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.


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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