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Gerber-Shiu analysis with two-sided acceptable levels. (English) Zbl 1364.91071
Summary: In this paper, insurer’s surplus process moved within upper and lower levels is analyzed. To this end, a truncated type of Gerber-Shiu function is proposed by further incorporating the minimum and the maximum surplus before ruin into the existing ones (e.g. [H. U. Gerber and E. S. W. Shiu, N. Am. Actuar. J. 2, No. 1, 48–78 (1998; Zbl 1081.60550); E. C. K. Cheung et al., Insur. Math. Econ. 46, No. 1, 117–126 (2010; Zbl 1231.91157)]). A key component in our analysis of this proposed Gerber-Shiu function is the so-called transition kernel. Explicit expressions of the transition function under two different risk models are obtained. These two models are both generalizations of the classical Poisson risk model: (i) the first model provides flexibility in the net premium rate which is dependent on the surplus (such as linear or step function); and (ii) the second model assumes that claims arrive according to a Markovian arrival process (MAP). Finally, we discuss some applications of the truncated Gerber-Shiu function with numerical examples under various scenarios.
##### MSC:
 91B30 Risk theory, insurance (MSC2010) 60K10 Applications of renewal theory (reliability, demand theory, etc.) 60K20 Applications of Markov renewal processes (reliability, queueing networks, etc.)
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