We propose a new way to model the time evolution of human mortality for all life span and for both males and females through a multivariate stochastic differential equation (SDE) cross-sectional model that incorporates the effect of environmental variability.

In previous work, we have modeled the time evolution of the crude death rates (CDR) of males and females of a given age using a bi-dimensional stochastic Gompertz model with correlated Wiener processes. When observing the data, we noticed, however, that there are simultaneous mortality ups and downs, not only when comparing the time series by sex, but also when comparing them by age. This behavior suggests the application of a multidimensional model with a correlated noise structure.

Thus, assuming that each CDR, by age group and sex, follows a one dimensional geometric brownian motion and that the brownian motions of different age groups and sexes are correlated, we propose the use of a so-called multidimensional geometric brownian motion. This multivariate model, with an appropriate simple correlation structure with relatively few parameters, is certainly much simpler that cohort modles covering the full age spectrum and outperforms them in terms of CDR forecasting. Its forecasts also outperform those based on individual (uncorrelated) modeling of each series. We have applied it to the evolution of CDR of the Portuguese population. The parameters are estimated by maximum likelihood, using numerical methods.