Three Numerical Schemes For Solving Neutral Stochastic Delay Differential Equations

In this paper, we discuss the convergence and stability of three numerical schemes for solving neutral stochastic delay differential equations in the Stratonovich integral sense. Three numerical schemes are:a Predictor-Corrector scheme, a Mid-point scheme and a Milstein-Like scheme, which are deduced by the spectral expan-sion of the Brownian motion and they can be used to solve stochastic differential equations in the Stratonovich integral sense.We numerically confirm that the Predictor-Corrector scheme and the Midpoint scheme converge with half-order in the mean-square sense while the Milstein-Like scheme is of first-order convergence. The Predictor-Corrector scheme and the Mid-point scheme are still of half-order convergence for the commutative white noises without delay in the diffusion coefficients of neutral stochastic delay differential e-quations which differ from solving stochastic delay differential equations. Further-more, we theoretically prove that the Predictor-Corrector scheme is of a half order convergence in the mean-square sense.For the mean-square stability of these numerical schemes for neutral stochastic delay differential equations, a numerical example shows that the Milstein-Like scheme can not preserve the stability of the exact solution if the step size is large, while the Midpoint scheme can preserve the mean-square stability of the exact solution unconditionally. We propose a stability condition for the Predictor-Corrector scheme and prove that under the condition the scheme is mean-square stable.