Gerber–Shiu functionals for classical risk processes perturbed by an α-stable motion

We study the Gerber–Shiu functional of the classical risk process perturbed by a spectrally negative α-stable motion. We provide representations of the scale functions of the process as an infinite series of convolutions of given functions. This, together with a result from Biffis and Kyprianou (2010), allows us to obtain a representation of the Gerber–Shiu functional as an infinite series of convolutions. Moreover, we calculate the Laplace transform and derive a defective renewal equation for the Gerber–Shiu functional, thus extending previous work of Furrer (1998) and of Tsai and Willmot (2002). We also obtain asymptotic expressions for the joint tail distribution of the severity of ruin and the surplus before ruin.