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Ruin probabilities. (English) Zbl 0960.60003
Advanced Series on Statistical Science & Applied Probability. 2. Singapore: World Scientific. x, 385 p. (1997).
This book is devoted to risk theory and ruin probabilities in particular. Risk theory in general and ruin probabilities in particular are traditionally considered as part of insurance mathematics, and have been an active area of research from the days of Lundberg all the way up today. However, the practical relevance of the area has been questioned repeatedly. And, as the author writes: “One reason for writing this book is a feeling that the area has in the recent years achieved a considerable mathematical maturity, which has in particular removed one of the standard criticisms of the area, that it can only say something about very simple models and questions. Apart from these remarks, I have deliberately stayed away from discussing the practical relevance of the theory; if the formulations occasionally give a different impression, it is not by intention. Thus, the book is basically mathematical in its flavour.”
The book contains Preface, eleven chapters, Appendix, Bibliography with 384 items and Index. In brief: Chapters I and II are Introduction and Some general tools and results, respectively; Chapters III–VII introduce some of the main models and give a first derivation of some of their properties; Chapters IX and X then go in more depth with some of the special approaches for analyzing specific models and add a number of results on the models in Chapters III–VII; Chapter XI is Miscellaneous topics; Appendix contains some necessary results in renewal theory, Wiener-Hopf factorization, matrix-exponentials, some linear algebra and complements on phase-type distributions. The end of each chapter contains notes and references.
The Introduction contains descriptions of the risk process, claim size distributions, the arrival process, and a summary of main results methods, models and topics to be studied in the rest of the book. In this chapter, a risk reserve process and ruin probabilities with infinite and finite horizon for it are introduced. They are the main topics of study of the present book. Chapter II collects and surveys some topics which repeatedly show up in the study of ruin probabilities. The martingale approach in Section 1 is essentially only used here. The likelihood ratio approach in Section 2 is basic for most of the models under study. The duality results in Section 3 are not crucial for the rest of the book, however, the topic is fundamental and the probability involved is rather simple and intuitive. Sections 4, 5 on random walks and Markov additive processes can be skipped until reading Chapter VI on the Markovian environment model. The ladder height formula in Theorem 6.1 is basic for the study of the compound Poisson model in Chapter III. Chapter III is devoted to the study of the compound Poisson model with Pollaczeck-Khinchin formula, change of measure via exponential families, Lundberg conjugation and various approximations for the ruin probabilities and estimation of the adjustment coefficient.
Chapter IV is concerned with the finite time ruin probabilities. Only the compound Poisson case is treated. Generalizations to other models are either discussed in the notes and references or in relevant chapters. Chapter V studies the risk process with renewal arrivals. The renewal model is often referred to as Sparre Andersen process, after E. Sparre Andersen whose 1959 paper was the first to treat renewal assumptions in risk theory in more depth. Chapter VI contains the risk theory in a Markovian environment. The author assumes that arrivals are not homogeneous in time but determined by a Markov process with finite state space. This latest process describes the environmental conditions for the risk process. In Chapter VII the author assumes that accumulated claims or total claims process is a compound Poisson process, however, the premium charged is assumed to depend upon the current reserve process \(R_{t}\) so that the premium rate is \(p(r)\) when \(R_{t}=r.\) Ruin probabilities for such models are studied. Chapter VIII deals with matrix-analytic methods in the risk theory. Ruin probabilities in the presence of heavy tails are considered in Chapter IX. Chapter X gives a summary of some basic methodology in simulation and Monte Carlo methods. Miscellaneous topics, beginning from Bernoulli random walk, Brownian motion and two-barrier problem and ending with reinsurance, are considered in Chapter XI.
The book is an excellent handbook in the risk theory and brilliant encyclopedia in ruin probabilities results. This book will be useful both for graduate students who deal with stochastic models in insurance mathematics, experts in risk processes and also for those specialists who apply the methods of risk theory in practice.

60-02 Research exposition (monographs, survey articles) pertaining to probability theory
60J25 Continuous-time Markov processes on general state spaces
91B30 Risk theory, insurance (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