This paper concerns an insurance firm’s surplus process observed at renewal inspection times, with a focus on assessing the probability of the surplus level dropping below zero. For various types of inter-inspection time distributions, an explicit expression for the corresponding transform is given. In addition, Cramér–Lundberg-type asymptotics are established. Also, an importance sampling-based Monte Carlo algorithm is proposed, and is shown to be logarithmically efficient.

In this paper, the instantaneous availability of a system maintained under periodic inspection is investigated using random walk models. Two cases are considered. In the first model, the system is repaired or modified and it is assumed to be as good as new upon periodic inspection and maintenance. In the second model, the system is not modified after the inspection if the system is still working, and the condition of the system is assumed to be the same as that before the inspection. For both models the failures only can be found through the inspection. Perfect repair or replacement of a failed system is assumed to be carried out, but the time it takes can be constant or of a random length. The relationship between this problem and the random walk model in a two-dimensional plane is described. Several new results are also shown.