An Effective-State Approach to Mean-Square Stability of Neutral Stochastic Differential Wquations with Delay
Neutral stochastic dierential equations represent an important class of mathematical models in which the present dynamics depend on both the current state and delayed memory eects under random perturbations. In this paper, a class of neutral stochastic dierential equations with delay is studied through an eective-state formulation. Instead of analyzing only the original process x(t), the neutral expression z(t) = x(t)-u(x(t-δ(t)), t) is introduced as the main dynamic quantity. This eective state separates the current state, the delayed state and the neutral memory term, which makes it suitable for stability analysis. A Lyapunov-type approach is used to formulate sucient conditions for boundedness and mean-square stability of the eective state. A scalar neutral stochastic model with delay is also considered as a motivating example. The proposed framework provides a basis for further numerical approximation and computational stability testing of neutral stochastic systems.