Mathematical Methods in Risk Theory
From the reviews: "The huge literature in risk theory has been carefully selected and supplemented by personal contributions of the author, many of which appear here for the first time. The result is a systematic and very readable book, which takes into a
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Preface
V1
lopment are recalled here for the reader who is moderately familiar with the theory on an intermediate level (without the use of measure theory). Chapter 2 treats the risk process and at the same time tlie tools of the theory of stochastic processes are elucidated. Chapter 3 explains the concept of the collective and develops the related risk quantities. Consequences of the mathematical model form the content of the second part. Chapter 4 deals with premium calculation and Chapter 5 with the retention problern. Finally, the real operational problems are taken up in Chapter 6, the subject of which are the risk carrier's stability criteria. In addition to the probability of ruin criterion, the dividend policy and utility criteria are also discussed. The general tendency toward forming a bridge between economic and actuarial theory is particularly visible in this last chapter. It should also be mentioned explicitly what the book does not treat (at least in so far as what the reader may have expected). No statistical estimatioiz methods are brought up in connection with the mathematical models developed here. I believe that the Separation of the trains of thought into a) construction of models and b) measurement of the Parameters which appear in the models should be preserved -as in the classical actuarial mathematics of life insurance. The present book therefore deals intentionnally only with point a). The publication of this book would not have been possible without the vigorous Support of my colleagues. Dr. H.-U. Gerber assisted with a preliminary version of the manuscript and also made a real contribution to the content of section 6.4 with his dissertatioii. My assistants, Messrs. F. Pfenninger and W. Maurer undertook a careful examination of the manuscript and the corrections. I should particularly like to thank the translator, Mr. C. E. Brooks, F. S.A. for his cooperation. Through him it has been possible to publish the book in the language which will make it accessible to the largest possible circle of readers. My appreciation is also due to Mrs. E. Minzloff, secretary at the Mathematical Research Institute of the Federal Institute of Technology for typing the manuscript in final form. Zürich, September 15, 1970 Hans Bühlmann
In nzemory of my father
Table of Contents Part I . The Theoretical Model Chapter 1 : Probability Aspects of Risk
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1.1. Random variables explained by the example of claim amount . . . . . . 1.1.1. Definition . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2. Classification and examples of distribution functions . . . . . . . 1.l. 3. Expected values . . . . . . . . . . . . . . . . . . . . . . . 1.1.4. Characteristics of a probability distribution and auxiliary functions 1.1.5. Chebyshev's Inequality . . . . . . . . . . . . . . . . . . . . 1.2. Sequences of random variables explained by the example of claim amount reproductions . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1. Multi-dimensional distributions and auxiliary fun
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