Non-life insurance mathematics. An introduction with stochastic processes.

*(English)*Zbl 1033.91019
Universitext. Berlin: Springer (ISBN 3-540-40650-6/pbk). xi, 235 p. (2004).

The author’s aim to bring some of the standard stochastic models of non-life insurance mathematics to the attention of a wide audience (including actuaries and other applied scientists) can definitely be reached by this textbook. It is well-written and gives - in a first and main part - a detailed discussion of the releveant stochastic processes such as claim number processes, renewal processes, Poisson and mixed Poisson processes, the total claim amount (i.e. compound processes), risk processes and ruin probability, including the famous Lundberg inequality and the Cramér-Lundberg asymptotics. In addition, it also deals with some exploratory statistical tools and aspects of suitable modeling, illustrated both by real non-life insurance data and results of computer simulations. Special attention is given to heavy-tail phenomena, which are of key interest for an appropriate analysis of, e.g., reinsurance problems. So, there is a good balance between the necessary mathematical theory and methods and their applications in non-life insurance practice.

More than 90 policies are the object of main interest. Collective risk theory, however, requires homogeneous portfolios to a large extent and/or long-time experience. In case of heterogeneous portfolios, individual risk models can serve to take the claim history of a policy into account. For the latter reason, but also as a tribute to Hans Bühlmann’s fundamental contributions to this area, the second part of the book gives a brief introduction to experience rating. Three major models of credibility theory are discussed, i.e. heterogeneity, the Bühlmann and the Bühlmann-Straub model, and give a first, but self-contained idea of how dependencies of the claim structure inside a policy and between policies could be described.

Many figures and tables in this book help in illustrating and visualizing the developed theory. Moreover, every section ends with an extensive collection of exercises which are considered an integral part of the text and should help the reader in accessing the theory. The book has primarily been written for students and lectureres of actuarial mathematics, but should indeed be of interest to a wider audience of scientists who want to become familiar with applications of probability theory and stochastic processes in various fields of research.

The level of the text is not too advanced, but requires some basic background in probability and measure theory and in stochastic processes. Markov process theory and martingales are avoided as much as possible, and so are many analytical tools such as Laplace-Stieltjes transforms. Instead, the author focuses on a more intuitive probabilistic understanding of the underlying random process structures. Under the given restrictions, however, proofs of results are given whereever possible. Moreover, the interested reader will find various comments sections with references to more advanced literature and an extensive list of references for deeper studies.

More than 90 policies are the object of main interest. Collective risk theory, however, requires homogeneous portfolios to a large extent and/or long-time experience. In case of heterogeneous portfolios, individual risk models can serve to take the claim history of a policy into account. For the latter reason, but also as a tribute to Hans Bühlmann’s fundamental contributions to this area, the second part of the book gives a brief introduction to experience rating. Three major models of credibility theory are discussed, i.e. heterogeneity, the Bühlmann and the Bühlmann-Straub model, and give a first, but self-contained idea of how dependencies of the claim structure inside a policy and between policies could be described.

Many figures and tables in this book help in illustrating and visualizing the developed theory. Moreover, every section ends with an extensive collection of exercises which are considered an integral part of the text and should help the reader in accessing the theory. The book has primarily been written for students and lectureres of actuarial mathematics, but should indeed be of interest to a wider audience of scientists who want to become familiar with applications of probability theory and stochastic processes in various fields of research.

The level of the text is not too advanced, but requires some basic background in probability and measure theory and in stochastic processes. Markov process theory and martingales are avoided as much as possible, and so are many analytical tools such as Laplace-Stieltjes transforms. Instead, the author focuses on a more intuitive probabilistic understanding of the underlying random process structures. Under the given restrictions, however, proofs of results are given whereever possible. Moreover, the interested reader will find various comments sections with references to more advanced literature and an extensive list of references for deeper studies.

Reviewer: Josef Steinebach (Köln)

##### MSC:

91B30 | Risk theory, insurance (MSC2010) |

60G35 | Signal detection and filtering (aspects of stochastic processes) |

60K05 | Renewal theory |

91-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to game theory, economics, and finance |