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We consider in this paper a risk reserve process where the claims and gainsarrive according to two independent Poisson processes. While the gain sizes arephase-type distributed, we assume instead that the claim sizes are phase-typeperturbed by a heavy-tailed component; that is, the claim size distribution isformally chosen to be phase-type with large probability $1-\epsilon$ andheavy-tailed with small probability $\epsilon$. We analyze the seminalGerber-Shiu function coding the joint distribution of the time to ruin, thesurplus immediately before ruin, and the deficit at ruin. We derive its valueas an expansion with respect to powers of $\epsilon$ with known coefficientsand we construct approximations from the first two terms of the aforementionedseries. The main idea is based on the so-called fluid embedding that allows toput the considered risk process into the framework of spectrally negativeMarkov-additive processes and use its fluctuation theory developed in Ivanovsand Palmowski (2012).

Phase-type approximations perturbed by a heavy-tailed component for the Gerber-Shiu function of risk processes with two-sided jumps