Special lognormal diffusion processes with binomial random catastrophes and applications to economic data

This work introduces and analyzes lognormal diffusion processes subject to random catastrophes, i.e. random events which cause jumps and reset the process to a possibly different random state. The model incorporates a binomial distribution for the restarting points and employs stochastic techniques to describe cycles between catastrophes. A maximum likelihood approach is developed to estimate the model parameters and it is applied both to simulated and real data. More specifically, we perform a simulation study based on 50 replications and 500 sample paths, both in the case in which the size of the binomial distribution is known and in the case in which it is unknown. Moreover, we provide a real application to GDP (gross domestic product) trajectories of five European countries affected by the economic crises of 2009 and 2020. The analysis demonstrates the model's effectiveness in mimicking complex phenomena characterized by growth dynamics interrupted by random external events.