Generalized Poisson models and their applications in insurance and finance. (English) Zbl 1041.60004

Bening, Vladimir E.; Korolev, Victor Yu.

Modern Probability and Statistics. Utrecht: VSP (ISBN 90-6764-366-1/hbk; 978-3-11-093601-8/ebook). xix, 434 p. (2002).

To quote from the preface: \`\` This book can be regarded as an ode to the Poisson distribution. …It presents the description of the present state of the art in the field of compound Cox processes and their applications in insurance and finance. Along with the review of the well-known classical results on compound and mixed Poisson processes and risk theory, it contains many new results obtained by the authors recently. Among new theoretical results presented in the book we should mention new convergence criteria, convergence rate estimates, asymptotic expansions for quantiles of stochastic processes and many others. …The main idea of the book is to concentrate on generalized Poisson models oriented at their exploitation in applied problems, first of all, in insurance and finance.
Among applied problems considered in the book, four deserve to be mentioned especially. The first of them is the problem of prediction of stock prices. …the theory based on the asymptotic properties of compound Cox processes can explain how heavy-tailed distributions, for example, stable laws can occur as the distributions of stock price increments even when jumps of the stock price have finite variances. …
The second problem is connected with the description of the asymptotic behavior of the so-called generalized risk processes. These processes are natural generalizations of the classical risk process with constant premium rate and Poisson flow of claims. …the generalized risk processes which are obtained from classical risk processes by means of stochastic change of time. In the book, the asymptotic behavior of these processes is investigated in full detail. The results presented in the book provide serious theoretical grounds for the construction of reasonable (asymptotic) approximations for the distribution of the surplus of an insurance company.
The third problem is that of statistical estimation of the probability of ruin for a generalized risk process. …each of the analytical (asymptotic) estimate requires information on the behavior of the tails of the distributions of claims whereas in practice this information can hardly be obtained since statistical inference concerning these distributions is based on samples with finite sizes. Therefore, from the practical viewpoint it is very reasonable to follow another approach and to construct direct statistical estimators for the ruin probability based on the pre-history of the risk process. This problem is principally new. …
The fourth problem is related to a non-traditional approach to the determination of optimal values of basic parameters of a risk process, namely, the starting capital of an insurance company and premium rate. …This problem is a re-formulation of a problem of optimal inventory control also considered in the book.\'\'
A good overview of the contents of the book is given by the chapter titles: 1) Basic notions of probability theory. 2) Poisson process. 3) Convergence of superpositions of independent stochastic processes: This includes the asymptotic theory of random sequences with random indices and in particular necessary and sufficient conditions for convergence in distribution of random sequences with independent random indices. In general, the limit is a mixture of a distribution function over both location and scale, and examples show that a bad random number of even light-tailed summands can converge to a heavy-tailed limit. 4) Compound Poisson distributions: This contains results about asymptotic normality, asymptotic expansions (also for quantiles), Esscher transforms, and convergence rates in local limit theorems. 5) Classical risk processes: This chapter contains the Pollaczek-Khinchin-Beekman formula for the Laplace transform of the ruin probability, approximations and asymptotic expansions of the ruin probability with small safety loading, empirical approximations (De Vylder, Beekman-Bowers, diffusion, Cramér-Lundberg, bounds), the asymptotic behavior of the surplus in general risk processes, and a problem of inventory control with an application to the optimization of initial capital. 6) Doubly stochastic Poisson processes (Cox processes): This shorter chapter starts with mixed Poisson processes, gives the definition and elementary properties of doubly stochastic Poisson processes, and studies their asymptotic behavior for large times. 7) Compound Cox processes with zero mean: This studies, for this class of processes, similar questions as in Chapters 3 and 4. 8) Modeling evolution of stock prices by compound Cox processes: This contains a construction of models and a study of some of their distributional properties. 9) Compound Cox processes with nonzero mean: With different normalizations due to the lack of centering, this studies the same basic questions as Chapter 7. 10) Functional limit theorems for compound Cox processes: While many limit theorems in the book are only given for one-dimensional marginal distributions, this chapter studies the functional convergence and explicitly identifies the limiting Lévy processes under different sets of assumptions. 11) Generalized risk processes: This chapter replaces the classical compound Poisson processes with constant drift by a time-changed version and then studies the convergence of one-dimensional marginals, convergence rates, asymptotic expansions etc., similarly as in Chapter 3. 12) Statistical inference concerning the parameters of risk processes: This gives novel estimates and (asymptotic) confidence intervals for ruin probabilities for classical and generalized risk processes.
All in all, this book contains a wealth of material that will make it very useful to researchers working with Poisson models and their applications, in particular in insurance and finance. Perhaps the only serious handicap of the book is its price.

Reviewer: Martin Schweizer (Zürich)

MSC:

60-02	Research exposition (monographs, survey articles) pertaining to probability theory
62P05	Applications of statistics to actuarial sciences and financial mathematics
60Fxx	Limit theorems in probability theory
60Gxx	Stochastic processes
91B28	Finance etc. (MSC2000)
91B30	Risk theory, insurance (MSC2010)

Keywords:

Poisson models; Cox processes; insurance; finance; asymptotic distributions; limit theorems; asymptotic expansions; ruin probability