The process of ageing in humans is highly relevant across many scientific fields, including actuarial science. The pattern of mortality over age and its trend over time represent a key concern for pension funds and insurance companies, that have to manage their longevity risk exposure. The Gompertz mortality law is a widely used and convenient mathematical formula for describing how mortality evolves over age. Its peculiar feature is the linear relationship between the logged hazard rate and the age, being consistently verified across most countries around the world with respect to adult ages. Projections over long and very long time horizons are challenging, due to the increasing riskiness as the forecasting horizon goes forward and thus as the upper end of the population age distribution is concerned, e.g., in the pool of annuitants and pensioners. A consistent stream of literature has demonstrated the theoretical suitability and the empirical appropriateness of affine diffusion processes for modelling mortality intensities, also providing stochastic extensions of the Gompertz mortality law. In our paper, we represent the stochastic time-dynamics of logged central death rates, for any fixed age, by means of a Ornstein-Uhlenbeck process (OU) whose drift coefficient involves the movement of the process towards a Gompertz-type long-term mean. The main focus concerns the expectations that we can form on the asymptotic behaviour of central death rates and on the age distribution of these expectations.
Dealing with asymptotic distributions enables us to understand whether expected central death rates exhibit an expontial increase with age in the long run. A cross-country data-driven analysis, framed in the devised theoretical context, provides insights into the systematicity of the Gompertz long run mean reversion.
