Numerical analysis of the balanced implicit methods for stochastic pantograph equations with jumps

This paper deals with a family of balanced implicit methods with linear interpolation for the stochastic pantograph equations with jumps. In this paper, the strong mean-square convergence theory is established for the numerical solutions of the system. It is shown that the balanced implicit methods, which are fully implicit methods, give strong convergence rate of at least 1/2. For a linear scalar test equation, the balanced implicit methods are shown to capture the mean-square stability for all sufficiently small time-steps under appropriate conditions. Furthermore, weak variants are also considered and their mean-square stability analyzed. Several numerical experiments are given for illustration and show that the fully implicit methods are superior to those of the explicit methods in terms of mean-square stabilities.