Strong Convergence of Euler Approximations of Stochastic Differential Equations with Delay under Local Lipschitz Condition

TL;DR

This paper proves the strong convergence of Euler schemes for stochastic delay differential equations under relaxed conditions, including polynomial growth and continuity, and provides convergence rates supported by numerical simulations.

Contribution

It extends existing convergence results by relaxing assumptions on coefficients and establishes convergence rates for Euler approximations of stochastic delay differential equations.

Findings

Proved strong convergence under general conditions.

Established convergence rates under Lipschitz and polynomial growth conditions.

Validated results with numerical simulations.

Abstract

The strong convergence of Euler approximations of stochastic delay differential equations is proved under general conditions. The assumptions on drift and diffusion coefficients have been relaxed to include polynomial growth and only continuity in the arguments corresponding to delays. Furthermore, the rate of convergence is obtained under one-sided and polynomial Lipschitz conditions. Finally, our findings are demonstrated with the help of numerical simulations.