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

TL;DR

This paper introduces explicit numerical schemes for nonlinear stochastic differential equations that ensure strong convergence and asymptotic stability without restrictive conditions, supported by theoretical proofs and simulations.

Contribution

The paper presents new explicit schemes for nonlinear SDEs that guarantee convergence and stability under minimal assumptions, extending existing methods.

Findings

Numerical solutions converge strongly to exact solutions in finite time.

Explicit schemes can reproduce LaSalle-type stability results for SDEs.

Simulations confirm theoretical convergence and stability results.

Abstract

In this article we introduce several kinds of easily implementable explicit schemes, which are amenable to Khasminski's techniques and are particularly suitable for highly nonlinear stochastic differential equations (SDEs). We show that without additional restriction conditions except those which guarantee the exact solutions possess their boundedness in expectation with respect to certain Lyapunov functions, the numerical solutions converge strongly to the exact solutions in finite-time. Moreover, based on the nonnegative semimartingale convergence theorem, positive results about the ability of explicit numerical approximation 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 and simulations are provided to support the theoretical results and to demonstrate the validity of the approach.