The Mean Square Convergence Of Euler-methods For Two Kinds Of Stochastic Delay Differential Equations

Stochastic delay differential equations are becoming increasingly used in various ?elds, such as environmental science, economics, control science and engineering, systems engineering, which have attracted much more attention recently. However, with the deepening of research, researchers have found that some stochastic delay differential equations are also in?uenced by the rate of change in the past state. This led to the emergence of neutral stochastic delay differential equations. As is known to all, the exact solutions of the two kinds of equations are diffcult to obtain, so numerical schemes are the main way to get its solutions. Therefore, it is necessary to study the mean square convergence of such equations.This paper is organized as follows:The ?rst chapter introduces the mean-square convergence of the numerical methods for the stochastic delay differential equations and the neutral stochastic delay differential equations. Some preliminaries and main contents of this article are also presented in this section.The second chapter discusses the mean square convergence of strong predictorcorrector Euler method for stochastic delay differential equations. Besides, some numerical examples are presented to validate the theoretical results.The third chapter studies the mean-square convergence of fully implicit Euler method for neutral stochastic delay differential equations. We also validate the theoretical results by some numerical examples.