On the Stability and Boundedness of Solutions to Fourth-Order Nonlinear Stochastic Differential Equations with Variable Delay

Abstract

This paper establishes sufficient conditions for the stochastic asymptotic stability (SAS) and uniform stochastic boundedness (USB) of solutions to a class of fourth-order stochastic delay differential equations. By defining an appropriate Lyapunov–Krasovskii functional (LKF), two new theorems are proved that guarantee these stability and boundedness properties. As an application of the theoretical findings, two illustrative examples are presented to demonstrate the effectiveness of the established criteria. Finally, the results provide a novel and meaningful contribution to the qualitative analysis of higher-order stochastic delay differential equations and are expected to support further theoretical developments and applied investigations. © 2025 Ilirias Research Institute, Prishtinë, Kosovë.