Stability of the Split-Step θ-Method for Stochastic Pantograph Systems with Markovian Switching and Jumps

This study focuses on analyzing the almost sure exponential stability of the split-step θ-method (SSθ-method) when applied to stochastic pantograph differential equations characterized by Markovian switching and jump processes. Initially, we establish the almost sure exponential stability of the system’s trivial solution. Subsequently, under an additional sufficient condition, it is demonstrated that the discrete solutions generated by the SSθ-method also exhibit this stability property. Finally, a computational experiment is conducted to support the theoretical results.