# zbMATH — the first resource for mathematics

Optimal dynamic premium control in non-life insurance. Maximizing dividend pay-outs. (English) Zbl 1039.91042
Let the controlled reserve $$\{R^{\beta,b}_{t}\}$$ of a non-life insurance company be given by $R^{\beta,b}_{t}= x+\int_{0}^{t}f(\beta(s))\,ds- \sum_{i=1}^{N^{\beta}_{t}} U_{i}-\int_{0}^{t}\,dL^{\beta,b}(s),\text{ where } L^{\beta,b}(t)= (x-b)^{+}+ \int_{0}^{t}f(\beta(s)) I(R^{\beta,b}_{s}=b)\,ds;$ $$f(\beta)=(1+h(\beta))\beta\mu$$; $$x\in R$$ is the initial reserve; $$\{N_{t}^{\beta}\}$$ is a Poisson process with intensity $$\beta$$; the random variables $$U_{i},\;i=1,2,\ldots$$ are i.i.d. with mean $$\mu$$ claim sizes independent on $$N_{t}$$. The company distributes dividends to shareholders according to a barrier strategy. The objective for the company is to find an optimal intensity process $$\{\beta^{*}(t)\}$$ and a barrier $$b^{*}$$ which maximizes the expected discounted future pay-out of dividend until the time of ruin. In this paper it is shown how to use a solution of the Bellman equation to construct the optimal premium policy and the optimal barrier. For exponential claim size distributions the closed form solution to the Bellman equation is found. For general claim size distributions the solution and optimal controls can be approximated numerically by a dynamic programming approach. The author also investigates the possibilities of the De Vylder approximation of the optimal control in the general case by using the closed form solution of an approximating problem with exponential claim size distributions.

##### MSC:
 91B30 Risk theory, insurance (MSC2010)
Full Text:
##### References:
  Asmussen S., Finance & Stochastics 4 (3) pp 299– (2000) · Zbl 0958.91026 · doi:10.1007/s007800050075  Bühlmann H., Mathematical methods in risk theory (1970) · Zbl 0209.23302  Buzzi R., Optimale Dividendestrategien für den Risikoprozess mit austauschbaren Zuwächsen (1974)  De Finetti B., Transactions of the 15th international Congress of Actuaries, New York 23 pp 433– (1957)  De Vyider F., Scand. Act. J. 1978 pp 114– (1978) · doi:10.1080/03461238.1978.10419484  Fleming W., Deterministic and stochastic optimal control (1975) · Zbl 0323.49001 · doi:10.1007/978-1-4612-6380-7  Gerber H., Mitt. Ver. Scwiez. Versich. 69 pp 185– (1969)  Gerber H., S.S. Huebner Foundation Monographs, in: An introduction to mathematical risk theory (1979)  Grandell J., Aspects of risk theory (1990) · Zbl 0717.62100  Hipp C., Insurance: Mathematics and Economics 27 pp 215– (2000)  Hipp C., Insurance: Mathematics and Economics 26 pp 185– (2000)  Højgaard B., Scand. Act. J. 1998 (2) pp 166– (1998) · Zbl 1075.91559 · doi:10.1080/03461238.1998.10414000  Højgaard B., Insurance: Mathematics and Economics 22 pp 41– (1998)  Højgaard B., Mathematical Finance 9 (2) pp 153– (1999) · Zbl 0999.91052 · doi:10.1111/1467-9965.00066  Paulsen J., Insurance: Mathematics and Economics 20 pp 215– (1997)  Schmidli H., Scand. Act. J. 2001 (1) pp 51– (2001) · Zbl 0971.91039 · doi:10.1080/034612301750077338
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.