Stability analysis of stochastic systems with fractional dynamics

This paper investigates the stability properties of stochastic functional differential equations driven by fractional Brownian motion (FBM), a natural extension of classical Brownian motion that incorporates memory effects through the Hurst parameter. By constructing appropriate Lyapunov functionals, we derive sufficient conditions for global exponential mean square stability and asymptotic stochastic stability. The analysis is applied to a delayed stochastic fractional Black–Scholes model, where explicit criteria for both stability types are established. Numerical simulations are presented to validate the theoretical results and illustrate the influence of the Hurst index on the stability behavior of the system.