Risk theory.

*(English)*Zbl 1422.91009
Springer Actuarial. Lecture Notes. Cham: Springer (ISBN 978-3-319-72004-3/pbk; 978-3-319-72005-0/ebook). xii, 242 p. (2017).

The presented book gives a wide introduction to the problems which arise in non-life insurance. By providing an overview of classical actuarial methods, the author of the book pays a great attention to the ruin models, which are particularly interesting from the mathematical point of view. The book consists of nine chapters and seven appendices.

The first chapter deals with risk in a single time period. It is supposed that such risk \(S\) can be expressed by a compound sum of independent random variables. The compound binomial, compound Poisson, compound mixed Poisson, and compound negative binomial models are discussed in detail. In addition, various possible approximations of the sum \(S\) are described together with the principal premium calculation principles and risk measures.

In the second chapter of the book the elements of utility theory are presented. The properties of utility functions are derived, the possible utility functions are presented, and the expected utility principle is described. There are considered both the insurer’s and the policyholder’s view to the zero utility premium. Results for Pareto-optimal risk exchanges complete the section.

The third chapter deals with Bayesian methods for the calculation of the net premium of a collective insurance contract together with the bonus-malus systems to get a simple approximation to the credibility premium.

The classical methods for the claim reserving are considered in the forth chapter. The chain-ladder, the loss-development, the additive, the Cape Cod method, and the Bornhuetter-Ferguson method are discussed and compared to each other.

In Chapter 5, the classical risk process \(C_t\) is investigated to describe the behaviour of the surplus of an insurance company,

\[ C_t=u+ct-\sum_{i=1}^{N_t}Y_i, \] where \(u\) is the initial capital, \(c\) is the premium rate, \(N_t\) is a homogeneous Poisson process with some positive intensity and \(\{Y_1,Y_2,\ldots\}\) is a sequence of independent and identically distributed random claims. According to this model, the aggregate claims in any interval have a compound Poisson distribution. The ruin probability, the time of ruin, the capital prior to ruin, the deficit to ruin and other critical characteristics of the classical risk model are considered in detail.

The sixth chapter deals with the simplest generalisation of the classical risk model by assuming that the claim number process \(N_t\) is a renewal process. In this case, the risk process \(C_t\) is no longer Markovian because the distribution of the time of the next claim depends on the past via the time of the last claim. The Lundberg inequality, the Cramér-Lundberg approximation, diffusion approximation and asymptotic formula for the ruin probability in the case of subexponential claim size distributions are considered. If it is assumed that the claim arrival process \(N_t\) is a double stochastic point process, then the corresponding risk process \(C_t\) is called Ammeter risk process. This process is introduced and studied in the seven chapter. In the case of the Ammeter process, many assertions describing the ruin probability are similar to the corresponding assertions for the classical risk model.

In the eighth chapter, the possibilities of the change measure techniques and the martingale method are demonstrated for the classical, the Ammeter, and the risk renewal models. The ninth and last chapter deals with the Markov modulated risk model. Firstly, a continuous-time Markov chain is constructed on a finite state space. This Markov chain represents an environment process for the risk business. The environment determines the intensity level. Moreover, the claim size distributions may be different for different states of an environment. For the ruin probability, the Lundberg inequality, the Cramér-Lundberg approximation and asymptotic formula for subexponential claim sizes are derived in the case the Markov modulated risk model. At the end of the book, one can find of several appendices. Some useful information is presented on stochastic processes, martingales, renewal processes, random walks, subexponential distributions and concave functions.

The first chapter deals with risk in a single time period. It is supposed that such risk \(S\) can be expressed by a compound sum of independent random variables. The compound binomial, compound Poisson, compound mixed Poisson, and compound negative binomial models are discussed in detail. In addition, various possible approximations of the sum \(S\) are described together with the principal premium calculation principles and risk measures.

In the second chapter of the book the elements of utility theory are presented. The properties of utility functions are derived, the possible utility functions are presented, and the expected utility principle is described. There are considered both the insurer’s and the policyholder’s view to the zero utility premium. Results for Pareto-optimal risk exchanges complete the section.

The third chapter deals with Bayesian methods for the calculation of the net premium of a collective insurance contract together with the bonus-malus systems to get a simple approximation to the credibility premium.

The classical methods for the claim reserving are considered in the forth chapter. The chain-ladder, the loss-development, the additive, the Cape Cod method, and the Bornhuetter-Ferguson method are discussed and compared to each other.

In Chapter 5, the classical risk process \(C_t\) is investigated to describe the behaviour of the surplus of an insurance company,

\[ C_t=u+ct-\sum_{i=1}^{N_t}Y_i, \] where \(u\) is the initial capital, \(c\) is the premium rate, \(N_t\) is a homogeneous Poisson process with some positive intensity and \(\{Y_1,Y_2,\ldots\}\) is a sequence of independent and identically distributed random claims. According to this model, the aggregate claims in any interval have a compound Poisson distribution. The ruin probability, the time of ruin, the capital prior to ruin, the deficit to ruin and other critical characteristics of the classical risk model are considered in detail.

The sixth chapter deals with the simplest generalisation of the classical risk model by assuming that the claim number process \(N_t\) is a renewal process. In this case, the risk process \(C_t\) is no longer Markovian because the distribution of the time of the next claim depends on the past via the time of the last claim. The Lundberg inequality, the Cramér-Lundberg approximation, diffusion approximation and asymptotic formula for the ruin probability in the case of subexponential claim size distributions are considered. If it is assumed that the claim arrival process \(N_t\) is a double stochastic point process, then the corresponding risk process \(C_t\) is called Ammeter risk process. This process is introduced and studied in the seven chapter. In the case of the Ammeter process, many assertions describing the ruin probability are similar to the corresponding assertions for the classical risk model.

In the eighth chapter, the possibilities of the change measure techniques and the martingale method are demonstrated for the classical, the Ammeter, and the risk renewal models. The ninth and last chapter deals with the Markov modulated risk model. Firstly, a continuous-time Markov chain is constructed on a finite state space. This Markov chain represents an environment process for the risk business. The environment determines the intensity level. Moreover, the claim size distributions may be different for different states of an environment. For the ruin probability, the Lundberg inequality, the Cramér-Lundberg approximation and asymptotic formula for subexponential claim sizes are derived in the case the Markov modulated risk model. At the end of the book, one can find of several appendices. Some useful information is presented on stochastic processes, martingales, renewal processes, random walks, subexponential distributions and concave functions.

Reviewer: Jonas Šiaulys (Vilnius)

##### MSC:

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

91B30 | Risk theory, insurance (MSC2010) |

91B16 | Utility theory |

60J75 | Jump processes (MSC2010) |

60K10 | Applications of renewal theory (reliability, demand theory, etc.) |