Some Numerical Solutions Of Stochastic Differential Equations

The theory of stochastic differential equations was widely applied in the fields of econ-omy,biology,physics and so on. But,because of the complexity of stochastic systems,exceptsome special SDEs,in general it is hard to obtain the explicit solution for a general givenSDEs. Thus constructing effective numerical methods is particularly important.This paperfirstly introduces two kinds of the most common numerical methods: EM method and Mil-stein method,and this article is to introduce another numerical algorithm-double implicitMilstein method in the base of Milstein method,then it discusses the convergence and thestability of the double implicit Milstein method;then in this paper introduces the applicationof stochastic differential equations in the economy,introducing an interest rate model-CIRmodel,we can use the double implicit Milstein method to simulate;the paper consists of fivefollowing parts.In the first chapter,we outlined the history of the development of stochastic differentialequations,and analysised the significance of numerical solution of stochastic differentialequations;In the second chapter,we gave prior knowledge of stochastic differential equations;In the third chapter,we introducted the classical Euler method and Milstein methodbriefly,used the numerical methods to solve the problem of stochastic differential equa-tions,and verified the stability of the semi-implicit Milstein much better than the displaymethod by computer simulation;In the fourth chapter,we studied the another numerical methods-the double implicitMilstein method,discussed the Existence of Positive Solutions,and proved the convergenceand stability,by using the computer simulation,we compared the mean square stability ofthe double implicit Milstein method and semi-implicit Milstein method;Chapter5was the application of stochastic differential equations in Financial Eco-nomics,we discussed the CIR model,and used the double implicit Milstein method to provethe conclusion.