## Optimal proportional reinsurance policies in a dynamic setting.(English)Zbl 0971.91039

A dynamic proportional reinsurance strategy $$b_t$$ is considered, where $$b_t$$ is the proportion of the portfolio which the company is willing to keep at time $$t$$. (So $$1-b_t$$ is the reinsured part of the portfolio). The aim in choosing $$b_t$$ is to minimize the ruin probability. It is shown that for the diffusion model the optimal $$b_t$$ is constant. Under the Cramér-Lundberg model a functional equation (containing integrals, derivatives and infimum) is derived which connects the optimal strategy and the survival probability. Examples of numerical solution are presented for exponentially and Pareto distributed claim sizes.

### MSC:

 91B30 Risk theory, insurance (MSC2010) 93E20 Optimal stochastic control
Full Text:

### References:

 [1] Dassios A., Comm. Statist. Stochastic Models 5 pp 181– (1989) · Zbl 0676.62083 [2] Davis M. H. A., J. Roy. Statist. Soc. Ser. B 46 pp 353– (1984) [3] Embrechts P., Insurance Math. Econom. 1 pp 55– (1982) · Zbl 0518.62083 [4] Ethier S. N., Markov processes (1986) [5] Grandell J., Stochastic Process. Appl. 8 pp 243– (1979) · Zbl 0401.62079 [6] Grandell J., Aspects of risk theory (1991) · Zbl 0717.62100 [7] Højgaard B., Scand. Actuarial J. pp 166– (1998) · Zbl 1075.91559 [8] Iglehart D. L., J. Appl. Probab. 6 pp 285– (1969) · Zbl 0191.51202 [9] Rolski T., Stochastic processes for insurance and finance (1999) · Zbl 0940.60005 [10] Schmidli H., Comm. Statist. Stochastic Models 10 pp 365– (1994) · Zbl 0793.60095 [11] Schnieper R., Schweiz. Verein. Versicherwigsmath. Mitt. 90 pp 129– (1990) [12] Siegmund D., Adv. in Appl. Probab. 11 pp 701– (1979) · Zbl 0422.60053 [13] Waters H. R., Insurance Math. Econom. 2 pp 17– (1983) · Zbl 0505.62085
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.