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

Muciek, Bogdan Krzysztof

J. Appl. Probab. 39, No. 2, 261-270 (2002).

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.

Reviewer: Peter Neumann (Dresden)

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

Keywords:

risk reserve process; optimal stopping; dynamic programming; interest rates