## Optimal stopping of a risk process: Model with interest rates.(English)Zbl 1011.62111

The following problem in risk theory is considered. An insurance company, endowed with an initial capital $$a\geq 0$$, receives premiums and pays out claims that occur according to a renewal process $$\{N(t)$$, $$t\geq 0\}$$. The times between consecutive claims are i.i.d. The sequence of successive claims is a sequence of i.i.d. random variables. The capital of the company is invested at interest rate $$\alpha\in[0,1]$$, claims, due to inflation, increase at rate $$\beta\in[0,1]$$. The aim is, also for preventing bankruptcy, to find the stopping time that maximizes the expected capital of the company, or better some value of a bounded utility function.
A dynamic programming method is used to find the optimal stopping time. Outgoing from a fixed number of claims, it succeeded to treat the case of an infinite number of claims likewise.

### MSC:

 62P05 Applications of statistics to actuarial sciences and financial mathematics 60G40 Stopping times; optimal stopping problems; gambling theory 91B30 Risk theory, insurance (MSC2010) 60K10 Applications of renewal theory (reliability, demand theory, etc.) 62L15 Optimal stopping in statistics 90C39 Dynamic programming
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