  We consider in this paper a risk reserve process where the claims and gains
arrive according to two independent Poisson processes. While the gain sizes are
phase-type distributed, we assume instead that the claim sizes are phase-type
perturbed by a heavy-tailed component; that is, the claim size distribution is
formally chosen to be phase-type with large probability $1-\epsilon$ and
heavy-tailed with small probability $\epsilon$. We analyze the seminal
Gerber-Shiu function coding the joint distribution of the time to ruin, the
surplus immediately before ruin, and the deficit at ruin. We derive its value
as an expansion with respect to powers of $\epsilon$ with known coefficients
and we construct approximations from the first two terms of the aforementioned
series. The main idea is based on the so-called fluid embedding that allows to
put the considered risk process into the framework of spectrally negative
Markov-additive processes and use its fluctuation theory developed in Ivanovs
and Palmowski (2012).
