##
**Aspects of risk theory.**
*(English)*
Zbl 0717.62100

Springer Series in Statistics. Probability and its Applications. New York etc.: Springer-Verlag. x, 175 p. DM 78.00 (1991).

In collective risk theory, a class of stochastic processes is considered which can describe the actual capital that an insurance company has at its disposal and which is governed by random arrival epochs of claims and by radom amounts of money to be paid by the company at these arrival epochs. The classical risk model assumes that the claims arrive according to a homogeneous Poisson process and that the costs of the claims can be described by i.i.d. random variables which are independent of the arrival epochs. Between consecutive arrival epochs of claims the capital is assumed to increase with a constant (and deterministic) rate. An important characteristic of these so-called risk processes is the ruin probability \(\Psi\) (0), i.e. the probability that at any time the actual capital of the company will be less than the initial capital. Furthermore, one is interested in the ruin probability \(\Psi\) (u) that at any time the actual capital undercuts the initial capital by more then u units.

The ruin probability \(\Psi\) (0) can be easily expressed by the intensity of the underlying Poisson process describing the arrival epochs of claims, by the expected costs of a single claim, and by the rate of income. In distinction to this, for \(u>0\) the determination of \(\Psi\) (u) is in general much more complicated because, for \(u>0\), \(\Psi\) (u) depends on the cost distribution not only via its expectation. As a consequence of this, bounds and approximation formulas for \(\Psi\) (u) have been derived in the literature, e.g. the Cramér-Lundberg approximation \[ (1)\quad \min_{u\to \infty}e^{Ru}\Psi (u)=C \] where the Lundberg exponent R is given by a certain functional equation, and the Lundberg inequality \[ (2)\quad \Psi (u)\leq e^{-Ru}\text{ for } every\quad u\geq 0. \] Besides a comprehensive explanation of such results for the classical risk model, the present book mainly deals with the question how the consideration of more general sequences of arrival epochs is reflected by the ruin probabilities. Among others it is pointed out that for several classes of non-Poisson arrival processes the above mentioned insensitivity property of \(\Psi\) (0), with respect to the cost distribution provided that its expectation is fixed, as well as (1) and (2) remain valid at least in a modified form.

In Chapter 1 the ruin probabilities \(\Psi\) (u), \(u\geq 0\), are determined for the classical risk model. For exponentially distributed costs and for further cost distributions, \(\Psi\) (u) is given by analytical formulas or in a form suited for numerical calculations. Furthermore, a method of statistical estimation of the Lundberg exponent R is discussed. Beginning in Chapter 2, risk models with more general arrival processes are considered. In connection with this the notion of a random point process turns out to be a suitable tool. Chapter 2 may be seen as a brief introduction to this branch of probability theory. In Chapter 3, the case is considered that the claims arrive according to a renewal process, where both the ordinary renewal model and the stationary renewal model are investigated. It is shown how the general results simplify when the costs are exponentially distributed. Furthermore, for special inter- occurrence time distributions, R and \(\Psi\) (u) are numerically calculated.

In Chapter 4 main emphasis is put on Lundberg inequalities in the case of a Cox arrival process which is very natural as a model for risk fluctuations. The stochastic intensity is assumed to be a Markov process, a process with “independent jumps”, and a Markov renewal process, respectively. Numerical illustrations show how the Lundberg exponents obtained in the Cox case differ from those in the Poisson case. Finally, in Chapter 5 it is assumed that the arrival process is a general stationary point process. It is proved that also in this case the ruin probability \(\Psi\) (0) depends only on the intensity of the underlying point process and on the expectation of the cost distribution. In an appendix, finite time ruin probabilities are discussed both for the classical model and for models with the more general arrival processes considered in the preceding chapters.

As indicated in the title, the book is a monograph on certain aspects of risk theory and not a textbook. It is thought for the actuary who has a good knowledge of classical risk theory and wants to get acquainted with more general stochastic models which can give a more realistic description of the claim behavior. Simultaneously, the book is thought for the probabilist who wants to get an introduction to modern risk theory and to form an idea of the use of point processes and martingale techniques in this theory. The book is well-written and a valuable increase in the literature on applications of stochastic models in risk theory.

The ruin probability \(\Psi\) (0) can be easily expressed by the intensity of the underlying Poisson process describing the arrival epochs of claims, by the expected costs of a single claim, and by the rate of income. In distinction to this, for \(u>0\) the determination of \(\Psi\) (u) is in general much more complicated because, for \(u>0\), \(\Psi\) (u) depends on the cost distribution not only via its expectation. As a consequence of this, bounds and approximation formulas for \(\Psi\) (u) have been derived in the literature, e.g. the Cramér-Lundberg approximation \[ (1)\quad \min_{u\to \infty}e^{Ru}\Psi (u)=C \] where the Lundberg exponent R is given by a certain functional equation, and the Lundberg inequality \[ (2)\quad \Psi (u)\leq e^{-Ru}\text{ for } every\quad u\geq 0. \] Besides a comprehensive explanation of such results for the classical risk model, the present book mainly deals with the question how the consideration of more general sequences of arrival epochs is reflected by the ruin probabilities. Among others it is pointed out that for several classes of non-Poisson arrival processes the above mentioned insensitivity property of \(\Psi\) (0), with respect to the cost distribution provided that its expectation is fixed, as well as (1) and (2) remain valid at least in a modified form.

In Chapter 1 the ruin probabilities \(\Psi\) (u), \(u\geq 0\), are determined for the classical risk model. For exponentially distributed costs and for further cost distributions, \(\Psi\) (u) is given by analytical formulas or in a form suited for numerical calculations. Furthermore, a method of statistical estimation of the Lundberg exponent R is discussed. Beginning in Chapter 2, risk models with more general arrival processes are considered. In connection with this the notion of a random point process turns out to be a suitable tool. Chapter 2 may be seen as a brief introduction to this branch of probability theory. In Chapter 3, the case is considered that the claims arrive according to a renewal process, where both the ordinary renewal model and the stationary renewal model are investigated. It is shown how the general results simplify when the costs are exponentially distributed. Furthermore, for special inter- occurrence time distributions, R and \(\Psi\) (u) are numerically calculated.

In Chapter 4 main emphasis is put on Lundberg inequalities in the case of a Cox arrival process which is very natural as a model for risk fluctuations. The stochastic intensity is assumed to be a Markov process, a process with “independent jumps”, and a Markov renewal process, respectively. Numerical illustrations show how the Lundberg exponents obtained in the Cox case differ from those in the Poisson case. Finally, in Chapter 5 it is assumed that the arrival process is a general stationary point process. It is proved that also in this case the ruin probability \(\Psi\) (0) depends only on the intensity of the underlying point process and on the expectation of the cost distribution. In an appendix, finite time ruin probabilities are discussed both for the classical model and for models with the more general arrival processes considered in the preceding chapters.

As indicated in the title, the book is a monograph on certain aspects of risk theory and not a textbook. It is thought for the actuary who has a good knowledge of classical risk theory and wants to get acquainted with more general stochastic models which can give a more realistic description of the claim behavior. Simultaneously, the book is thought for the probabilist who wants to get an introduction to modern risk theory and to form an idea of the use of point processes and martingale techniques in this theory. The book is well-written and a valuable increase in the literature on applications of stochastic models in risk theory.

Reviewer: V.Schmidt

### MSC:

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

62-02 | Research exposition (monographs, survey articles) pertaining to statistics |

91B30 | Risk theory, insurance (MSC2010) |

90-02 | Research exposition (monographs, survey articles) pertaining to operations research and mathematical programming |

60G55 | Point processes (e.g., Poisson, Cox, Hawkes processes) |

60K20 | Applications of Markov renewal processes (reliability, queueing networks, etc.) |

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