## 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)

Zbl 1074.91026
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