Mr. Chairman, Honourable Colleagues!

Let me thank the organisers of this symposium for the honour they have done me by inviting me to present my report before this eminent audience. I would also like to express my thanks to all who are present here for the confidence shown to me as rapporteur.

The genesis and development of the theory of random processes is closely related to the problems of demography and insurance. Numerous problems stimulating the development of the theory of probability and the theory of random processes—the latter constituting one of the essential parts of the modern theory of probability—are connected with these two fields of research and activity.

The theory of random processes, being stimulated—similarly to other fields of mathematics—by concrete practical tasks, has created the general ideas and methods of research, achieving many valuable results. With the passage of time, it appeared that the theory of random processes can be usefully applied, by its conceptions, results and methods, to a number of scientific fields and practical activity. Its conceptions have found wide and important application to the earlier crystallized fields of human knowledge, as well as to the newly appearing ones.

These new fields of mathematics serve us as an example which once more corroborate the thesis, well known to us, according to which the course of the development of human knowledge starts from empirical observations, then passes up to abstract thinking and finally to practice.

1. Le fait que le 8-e Colloque ASTIN a lieu en Pologne donne une bonne occasion pour informer les membres de l'ASTIN sur l'organisation actuelle des assurances dans ce pays qui est d'ailleurs pareille à celle des assurances dans les autres pays socialistes. Dans le présent rapport sera présenté en particulier le problème des assurances du patrimoine national (propriété de l'État) qui comprennent aussi les risques d'assurance importants et complexes.

2. Après la II-e Guerre Mondiale la Pologne a connu de profonds changements sociaux et économiques. La grande et moyenne industries ont été nationalisées. On a réalisé la réforme agraire. Le transport et l'échange des marchandises ont été confiés à des entreprises d'Etat et coopératives. Tout le système financier, y compris les banques et les assurances, a été nationalisé, etc. Actuellement en Pologne les activités d'assurance sont effectuées par deux établissements.

Panstwowy Zakład Ubezpieczeń — en abréviation PZU — (Assurance Nationale Polonaise);

Towarzystwo Ubezpieczeń i Reasekuracji „WARTA” — en abréviation „WARTA” (Compagnie d'Assurances et de Réassurances SA „WARTA”).

3. La PZU ne s'occupe que des assurances du pays, c.á.d. des assurances des objets situés en Pologne et des personnes domiciliées en Pologne.

La „WARTA” s'occupe de toutes assurances extérieures liées aux paiements en devises (contractées pour la plupart en monnaie étrangère), ainsi done avant tout des assurances des marchandises dans le transport maritime, des assurances des moyens de transport maritime et fluvial, des assurances d'aviation, des assurances des véhicules automobiles etc.

In several different countries there has lately been a very unfavourable development as far as fire losses are concerned. There have been many and very bad big fires which have especially affected industry and trade. The facts which can be considered to have caused this development are well-known. Bigger and bigger and more concentrated units of production, business enterprises, and storing form an accumulation of values which in itself implies increasing risks. On account of production-technical and business-economical conditions it has not always been possible to divide the buildings into sections desirable from a fire service point of view. The new production methods have sometimes caused new fire risks.

The Northern Central Office for Fire Insurance Statistics from Mutual Companies (CNÖB) has made an investigation in order to see to what extent the international trend towards increasing fire losses has any equivalence in the Northern countries (Denmark, Finland, Norway, and Sweden). I.a. there was made a graphical representation of how the observed rate of net risk premium of the portfolio has developed during the period 1946–1966, the result of which may be seen from fig. 1. No increasing trend can be observed from this. After having been decreasing during the immediate postwar time, the observed rate of net risk premium has been fairly constant from the beginning of the fifties, perhaps even somewhat decreasing except during the later years. The conclusion is that in so far facts as mentioned above have influenced the development of claims of the portfolios or parts of these in a deteriorating way, there must in that case have been circumstances which have worked as neutralizers.

On peut penser que la dimension des sinistres est la résultante d'un très grand nombre de causes indépendantes à effets positifs.

Soient

les variables aléatoires indépendantes qui représentent les facteurs élémentaires exerçant leur action dans l'ordre indiqué par les indices.

Si Xν est l'ampleur du sinistre dû aux facteurs ξ1, ξ2, …., ξν nous pouvons supposer que l'augmentation causée par ξν+1 soit proportionelle à ξν+1 et à une certaine fonction, g(Xν) de la dimension Xν.

C'est à dire, nous supposons qu'il y ait les relations:

Si chaque facteur n'apporte qu'un faible contribut à l'augmentation du sinistre, nous pouvons alors poser approximativement aussi:

Comme X varie de Xo à Xn (et que X = Xn désigne la dimension extrème du sinistre) le second membre de la (2) décrit une variable aléatoire, fonction de X.

Puisque par hypothèse ξ1,, ξ2, …., ξn sont des variables aléatoires indépendante, si n est suffisament grand, d'après le théorème de la limite des probabilités il s'insuit que la variable aléatoire au second membre de la (2), est, à la limite, distribuée suivant la loi normale. Si l'on pose g(t) = t − c, la variable aléatoire normale qui décrit la dimension du sinistre pourra être posée dans la forme In(X − c).

We are using the following terminology—essentially following Feller:

a) Compound Poisson Variable

This is a random variable

where X1, X2, … Xn, … independent, identically distributed (X0 = o) and N a Poisson counting variable

hence

(common) distribution function of the Xj with j ≠ 0 or in the language of characteristic functions

b) Weighted Compound Poisson Variable

This is a random variable Z obtained from a class of Compound Poisson Variables by weighting over λ with a weight function S(λ)

hence

or in the language of characteristic functions

Let [Z(t); t ≥ o] be a homogeneous Weighted Compound Poisson Process. The characteristic function at the time epoch t reads then

It is most remarkable that in many instances φt(u) can be represented as a (non weighted) Compound Poisson Variable. Our main result is given as a theorem.

As is well-known, in the early 60's a Swedish committee set to work at the numerical calculation of the distribution function of the total amount of claims and of the related stop loss premiums in the Poisson and Polya cases (Bohman and Esscher [6]). Since the characteristic function for the said distribution function was easily available in terms of the characteristic function for the distribution function of an individual claim, the committee chose to base the numerical calculations on the C-method by H. Bohman (Bohman [5]). The calculation of the ruin probability for a finite or infinite period was not considered by the committee.

The last-mentioned problem has now been taken up by a new committee formed by the Swedish Council for Actuarial Science and Insurance Statistics. The committee—consisting of H. Bohman, J. Jung, N. Wikstad and the present author—has to consider several aspects of the practical applicability of the collective risk theory. However, without possibilities of calculating—at least approximately—the ruin probability for a finite period the applicability of the existing ruin theories seems to be rather limited, so the committee has looked around for such possibilities. At the present stage the committee is considering the classical Poisson theory and Sparre Andersen's generalization of this theory [2]. It is the hope of the committee that, at a later stage, also the Polya theory and the theory recently presented by Segerdahl [11] combining the Sparre Andersen theory and the Polya theory may be treated.

It is fantastic how the computer has changed our attitude to numerical problems. In the old days when our numerical tools were paper, pencil, desk calculator and logarithm tables we had to stay away from formulas and methods which led to too lengthy calculations. A consequence is that we have a tendency to think of numerical analysis in terms of the classical tools. If we go back to the results of earlier writers it seems, however, very likely that many results and formulas developed by them which had earlier a theoretical interest only could nowadays be applied successfully in numerical analysis.

As an example I take the ruin probability ψ(x). The Laplace transform of ψ(x) is given by the following expression

where c > 1. In fact (c — 1) is equal to the “security loading”. The function p(y) is equal to the Laplace transform of the claim distribution. We assume that the mean claim amount is equal to one, i.e. p′(0) = — 1.

In his book from 1955 [1] Cramer points out that this formula will be more easy to handle if the claim distribution is an exponential polynomial. In this case we have

where

Cramér's results are given on pages 81-83 in his book. We reproduce them here with a slight change of notations only.