Convergence And Stability Of Two-parameter (α,β)-Method For Stochastic Delay Differential Equations

Due to the practical problems are often contain both random and delay the two factors, so stochastic delay differential equations can more truly simulate the actual problems in reality. In recent decades, stochastic delay differential equations have been widely applied to economics, medical, logistics, transportation, population dynamics and other fields.However, in the general case, the analytical solution of the stochastic delay dif-ferential equations is difficult to obtain, so the study of its structure and properties of numerical methods is very necessary. At present, many scholars have studied the nu-merical method of stochastic differential equations, and a lot of research achievements have been made, which included a two-parameter (α,β)-method, but failed to find the literature of two-parameter (α,β)-method for stochastic delay differential equations, so this paper will focus on research of two-parameter (α,β)-method for stochastic delay differential equations.This paper’s work is as follows:The first chapter, reviews the research background and the main research results.The second chapter,mainly introduces the model and related concepts’ of stochas-tic delay differential equation, and proved that the stochastic delay differential equa-tions satisfy certain constraint conditions,the mean square convergence of the method is 3/2 order.The third chapter, research on the mean square stability of two-parameter(α,β)-method for stochastic delay differential equations,and obtain a sufficient condition of the mean square stability of two-parameter (α,β)-method for stochastic delay differen-tial equations.The fourth chapter, through numerical experiments to verify the convergence and stability conclusion of numerical method in the second chapter and the third chapter,the numerical test results show that the theoretical result is correct.