θ-Maruyama methods for nonlinear stochastic differential delay equations

In this paper, mean-square convergence and mean-square stability of θ  -Maruyama methods are studied for nonlinear stochastic differential delay equations (SDDEs) with variable lag. Under global Lipschitz conditions, the methods are proved to be mean-square convergent with order 12, and exponential mean-square stability of SDDEs implies that of the methods for sufficiently small step size h>0h>0. Further, the exponential mean-square stability properties of SDDEs and those of numerical methods are investigated under some non-global Lipschitz conditions on the drift term. It is shown in this setting that the θ  -Maruyama method with θ=1θ=1 can preserve the exponential mean-square stability for any step size. Additionally, the θ  -Maruyama method with 12≤θ≤1 is asymptotically mean-square stable for any step size, provided that the underlying system with constant lag is exponentially mean-square stable. Applications of this work to some special problem classes show that the results are deeper or sharper than those in the literature. Finally, numerical experiments are included to demonstrate the obtained theoretical results.