Some results on degradation processes with repairs and catastrophes

A classical assumption in the theory of degradation models is that the quality of a device is described by a non-negative diffusion process {X(t), t ≥ 0} starting at X(0) = x0 > 0. Assuming that large values of X(t) correspond to a better quality, the level X(t) = 0 stands for zero quality, i.e. for the lack of functionality of the device. Consequently, the first-passage time T of {X(t)} through state zero gives the failure time of the device. We shall denote its probability density function (p.d.f.) by