Almost sure exponential stability of numerical solutions for stochastic pantograph differential equations

The sufficient conditions of the almost sure exponential stability of the exact solution for the stochastic pantograph differential equation are considered, with a Khasminskii-type condition. The almost sure exponential stability of the numerical solutions by the Euler–Maruyama method and the backward Euler–Maruyama method is also discussed, based on the discrete semimartingale convergence theorem. We present the sufficient conditions for the stability of the Euler–Maruyama method, with one extra condition when compared with the exact solution. We show that the backward Euler–Maruyama method can share almost the same conditions for the almost sure exponential stability as the exact solution.