Exploring nonlinear chaotic systems with applications in stochastic processes

This manuscript explores the stability theory of several stochastic/random models. It delves into analyzing the stability of equilibrium states in systems influenced by standard Brownian motion and exhibit random variable coefficients. By constructing appropriate Lyapunov functions, various types of stability are identified, each associated with distinct stability conditions. The manuscript establishes the necessary criteria for asymptotic mean-square stability, stability in probability, and stochastic global exponential stability for the equilibrium points within these models. Building upon this comprehensive stability investigation, the manuscript delves into two distinct fields. Firstly, it examines the dynamics of HIV/AIDS disease persistence, particularly emphasizing the stochastic global exponential stability of the endemic equilibrium point denoted as , where the underlying basic reproductive number is greater than one ( ). Secondly, the paper shifts its focus to finance, deriving sufficient conditions for both the stochastic market model and the random Ornstein–Uhlenbeck model. To enhance the validity of the theoretical findings, a series of numerical examples showcasing stability regions, alongside computer simulations that provide practical insights into the discussed concepts are provided.