On the expected discounted penalty function for a perturbed risk process driven by a subordinator

The Expected Discounted Penalty Function (EDPF) was introduced in a series of now classical papers [Gerber, H.U., Shiu, E.S.W., 1997. The joint distribution of the time of ruin, the surplus immediately before ruin, and the deficit at ruin. Insurance: Math. Econ. 21, 129–137; Gerber, H.U., Shiu, E.S.W., 1998a. On the time value of ruin. N. Am. Actuar. J. 2 (1), 48–78; Gerber, H.U., Shiu, E.S.W., 1998b. Pricing perpetual options for jump processes. N. Am. Actuar. J. 2 (3), 101–112]. Later, Gerber and Landry [Gerber, H.U., Landry, B., 1998. On a discounted penalty at ruin in a jump–diffusion and the perpetual put option. Insurance: Math. Econ. 22, 263–276] extended the concept to the perturbed case. Recent papers have extended these results in more general settings [for instance Tsai and Willmot [Tsai, C.C.L., Willmot, G.E., 2002. A generalized defective renewal equation for the surplus process perturbed by diffusion. Insurance: Math. Econ. 30, 51–66], Li and Lu [Li, S., Lu, Y., 2005. On the expected discounted penalty function for two classes of risk processes. Insurance: Math. Econ. 36, 179–193] and Li and Garrido [Li, S., Garrido, J., 2005. On the Gerber–Shiu functions in a Sparre–Andersen risk model perturbed by diffusion. Scand. Actuar. J. 161–186]]. In this note we present yet another generalization that has not been considered before in the literature. We present a perturbed risk process with a subordinator as the model for the aggregate claims. We generalize existing results [Tsai, C.C.L., Willmot, G.E., 2002. A generalized defective renewal equation for the surplus process perturbed by diffusion. Insurance: Math. Econ. 30, 51–66] on the EDPF for the subordinator case.