Convergence rate and stability of the split-step theta method for stochastic differential equations with piecewise continuous arguments

In this paper, we investigate the strong convergence rate of the split-step theta (SST) method for a kind of stochastic differential equations with piecewise continuous arguments (SDEPCAs) under some polynomially growing conditions. It is shown that the SST method with θ ∈ [1/2,1] is strongly convergent with order 1/2 in pth(p ≥ 2) moment if both drift and diffusion coefficients are polynomially growing with regard to the delay terms, while the diffusion coefficients are globally Lipschitz continuous in non-delay arguments. The exponential mean square stability of the improved split-step theta (ISST) method is also studied without the linear growth condition. With some relaxed restrictions on the step-size, it is proved that the ISST method with θ ∈ (1/2,1] is exponentially mean square stable under the monotone condition. Without any restriction on the step-size, there exists θ^∗ ∈ (1/2,1] such that the ISST method with θ ∈ (θ^∗,1] is exponentially stable in mean square. Some numerical simulations are presented to illustrate the analytical theory.