The Moment Estimations Of Approximate Solutions For Stochastic Differential Equations

In the present article, we mainly study stochastic differential equations of Ito type and discuss the approximate solutions, that is, Euler-Maruyama approximate solutions and Caratheodory approximate solutions.In the first chapter, we give the introduction of this article. And then we introduce the background of stochastic differential equations and their unique solutions. Finally, we point out that it is necessary to study the approximate solutions of SDEs.In the second and third chapter, we will give some knowledge and important theorems which have been proved. Next, according to the expressions of approximate solutions, we will calculate some conclusions of moment estimations of stochastic differential equations.In the fourth chapter, based on the upper conclusions we have just obtained, we will find the relationship between the approximate solutions and the unique solutions.In the fifth chapter, we will discuss a special kind of stochastic differential equations. that is, perturbed stochastic differential equations. Finally, we are likely to show the convergence between approximate solutions and the unique solutions with the help of some calculations.