Convergrnce And Stability Of Split-step θ-methods For Stochastic Differential Equations With Piecewise Continuous Arguments

In recent years, as a special class of stochastic differential equations, stochastic differential equations with piecewise continuous arguments(SEPCAs) play an important role in many fields, such as biological systems, control theory and signal processing et al.. Therefore, these equations attract a lot of attention. Since there is in general no explicit solution to an SEPCA under the non-globally Lipschitz conditions,the numerical properties of SEPCAs are required in practice.In this paper, we investigate the exponential stability of the SEPCAs. We also study the convergence and exponential stability of the split-step theta(SST) methods for SEPCAs under non-globally Lipschitz conditions.Throughout this paper, we use the SST methods to solve the SEPCAs. We have known that if the drift coefficient of the SEPCA satisfies the local Lipschitz and the monotone condition and the diffusion coefficient satisfies global Lipschitz condition,then there exists a unique solution for an SEPCA. Under the condition above, if the drift coefficient of the SEPCA also satisfies the one-sided Lipschitz condition, then the SST methods are convergent. In order to obtain the convergence of the SST methods,boundedness of the numerical methods in mean square is proved, which is the key to prove the convergence of the SST methods.We also investigate the exponentially mean square stability of the SEPCAs and the SST methods. A sufficient condition is given to guarantee the exponentially mean square stability of the SEPCAs. Under this sufficient condition, the SST methods preserve the exponentially mean square stability with small step-sizes. We also show that the decay rate is also preserved.Moreover some numerical experiments are provided to illustrate the theory we obtained. These values are used to verify the results obtained in the theory.