带跳随机比例微分方程补偿分步θ方法的收敛性和稳定性

Convergence and Stability of Compenseted Split-Step Theta Methods for Stochastic Pantograph Differential Equations with Jumps

我们探讨了带跳的随机比例微分方程的补偿分步θ方法。首先,证明了该数值方法收敛,并且收敛阶为12,其次证明了解析解的均方稳定性,并且证明了补偿分步θ方法能保持解析解的均方稳定性,最后给出数值算例验证理论结果的正确性。

A semi-implicit compensated split-step θ method for stochastic pantograph differential equations with Poisson jumps is investigated in the paper. Firstly, the convergence of the numerical method is discussed, and it is proved that the convergence order is 1/2. Secondly, the mean-square stability of the analytical solution is proved, and it is found that the compensated split-step θ methods can maintain the mean square stability of the analytical solution under some conditions. Finally, two numerical examples are given to verify the correctness of the theoretical results.