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

TL;DR

This paper introduces explicit truncated Euler-Maruyama schemes for nonlinear stochastic delay differential equations, proving their convergence and stability under certain conditions, supported by numerical experiments.

Contribution

It proposes new explicit approximation schemes for SDDEs with proven convergence and stability properties, extending existing methods.

Findings

Numerical solutions are bounded and converge in qth moment.

The schemes achieve a 1/2 order convergence rate.

Solutions are exponentially stable in mean square and P-1.

Abstract

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.