Stochastic processes for insurance and finance.

*(English)*Zbl 0940.60005
Chichester: Wiley. xviii, 654 p. (1999).

This monograph is intended for research workers and graduate students who are interested in a survey on modern tools of applied stochastics with applications to insurance and financial mathematics. It presents in a concentrated but well readable way the basic notions from probability theory, in particular from stochastic processes, and shows how they are used to treat problems in insurance and mathematical finance. The book starts with selected notions from ruin theory, like claim number process, claim size process, solvability of portfolios, reinsurance, ruin problems, and with related financial topics (Chapter 1). In Chapter 2 the authors present facts from the theory of one-dimensional probability distributions, having in mind the applications to insurance and finance. In particular, the reader finds heavy-tailed distributions and their statistical treatment. The next chapters introduce the reader to basic concepts of (in particular) non-life-insurance like premium principles (Chapter 3) and aggregate claim amounts, the individual and the collective model (Chapter 4). The dynamic aspects related to Chapter 4 are known as risk processes and are contained in Chapter 5. Then the authors turn back to more ambitious tools of probability theory, and relate them to the topics mentioned above. They study renewal processes and random walks (Chapter 6), Markov chains (Chapter 7), continuous-time Markov (jump) processes (Chapter 8), martingales with discrete and with continuous time (Chapters 9 and 10), piecewise deterministic Markov processes (Chapter 11), point processes (Chapter 12) and diffusion processes (Chapter 13). Some of them are already classical fields of processes and have often been presented in monographs, others are more or less recent models known to specialists only so far (phase type distribution, piecewise deterministic Markov processes). The applications of these processes in insurance and finance are illustrated e.g. within the classical risk process (ruin theory) and some of its generalizations, the Black Scholes model and simple interest rate models.

It can be considered as a particular value of this book that it provides a condensed but nevertheless selfcontained survey on the mathematical foundations of many applications of stochastics. Two of these applications are treated to some extend, insurance and mathematical finance. Of course, both of them are not exhausted, in particular the second one. This would have exceeded the framework of this book. The presentation is mathematically precise and, therefore, demanding. It should be understandable for at least graduate students of mathematics, but also for interested readers of other sciences, and it can be recommended to everybody, who wants to have a look into the mathematical machinery of applied stochastics. Bibliographical notes at the end of each chapter elucidate the origin of many notions and results and refer to more recent results of research. Exercises are missing, they will be presented separately by the authors in a forthcoming book.

It can be considered as a particular value of this book that it provides a condensed but nevertheless selfcontained survey on the mathematical foundations of many applications of stochastics. Two of these applications are treated to some extend, insurance and mathematical finance. Of course, both of them are not exhausted, in particular the second one. This would have exceeded the framework of this book. The presentation is mathematically precise and, therefore, demanding. It should be understandable for at least graduate students of mathematics, but also for interested readers of other sciences, and it can be recommended to everybody, who wants to have a look into the mathematical machinery of applied stochastics. Bibliographical notes at the end of each chapter elucidate the origin of many notions and results and refer to more recent results of research. Exercises are missing, they will be presented separately by the authors in a forthcoming book.

Reviewer: U.Küchler (Berlin)

##### MSC:

60-02 | Research exposition (monographs, survey articles) pertaining to probability theory |

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

62P05 | Applications of statistics to actuarial sciences and financial mathematics |

91B30 | Risk theory, insurance (MSC2010) |

91G80 | Financial applications of other theories |

91-02 | Research exposition (monographs, survey articles) pertaining to game theory, economics, and finance |