Distribution of the first ladder height of a stationary risk process perturbed by α-stable Lévy motion

We consider a risk model described by an ergodic stationary marked point process. The model is perturbed by a Lévy process with no downward jumps. The (modified) ladder height is defined as the first epoch where an event of the marked point process leads to a new maximum. Properties of the process until the first ladder height are studied and results of Dufresne and Gerber [Insurance: Math. Econ. 10 (1991) 51], Furrer [Scand. Actuarial J. (1998) 59], Asmussen and Schmidt [Stochastic Process. Appl. 58 (1995) 105] and Asmussen et al. [ASTIN Bull. 25 (1995) 49] are generalized.