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Non-Life Insurance Mathematics: An Introduction with the Poisson Process

Thomas Mikosch
Publication Date: 
Number of Pages: 
[Reviewed by
P. N. Ruane
, on

The growth of the insurance industry in the past hundred years has led to enormous developments in the academic discipline that applies mathematical methods to the assessment of risk in the financial sector. The said discipline is actuarial science, and there are now a large number of degree courses in this field. This book is the outcome of one such course, taught by the author at the University of Copenhagen.

Years ago, entry to the actuarial profession would have been preceded by a mathematics degree and possibly followed by the appropriate professional qualification. Nowadays, specialist undegraduate courses provide the necessary mathematical background together with introductions to theoretical aspects of finance and insurance.

However, it is surprising that, among thousands of reviews available on MAA Reviews, there are only twenty-some books listed under the heading of Actuarial Science. Of these, only six or so have been actually been reviewed, and not one of them pertains to the specialised field covered by this book. So, from that point of view alone, this book is likely to be greeted with some enthusiasm.

The brief summary of the book’s contents and purpose (on the rear cover) describes it as a mathematical introduction to non-life insurance, and it introduces the appropriate range of stochastic processes for this purpose. The subject of this analysis includes claim sizes, claim arrivals and total claim amounts. One of the central components of this analysis is also the most long-standing; namely, the Poisson process.

The book is organized into four parts, as follows:

  1. Collective Risk Models (four chapters, ending in one on Ruin Theory)
  2. Experience Rating (Bayes Estimation, Linear Bayes Estimation)
  3. Point Processes and Collective Risk Theory (General Poisson Process, Poisson Random Measure, Weak Convergence of Point Processes).
  4. Special Topics (Lévy Processes, Cluster Point Processes).

Not being a specialist in this field, I am nonetheless intrigued by the range and depth of mathematical ideas that are brought to bear on such aspects of risk theory. The book is based upon lectures given to 3rd year mathematics students, and it seems that a strong background in mathematical analysis and good knowledge of probability are the minimum pre-requisites.

Although the treatment is largely theoretical, and characterised by mathematical rigor, the theory is applied to many real world case studies. For instance, the arrival times for the Danish fire insurance claims for the decade 1980–1990 are analysed by means of the Poisson process, and the Pareto distribution is invoked for analysis of claim sizes. But I was also surprised to discover that the notion of Brownian motion is a particular example of the Levy process. Apparently, Brownian motion is among the simplest of the continuous-time stochastic processes - and not just the study of the movement of pollen grains suspended in water.

There are many exercises in this book, few of which could be classed as routine. In fact, the majority of them could serve as mini assignments, and their level of difficulty suggests that this book is suited to study at the advanced undergraduate level at the earliest.

Overall, this is a fascinating book on a subject I’d like to know more about.

In 1959, Peter Ruane worked as claims and renewal clerk for a large British motor insurance company. The theoretical basis of that work was confined to arithmetical calculations in £ s d.