Gerber-Shiu analysis with two-sided acceptable levels

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. Gerber and Shiu (1998), Cheung et al. (2010a)). 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.