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Erlangian approximations for finite-horizon ruin probabilities. (English) Zbl 1081.60028
The authors consider the probability \(\psi(u,T)\) of ruin before time \(T>0\) in the Cramér-Lundberg risk model, where \(u\) denotes the initial capital. F. Avram and M. Usabel [Insur. Math. Econ. 32, No. 3, 371–377 (2003; Zbl 1074.91026)] have shown that, if the individual claim amount distribution is of phase-type, and if \(T\) is an independent random variable with an exponential distribution, then \(\psi(u,T)\) can be evaluated via a matrix-exponential formula. In the present paper, an extension is given, when the distribution of \(T\) is of phase-type. It is shown, how the case of Erlang distributed \(T\) can be used to approximate \(\psi(u,T_0)\) for fixed \(T_0\).

MSC:
60G51 Processes with independent increments; Lévy processes
60K15 Markov renewal processes, semi-Markov processes
60K20 Applications of Markov renewal processes (reliability, queueing networks, etc.)
91B30 Risk theory, insurance (MSC2010)
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