Exit Problems for Reflected Markov-additive Processes with Phase–type Jumps

Let (X ,J ) denote a Markov-additive process with phase–type jumps (PH-MAP) and denote its supremum process by S. For some a > 0, let τ(a) denote the time when the reflected processY := S−X first surpasses the level a. Further, let τ−(a) denote the last time before τ(a) when X attains its current supremum. In this paper we shall derive the joint distribution of Sτ(a), τ−(a) and τ(a), where the latter two shall be given in terms of their Laplace transforms. Furthermore, we define scale functions for PH-MAPs and remark on some of their properties. This extends recent results for spectrally negative Lévy processes to the (dense) class of PH-MAPs. The result is of interest to applications such as the dividend problem in insurance mathematics and the buffer overflow problem in queueing theory. Examples will be given for the former.