Strong convergence and asymptotic stability of explicit numerical schemes for nonlinear stochastic differential equations

In this article we introduce a number of explicit schemes, which are amenable to Khasminski’s technique and are particularly suitable for highly nonlinear stochastic differential equations (SDEs). We show that without additional restrictions to those which guarantee the exact solutions possess their boundedness in expectation with respect to certain Lyapunov-type functions, the numerical solutions converge strongly to the exact solutions in finite-time. Moreover, based on the convergence theorem of nonnegative semimartingales, positive results about the ability of the explicit numerical scheme proposed to reproduce the well-known LaSalle-type theorem of SDEs are proved here, from which we deduce the asymptotic stability of numerical solutions. Some examples are discussed to demonstrate the validity of the new numerical schemes and computer simulations are performed to support the theoretical results.