The Numerical Method Of Stochastic Pantograph Differential Equations

This paper investigates stochastic pantograph differential equations (SPDEs), first, we show there exists a unique global solution of the equation if the drift coefficient and diffusion coefficient satisfy local Lipschitz condition and monotone condition. On this basis, we are going to investigate the strong convergence of the split-step backward Euler method for stochastic pantograph differential equations if the drift coefficient and diffusion coefficient satisfy local Lipschitz condition and one-side linear growth condition. Last, under the condition of almost surely asymptotic stability of the exact solution, we prove that the almost surely asymptotic stability of the numerical solution by using the discrete semi-martingale convergence theorem.