Stochastic growth models and sigmoidal processes are crucial due to their ability to describe phenomena commonly observed in nature. These models are particularly relevant in fields such as medicine and biology, where they are used to represent the spread of diseases, immune responses, and the growth of cellular populations. However, they also have significant applications in finance and physics (see, for example, [2]). This work (cf. [1]) focuses on the lognormal diffusion process subject to random catastrophes, random events which cause jumps and reset the process to a possibly different random state (cf. [3]). The primary contribution of this research is the assumption that the post-catastrophe recovery level follows a binomial distribution. Unlike traditional models where a system might revert to a fixed initial size, our approach allows the population to restart at a random level which reflects a certain survival probability for each element of the population. Furthermore, the model effectively captures real-world economic scenarios, such as the trajectories of GDP (Gross Domestic Product) in five European countries impacted by the crises of 2009 and 2020. The findings show that the model can realistically reproduce complex trajectories, displaying periods of gradual growth interspersed with sudden declines triggered by unpredictable external shocks.