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The Surplus Process As A Fair Game—Utilitywise

Published online by Cambridge University Press:  29 August 2014

Hans U. Gerber
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The concept of utility is twofold. One may think of utility:

1) as a tool to describe a “fair game”

2) as a quantity that ought to be maximized.

The first line of thought was initiated by Daniel Bernoulli in connection with the St. Petersburg Paradox. In recent decades, actuaries, economists, operations researchers and statisticians (this order is alphabetical) have been concerned mostly with optimization problems, which belong to the second category. Most of the actuarial models can be found in a paper by Borch [4] as well as in the texts by Beard, Pesonen and Pentikainen [3], Bühlmann [6], Seal [15], and Wolff [17].

We shall adopt the first variant and stipulate the existence of a utility function such that the surplus process of an insurance company is a fair game in terms of utility. This condition is naturally satisfied under the following procedure: a) a utility function is selected, possibly resulting from a compromise between an insurance company and supervising authorities, b) whenever the company makes a decision that affects the surplus, it should not affect the expected utility of the surplus.

Mathematically, this simply means that the utility of the surplus is a martingale. Therefore martingale theory (that was initiated by Doob) is the natural framework in which we shall study the model. We shall utilize one of the most powerful tools provided by this theory, the Martingale Convergence Theorem.

Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 8 , Issue 3 , September 1975 , pp. 307 - 322
Copyright
Copyright © International Actuarial Association 1975

References

[1]Andersen, E. S. (1957), “On the collective theory of risk in the case of contagion between the claims”. Trans. XV Intern. Cong. Actu., New York, 2, 219–227.Google Scholar
[2]Bauer, H. (1968), Wahrscheinlichkeitstheorie und Grundzüge der Masstheorie, De Gruyter, Berlin.CrossRefGoogle Scholar
[3]Beard, R. E., Pentikainen, T. and Pesonen, E.(1969), Risk Theory, Methuen, London.Google Scholar
[4]Borch, K.(1974), “Mathematical models in insurance”, The ASTIN Bull., 7, 192–202.CrossRefGoogle Scholar
[5]Breiman, L.(1968), Probability, Addison-Wesley, Reading, Mass.Google Scholar
[6]Buhlmann, H.(1970), Mathematical Methods in Risk Theory, Springer, Berlin.Google Scholar
[7]DeFinetti, B. (1940), “Il problema dei pieni”, Gior. Ist. Ital. Attu., 11, 1–88.Google Scholar
[8]Dubourdieu, J.(1952), Theorie Mathematique du Risque dans les Assurances de Repartition, Gauthier-Villars, Paris.Google Scholar
[9]Feller, W.(1966), An Introduction to Probability Theory and Its Applications, Volume 2, Wiley, New York.Google Scholar
[10]Ferra, C.(1964), “Considerazioni sulle funzioni di utilità in connessione con la teoria del rischio”, Gior. Ist. Ital. Attu., 27, 51–70.Google Scholar
[11]Gerber, H.(1973), “Martingales in Risk Theory”. Mitt. Ver. Schweiz. Vers. Math., 73, 205–216.Google Scholar
[12]Neveu, J. (1972), Martingales à Temps Discrets, Masson, Paris.Google Scholar
[13]Ottaviani, M.(1970), “Sulla probabilità di fallimento di una impresa di assicurazioni nel caso in cui il caricamento per il rischio vari in funzione dell'ammontare del fondo di garanzia”, Gior. Ist. Ital. Attu., 33, 59–70.Google Scholar
[14]Ottaviani, M.(1970), “Caricamenti per il rischio dipendenti dall'ammontare del fondo per il rischio e probabilità di fallimento”, Gior. Ist. Ital. Attu., 33 180–189.Google Scholar
[15]Seal, H. L. (1969), Stochastic Theory of a Risk Business, Wiley, New York.Google Scholar
[16]Segerdahl, C. O. (1959), “A survey of results in the collective theory of risk”, The Harald Cramer Volume, Almquist, Uppsala.Google Scholar
[17]Wolff, K. H. (1966), Methoden der Unternehmensforschung im Versicherungswesen, Springer, Berlin.CrossRefGoogle Scholar