# The Truncated Euler-Maruyama Method for Stochastic Differential Delay Equations

## TL;DR

This paper investigates the strong convergence of a truncated Euler-Maruyama numerical method for stochastic differential delay equations under generalized Khasminskii-type conditions, extending previous probabilistic convergence results.

## Contribution

It establishes the strong (L^p) convergence of the truncated EM method for SDDEs under broader conditions, filling a gap in existing numerical analysis.

## Key findings

- Proves strong convergence in L^p norm.
- Extends convergence results to generalized Khasminskii conditions.
- Provides theoretical foundation for numerical solutions of SDDEs.

## Abstract

The numerical solutions of stochastic differential delay equations (SDDEs) under the generalized Khasminskii-type condition were discussed by Mao [15], and the theory there showed that the Euler-Maruyama (EM) numerical solutions converge to the true solutions in probability. However, there is so far no result on the strong convergence (namely in L^p) of the numerical solutions for the SDDEs under this generalized condition. In this paper, we will use the truncated EM method developed by Mao [16] to study the strong convergence of the numerical solutions for the SDDEs under the generalized Khasminskii-type condition.