Strong Convergence Of The Stopped Euler-Maruyama Method For SDEs With Non-lipschitz Coefficients

Recently,the convergence of the numerical schemes for stochastic differential equations has been studied,in which the convergence of the Euler-Maruyama(EM)method for stochastic differential equations with global Lipschitz coefficients is classical.Now we aim to study the convergence of the stopped Euler-Maruyama method for stochastic differential equations with non-Lipschitz coefficients.The method of the stopped EM is defined on the basis of the method of EM,adding the stopping time.That is to say,when the approximations is the first time to get negative,it will stop,which will be told detailedly in the first chapter.Besides,from the definition of the method of the stopped EM it is obvious that the approximations are positive.The positive property has great importance to the application in many aspects.In order to introduce the proof of the convergence of numerical schemes for the stochas-tic differential equations better,the related basic knowledge will be described in detail.In the second chapter,we will show the definition and property of the Brownian motion and martingale,many kinds of the Ito formula,the history of the existence and uniqueness of the stochastic differential equations,the methods of the approximation,Markov prop-erty and inequalities.If we want to discuss the convergence of the numerical solutions to stochastic differential equations,we have to ensure the existence and uniqueness of the solutions,which have been widely studied.The conditions of the existence and uniqueness vary from global Lipschitz to local Lipschitz,and even to non-Lipschitz.The first change will be guaranteed by defining a stopping time.And by modified EM method,the second can be achieved,such as truncated EM method,?-EM method,semi-tamed EM method and so on.In the third chapter,we aim to prove the strong convergence of the stopped Euler-Maruyama method for SDEs with non-Lipschitz coefficients.Firstly,we will prove the existence and uniqueness of the solutions to stochastic differential equations with non-Lipschitz coefficients.It is easy to get the property of the existence.Using the conditions that the non-Lipschitz coefficients satisfy,we can have the uniqueness.Secondly,we begin to study the strong convergence.Dividing into two parts and by Holder inequality,B-D-G inequality,martingale inequality,elemental inequality and Gronwall inequality,we obtain the strong convergence and the rate of the Lp convergence and almost sure convergence.Finally,for a given stochastic differential equation with non-Lipschitz coefficients,we use R language to simulate the iterative process.Assume the initial values and the iterations,such as 102,103,104,the curves between time and iterative values can be made.By comparing and analyzing iterative values with true solution,we get the result that the more iterations,the more approximation effect.Moreover,the regression equation of the time and the errors between approximation solutions and true solutions can be established.The slope of the regression equation is the rate of convergence.From the line,we can get the rate of convergence.