The Strong Convergence and Stability of Explicit Approximations for Nonlinear Stochastic Delay Differential Equations

This paper focuses on explicit approximations for nonlinear stochastic delay differential equations (SDDEs). Under the weakly local Lipschitz and some suitable conditions, a generic truncated Euler-Maruyama (TEM) scheme for SDDEs is proposed, which numerical solutions are bounded and converge to the exact solutions in qth moment for q>0. Furthermore, the 1/2 order convergent rate is yielded. Under the Khasminskii-type condition, a more precise TEM scheme is given, which numerical solutions are exponential stable in mean square and P-1. Finally, several numerical experiments are carried out to illustrate our results.