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

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