First and Last Passage Times of Spectrally Positive L,vy Processes with Application to Reliability

We consider a wide class of increasing L,vy processes perturbed by an independent Brownian motion as a degradation model. Such family contains almost all classical degradation models considered in the literature. Classically failure time associated to such model is defined as the hitting time or the first-passage time of a fixed level. Since sample paths are not in general increasing, we consider also the last-passage time as the failure time following a recent work by Barker and Newby (Reliab Eng Syst Saf 94:33-43, 2009). We address here the problem of determining the distribution of the first-passage time and of the last-passage time. In the last section we consider a maintenance policy for such models.