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Analysis of a generalized penalty function in a semi-Markovian risk model. (English) Zbl 1483.91182

Summary: In this paper an extension of the semi-Markovian risk model studied by H. Albrecher and O. J. Boxma [Insur. Math. Econ. 37, No. 3, 650–672 (2005; Zbl 1129.91023)] is considered by allowing for general interclaim times. In such a model, we follow the ideas of E. C. K. Cheung et al. [Insur. Math. Econ. 46, No. 1, 117–126 (2010; Zbl 1231.91157)] and consider a generalization of the Gerber-Shiu function by incorporating two more random variables in the traditional penalty function, namely, the minimum surplus level before ruin and the surplus level immediately after the second last claim prior to ruin. It is shown that the generalized Gerber-Shiu function satisfies a matrix defective renewal equation. Detailed examples are also considered when either the interclaim times or the claim sizes are exponentially distributed. Finally, we also consider the case where the claim arrival process follows a Markovian arrival process. Probabilistic arguments are used to derive the discounted joint distribution of four random variables of interest in this risk model by capitalizing on an existing connection with a particular fluid flow process.

MSC:

91G05 Actuarial mathematics
60K10 Applications of renewal theory (reliability, demand theory, etc.)
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