Convergence and stability of split-step theta methods with variable step-size for stochastic pantograph differential equations

In this paper, we are interested in numerical methods with variable step-size for stochastic pantograph differential equations (SPDEs). SPDEs are very special stochastic delay differential equations (SDDEs) with unbounded memory. The problem of computer memory hold, when the numerical methods with constant step-size are applied to the SPDEs. In this work, we construct split-step theta (SSθ) methods with variable step-size for SPDEs. The boundedness and strong convergence of the numerical methods are investigated under a local Lipschitz condition and a coupled condition on the drift and diffusion coefficients. It is proved that, the SSθ methods with variable step-size for (Formula presented.) converge strongly to the exact solution. In addition, the strong order 0.5 is given under mild assumptions. The mean-square stability (MS-Stability) of the numerical methods with (Formula presented.) is given. Finally, some illustrative numerical examples are presented to show the efficiency of the methods, and how MS-Stability of SSθ methods depends on the parameter theta for both linear and nonlinear models.