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A unified analysis of claim costs up to ruin in a Markovian arrival risk model. (English) Zbl 1284.91214
Summary: An insurance risk model where claims follow a Markovian arrival process (MArP) is considered in this paper. It is shown that the expected present value of total operating costs up to default \(H\), as a generalization of the classical Gerber-Shiu function, contains more non-trivial quantities than those covered in [J. Cai et al., Adv. Appl. Probab. 41, No. 2, 495–522 (2009; Zbl 1173.91023)], such as all moments of the discounted claim costs until ruin. However, it does not appear that the Gerber-Shiu function \(\phi\) with a generalized penalty function which additionally depends on the surplus level immediately after the second last claim before ruin [E. C. K. Cheung et al., Scand. Actuar. J. 2010, No. 3, 185–199 (2010; Zbl 1226.60123)] is contained in \(H\). This motivates us to investigate an even more general function \(Z\) from which both \(H\) and \(\phi\) can be retrieved as special cases. Using a matrix version of Dickson-Hipp operator [R. Feng, “A matrix operator approach to the analysis of ruin-related quantities in the phase-type renewal risk model”, Mitt., Schweiz. Aktuarver. 2009, No. 1–2, 71–87 (2009), http://math.uiuc.edu/~rfeng/BSAA2009.pdf], it is shown that \(Z\) satisfies a Markov renewal equation and hence admits a general solution. Applications to other related problems such as the matrix scale function, the minimum and maximum surplus levels before ruin are given as well.

MSC:
91B30 Risk theory, insurance (MSC2010)
60K10 Applications of renewal theory (reliability, demand theory, etc.)
60J28 Applications of continuous-time Markov processes on discrete state spaces
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