Double optimal stopping of a risk process

Abstract

The following problem in risk theory is considered. An insurance company, endowed with an initial capital a>0, receives insurance premiums and pays out successive claims. The losses occur according to renewal process. At any moment, the company may broaden or narrow down the offer, what entails the change of the parameters. This change concerns the rate of income, the intensity of renewal process and the distribution of claims. After the change, the management wants to know the moment of the maximal value of the capital assets. Therefore, our goal is finding two optimal stopping times: the best moment of change the parameters and the moment of maximal value of the capital assets. We will use a dynamic programming method to calculate the expected capital at that times.

Keywords:

• Risk reserve process
• Optimal stopping
• Dynamic programming
• Capital assets
• Semi-Markov process
• Infinitesimal generator

Mathematics Subject Classification 2000::

• Primary 60G40; 60K99
• Secondary 90A46

Acknowledgements

The authors are grateful to an anonymous referee for pointing out errors in an earlier draft and for several suggestions that improved the presentation of this paper.

Notes

^† [email protected]

Additional information

Notes on contributors

Anna Karpowicz

† † [email protected]