Split-step θ-method for stochastic pantograph differential equations: Convergence and mean-square stability analysis

This paper introduces a split-step θ-method (SSθ-method) with variable step sizes for solving stochastic pantograph delay differential equations (SPDDEs). We establish the mean-square convergence of the proposed SSθ-method and show that it achieves a strong convergence order of order 1/2. Under certain assumptions, we prove that the SSθ-method is exponentially mean-square stable for θ≥0.5. Additionally, we analyze the asymptotic mean-square stability of the SSθ-method under a stronger assumption. Finally, numerical examples illustrate the effectiveness of the proposed methods.