In this paper, we study a new one-dimensional homogeneous stochastic process, termed the Square of the Brennan-Schwartz model, which is used in various contexts. We first establish the probabilistic characteristics of the model, such as the analytical expression solution to Ito’s stochastic differential equation, after which we determine the trend functions (conditional and non-conditional) and the likelihood approach in order to estimate the parameters in the drift. Then, in the diffusion coefficient, we consider the problem of parameter estimation, doing so by a numerical approximation. Finally, we present an application to population growth by the use of real data, namely the growth of the total population aged 65 and over, resident in the Arab Maghreb, to illustrate the research methodology presented.