A unified analysis of claim costs up to ruin in a Markovian arrival risk model

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 Cai etal. (2009), such as all moments of the discounted claim costs until ruin. However, it does not appear that the Gerber-Shiu function φ with a generalized penalty function which additionally depends on the surplus level immediately after the second last claim before ruin (Cheung etal., 2010a) is contained in H. This motivates us to investigate an even more general function Z from which both H and φ can be retrieved as special cases. Using a matrix version of Dickson-Hipp operator (Feng, 2009b), 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.