On the Global Stability of a Stochastic Logistic-Type Equation in x 0 > K Regime

We investigate a stochastic logistic-type differential equation with multiplicative noise acting on the intrinsic growth rate. While this equation has been widely studied as a prototypical nonlinear stochastic system, existing analytical results have largely focused on initial conditions below the equilibrium level. The asymptotic behavior for super-equilibrium initial data has remained insufficiently understood. In this work, we establish almost sure convergence to the equilibrium for all initial conditions satisfying x 0 > K. Using a Lyapunov-based approach, we prove that trajectories starting above the carrying capacity converge to the equilibrium regardless of the sign of the effective growth rate. In addition, we demonstrate that the extinction-like behavior sometimes observed in numerical simulations in this regime can arise from coarse time discretization rather than from the underlying continuous-time dynamics. Numerical experiments based on refined Milstein schemes confirm the stability behavior predicted by the theoretical analysis. These results provide a rigorous characterization of the dynamics for x 0 > K and highlight the importance of numerical consistency when simulating stochastic differential equations with multiplicative noise.