Mathematical Methods in Risk Theory (Grundlehren der mathematischen Wissenschaften)
Actuarial mathematics originated toward the end of the 17th century, when E. Halley’s famous mortality table permitted the mathematical treatment and calculation of annuity values for the first time. The underlying model expressed in mathematical language at this time has been so closely adhered to in life insurance techniques-the classical field of application for actuarial mathematics -that actuarial theory in this classical sense has often been characterized as closed in upon itself.
Nonetheless, a new orientation of actuarial mathematics has taken place in the past few decades which has introduced fundamental new accents. The stimulus for this development arose from areas of activity other than life insurance, viz. from casualty insurance or from the nonlife branches of insurance generally. The development was made possible by the powerful advances in probability theory and mathematical statistics since the 1930’s and has been iavorably influenced by a parallel emphasis on mathernatical methods in economic theory. If one seeks to characterize this “new” actuarial mathematics, one can best do so by saying that it undertakes to solve the technical problems of all branches of insurance and that it concerns itself particularly with the operational problems of the insurance enterprise. Characteristic of the present stage of development, however, is the fact that the current profusion of scientific publications in the field of actuarial mathematics deals above all with detached individual problems. An excellent general view of this diversity of publications is given by Carl Philipson’s bibliography . H. Seal’s book  also gives a well-rounded survey of the literature. The work of Beard, Pentikäinen and Pesonen  is to be recommended as an introduction on an easily understandable level to the new development in actuarial mathematics.
The present book is intentionally not oriented bibliographically. It attempts to create a synthesis out of a selection made by the author of modern scientific publications in the field of actuarial mathematics, with the goal of presenting a unified system of thought.
The construction is so arranged that the mathematical model of the events dealt with in insurance is presented in the first part. Chapter 1 explains the probability theoretical fundamentals of risk. The elements of probability theory which will be necessary for the subsequent development 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.
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