Using waveform relaxation methods to approximate neutral stochastic functional differential equation systems

This paper extends the waveform relaxation (WR) method to neutral stochastic functional differential equations. The linear convergence theory of the continuous time WR method is established in the mean square sense when the coefficients of stochastic differential equation systems satisfy the Lipschitz condition and the contractive mapping. The discrete time WR method based on the Euler scheme, which is used in an actual implementation, was also studied. It turns out that the sequence produced by this method converges linearly to the Euler approximate solution which is convergent. In addition, we prove that the preceding methods are convergent superlinearly if the neutral term does not been split. Finally, the theory is applied to a one-dimensional model problem and checked against results obtained by numerical experiments.