This article proposes a continuous time mortality model based on calendar years. Mortality rates belong to a mean-reverting random field indexed by time and age. In order to explain the improvement of life expectancies, the reversion level of mortality rates is the product of a deterministic function of age and of a decreasing jump-diffusion process driving the evolution of longevity. We provide a general closed-form expression for survival probabilities and develop it when the mean reversion level of mortality rates is proportional to a Gompertz–Makeham law. We develop an econometric estimation method and validate the model on the Belgian population.

A plantation is modelled in terms of discs of random size centred on the points of a two-dimensional lattice, together with rules for partitioning the union of these discs. The resulting sets are functions of random sequences, and some properties of these functions are deduced. Results are given which relate various properties of measurements on a plantation to measurements of individual plants and of plants in isolation. These include limit theorems for complete plantations and for samples. They also lead to a rigorous demonstration of certain well-known empirical relations between typical measurements and survival frequencies and to some new relations which are amenable to test. The model is a formulation of computer simulations which have had success in describing competition in plantations, but are too costly for routine forest management.

This paper deals with the computation of survival probabilities and extinction times for multitype positively regular branching processes. If all of the generating functions of the offspring distributions are of the linear fractional form and have the same denominator, explicit expressions may be obtained for all of their iterates. It is then possible to obtain formulae for survival probabilities and bounds on the mean time to extinction, given extinction, of a line descended from a single individual. If there are two types and the offspring distributions are bivariate Poisson, their generating functions may be bounded by linear fractional generating functions. It is then possible to compute upper and lower bounds on mean times to extinction, given extinction, and this is done for some special cases.