Existence and uniqueness of solutions for stochastic urban-population growth model

Urban-population growth model has attracted attention over the last few decades due to its usefulness in representing population dynamics, virus dynamics, and epidemics. Researchers have included stochastic perturbation in the urban-population growth model to improve the model, attempting to capture the random nature of real-time dynamics. When doing so, researchers have presented conditions to ensure that the corresponding stochastic solution is both positive and unique (in probability). This paper advances that knowledge by showing that the stochastic diffusion constant can be both positive and negative—previous results in the literature have required that such a constant be positive only. A numerical simulation illustrates the paper's findings.