the stable distributions on the positive numbers, the gamma, the degenerate and the inverse Gaussian distributions. The family is characterized by the Laplace transform, from which moments, convolutions, infinite divisibility, unimodality and other properties are derived. The density is complicated, but a simple saddlepoint approximation is provided. Weibull and Gompertz distributions are naturally mixed over some of the distributions. The family is natural exponential in one of the parameters. The distributions are relevant for application as frailty distributions in life table methods for heterogeneous populations. Desirable properties of such distributions are discussed. As an example survival after myocardial infarction is considered.

https://www.jstor.org/stable/info/2336215

Survival models for heterogeneous populations derived from stable distributions

Summary In the present paper the author proves that the lognormal distribution is infinitely divisible. This is achieved by showing that the lognormal is the weak limit of a sequence of probability distributions all of which are generalized Γ-convolutions and thus infinitely divisible. It is also proved that the lognormal itself is a generalized Γ-convolution.

On the infinite divisibility of the lognormal distribution

Summary That the Pareto distribution is infinitely divisible is a simple consequence of an important theorem by Goldie and Steutel if an observation by Thyrion is used. In the present paper the author investigates the Pareto distribution more closely and proves that it belongs to a well-known subclass of the class of infinitely divisible distributions, the so called class L. That is achieved by showing that the Pareto distribution can be viewed as a “generalized Γ-convolution”. The corresponding problem concerning the lognormal distribution is briefly touched upon.

On the infinite divisbility of the Pareto distribution

In Thorin (1977a) the class of generalized r-convolutions was introduced and considered. This class was further investigated in Thorin (1977b). We recall the definition:

An extension of the notion of a generalized Γ-convolution

In a paper [13] in this journal the author has further developed the ruin theory—first presented by E. Sparre Andersen [1] to the XVth International Congress of Actuaries, New York, 1957—in case the epochs of claims form a renewal process and the gross risk premium is positive. It turned out that much of the Cramer treatment [6] of the classical ruin theory (Poisson case) could be generalized to the more general situation. In the present paper the author points out that the identity shown by him in an earlier paper [12] for the Poisson case is also capable of being generalized. Some comments concerning the case with a negative gross risk premium are also included. I.a. the Cramer asymptotic formula for the ruin probability for an infinite period is generalized also in that case. Further information about the contents of the paper may be inferred from the following section headings. The paragraph 5.4 and the sections 6–7 are found in Part II to appear in SAT 1971:3–4.

Further remarks on the ruin problem in case the epochs of claims form a renewal process

In this article a summing up is made of the author's papers concerning the probability of ruin in a risk business. Results as well as proofs are reviewed. In certain cases not covered in the earlier papers a more systematic treatment is given. Primarily the probability of ruin for a finite time period is dealt with. The corresponding problem for an infinite period is treated as a limit case in a more cursory way. It is throughout assumed that the epochs of claims follow a renewal process, i.e. we adhere to E. Sparre Andersen's model. The case of an ordinary renewal process as well as the case of a stationary such one are treated for both positive and negative capital risks in the spirit of H. Cramer's treatment of the Poisson case. Since the author La. has aimed at facilitating the numerical calculation of ruin probabilities, an essential complement to the article in this respect is the numerical implementation of the author's formulas performed by Nils Wikstad. The reader is, therefore, referred...

Olof Thorin

Papers

On the infinite divisibility of the lognormal distribution

On the infinite divisbility of the Pareto distribution

An extension of the notion of a generalized Γ-convolution

Further remarks on the ruin problem in case the epochs of claims form a renewal process

Probabilities of ruin