A Cox risk process with a piecewise constant intensity is considered where the sequence (Li) of successive levels of the intensity forms a Markov chain. The duration σi of the level Li is assumed to be only dependent via Li. In the small-claim case a Lundberg inequality is obtained via a martingale approach. It is shown furthermore by a Lundberg bound from below that the resulting adjustment coefficient gives the best possible exponential bound for the ruin probability. In the case where the stationary distribution of Li contains a discrete component, a Cramér–Lundberg approximation can be obtained. By way of example we consider the independent jump intensity model (Björk and Grandell 1988) and the risk model in a Markovian environment (Asmussen 1989).

The Wright–Fisher model with varying population size is examined in the case where the selective advantage varies from generation to generation. Models are considered where the selective advantage is not always in favour of a particular genotype. Sufficient conditions in terms of the selection coefficients and the population growth are given to ensure ultimate homozygosity.

Given an indexed family of rate functions, we construct a family of independent random variables such that their minimum and the corresponding index form a marked point having the same rates. If the rates have common discontinuities, the random variables are conditionally independent, given a random allocation of these jumps. Between any two adjacent points of a marked point process there is a similar structure. Coherent functions in system lifetime theory provide examples.

Non-identifiability of multivariate lifetime distributions from mortality data is interpreted in this context.