Counting processes and their compensators are introduced at a heuristic level. The martingale property of stochastic integrals with respect to a compensated counting process leads to moment estimates and asymptotic normal distributions for statistics arising in multiple state, non-parametric and semi-parametric models. The place of survival models in actuarial education is discussed.

Stochastic comparison results for replacement policies are shown in this paper using the formalism of point processes theory. At each failure moment a repair is allowed. It is performed with a random degree of repair including (as special cases) perfect, minimal and imperfect repair models. Results for such repairable systems with schemes of planned replacements are also shown. The results are obtained by coupling methods for point processes.

First, some basic concepts from the theory of point processes are recalled and expanded. Then some notions of stochastic comparisons, which compare whole processes, are introduced. The use of these notions is illustrated by stochastically comparing renewal and related processes. Finally, applications of the different notions of stochastic ordering of point processes to many replacement policies are given.

We construct a risk process, where the law of the next jump time or jump size can depend on the past through earlier jump times and jump sizes. Some distributional properties of this process are established. The compensator is found and some martingale properties are discussed.