##
**Ruin probabilities.
2nd ed.**
*(English)*
Zbl 1247.91080

Advanced Series on Statistical Science & Applied Probability 14. Hackensack, NJ: World Scientific (ISBN 978-981-4282-52-9/hbk; 978-981-4282-53-6/ebook). xvii, 602 p. (2010).

The monography is devoted to a classical field of insurance mathematics, risk theory. It is the second considerably extended edition (from 385 to 602 pages) of the first edition published under the same title by Asmussen in 1997 (for a review, see Zbl 0960.60003). It includes five new chapters with recent research contributions to risk theory, i.e., level dependent risk processes, ruin probabilities for Lévy processes, Gerber-Shiu-functions, further models with dependence, stochastic control. The book presents an excellent overview of different kinds of mathematical models of risk processes, mathematical tools to treat them, and the calculation of the corresponding ruin probabilities. It is a mathematical book, all results presented are proven in sufficient detail. Motivations at the beginning of the chapters make it easier to understand the relevance of, e.g., the model assumptions. Detailed notes and references at the end of the sections get the reader a wide look out to the relevant literature. Last but not least, a great number of examples illustrate the theoretical results. Remarkable is the bibliography with 923 items. The reader is assumed to have some knowledge in probability theory and in stochastic processes.

The first chapter introduces basic notions of the main topic of the book, the risk reserving process, and summarizes the main results and methods. In chapter II, elements of random walks, martingales and Brownian motion with application to risk processes are presented in a very concentrated way. Chapter III includes advanced tools from the theory of stochastic processes necessary in the sequel (amongst other things, likelihood ratios, Markov additive processes, ladder height distributions). The duality between risk processes and certain models for queuing and storage is shown. In chapters IV–VIII the authors treat different models for the risk reserving process and their properties like compound Poisson processes, diffusion approximation, finite time horizon, Markovian environment, level dependent risk processes. In chapter IV the risk process is modeled as a compound Poisson process, the Cramer-Lundberg approximation of the ruin probability (infinite horizon) is derived, sensitivity estimates of the ruin probability in dependence of changing the premium rate, the claim size distribution and the arrival rate as well as estimates of the adjustment coefficient are provided. The same model but for finite time horizon is studied in chapter V. The authors answer the question of how a risk process behaves given it will lead to ruin. In chapter VI, the authors leave the Poisson case but keep the assumption that the arrival points of claims form a renewal process. Again the duality with queuing theory is elucidated. If the parameters of the risk process are not constant but follow a Markov process, than one says the process moves in a Markovian environment. This is the topic of the chapter VII. Finally, chapter VIII completes the series of models for risk reserving processes by including investment strategies of the insurance company, dividend payments, interest earning, tax payments. In chapters IX and X the authors consider further classes of distributions occurring as claim size distributions. Chapter IX treats the so-called phase-type distributions, which allow in general an algorithm treatment by using matrix calculus. They are interesting for anybody working in applied probability. Another class of distributions which have attracted the attention of applied mathematicians very much during the last decades are the heavy tailed distributions, in particular, the subexponential distributions. In chapter X the authors derive ruin probabilities for claim size distributions with heavy tails and draw connections to extreme value theory. Chapters XI to XVI are devoted to recent developments in risk theory. The topics are Lévy processes (instead of compound Poisson processes, chapter XI) and Gerber-Shiu-functions (functional of the deficit at ruin, surplus prior to ruin, time of ruin, chapter XII). In chapter XIII, the influence of different kinds of dependence to the adjustment coefficient and other quantities is studied. Large deviation theory is used for deriving asymptotic properties of the ruin probability. Stochastic control theory applied to risk reserved processes can be found in chapter XIV. In chapter XV, simulation methods are summarized, which are fitted to the peculiarity of the risk processes, e.g., light and heavy tailed distributions, rare events simulation, variance reduction, regenerative simulation. Chapter XVI starts with risk reserving processes having discrete time and discrete claim size distributions. They are better suited to computations. Then, short excursions to premium calculation principles and kinds of reinsurance close the chapter. In an appendix, the relevant elements of renewal theory, Wiener-Hopf factorization, matrix-exponentials, some linear algebra, complements on phase-type distributions and Tauber theorems are presented. The book is a milestone in the landscape of applied probability theory and can be recommended to all persons interested in applied mathematics.

The first chapter introduces basic notions of the main topic of the book, the risk reserving process, and summarizes the main results and methods. In chapter II, elements of random walks, martingales and Brownian motion with application to risk processes are presented in a very concentrated way. Chapter III includes advanced tools from the theory of stochastic processes necessary in the sequel (amongst other things, likelihood ratios, Markov additive processes, ladder height distributions). The duality between risk processes and certain models for queuing and storage is shown. In chapters IV–VIII the authors treat different models for the risk reserving process and their properties like compound Poisson processes, diffusion approximation, finite time horizon, Markovian environment, level dependent risk processes. In chapter IV the risk process is modeled as a compound Poisson process, the Cramer-Lundberg approximation of the ruin probability (infinite horizon) is derived, sensitivity estimates of the ruin probability in dependence of changing the premium rate, the claim size distribution and the arrival rate as well as estimates of the adjustment coefficient are provided. The same model but for finite time horizon is studied in chapter V. The authors answer the question of how a risk process behaves given it will lead to ruin. In chapter VI, the authors leave the Poisson case but keep the assumption that the arrival points of claims form a renewal process. Again the duality with queuing theory is elucidated. If the parameters of the risk process are not constant but follow a Markov process, than one says the process moves in a Markovian environment. This is the topic of the chapter VII. Finally, chapter VIII completes the series of models for risk reserving processes by including investment strategies of the insurance company, dividend payments, interest earning, tax payments. In chapters IX and X the authors consider further classes of distributions occurring as claim size distributions. Chapter IX treats the so-called phase-type distributions, which allow in general an algorithm treatment by using matrix calculus. They are interesting for anybody working in applied probability. Another class of distributions which have attracted the attention of applied mathematicians very much during the last decades are the heavy tailed distributions, in particular, the subexponential distributions. In chapter X the authors derive ruin probabilities for claim size distributions with heavy tails and draw connections to extreme value theory. Chapters XI to XVI are devoted to recent developments in risk theory. The topics are Lévy processes (instead of compound Poisson processes, chapter XI) and Gerber-Shiu-functions (functional of the deficit at ruin, surplus prior to ruin, time of ruin, chapter XII). In chapter XIII, the influence of different kinds of dependence to the adjustment coefficient and other quantities is studied. Large deviation theory is used for deriving asymptotic properties of the ruin probability. Stochastic control theory applied to risk reserved processes can be found in chapter XIV. In chapter XV, simulation methods are summarized, which are fitted to the peculiarity of the risk processes, e.g., light and heavy tailed distributions, rare events simulation, variance reduction, regenerative simulation. Chapter XVI starts with risk reserving processes having discrete time and discrete claim size distributions. They are better suited to computations. Then, short excursions to premium calculation principles and kinds of reinsurance close the chapter. In an appendix, the relevant elements of renewal theory, Wiener-Hopf factorization, matrix-exponentials, some linear algebra, complements on phase-type distributions and Tauber theorems are presented. The book is a milestone in the landscape of applied probability theory and can be recommended to all persons interested in applied mathematics.

Reviewer: Uwe Küchler (Berlin)

### MSC:

91B30 | Risk theory, insurance (MSC2010) |

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

60J75 | Jump processes (MSC2010) |

60K05 | Renewal theory |

60K15 | Markov renewal processes, semi-Markov processes |

60G44 | Martingales with continuous parameter |

60G50 | Sums of independent random variables; random walks |

60J60 | Diffusion processes |

60F10 | Large deviations |

60K37 | Processes in random environments |