The Interest Rate Model Based On Stochastic Differential Equation And Its Approximate Solution

Interest rate is an important lever in the international financial market.It is not only an important economic tool to effectively adjust the national monetary policy,but also to effectively control risks such as foreign investment,inflation and reducing unemployment rate,thus directly affecting the steady growth of the economies of various countries.In China's modern market economy,interest rate has always been the main price of liquidity,which is not only directly restricted by many decisive factors in the whole economy and society,but also has a great impact on the development of the whole society and economy,so it is more and more important to study interest rate and build an interest rate model.In this paper,the iterative calculation of approximate solution of interest rate model described by stochastic differential equation is studied.Firstly,the basic concept of interest rate is briefly given,its importance,development history and related description of important research achievements have been made.It is described that various models are constructed by stochastic differential equation in the process of interest rate research.A class of highly nonlinear stochastic interest rate models proposed by Ait-Sahalia are considered.The convergence of its solution is studied.Under the condition that the parameters satisfy local Lipschitz and Khasminskii-type,the existence,uniqueness and boundedness of the global positive solution are proved,and the truncated E-M method of approximate solution is given.By using Gronwall inequality and Ito formula,it is proved that the truncated E-M approximate solution is convergence in probability to its true solution.The convergence of continuous approximate solutions based on BEM method is studied.The basic framework of the article is as follows:In chapter 1,firstly,this paper briefly introduces the research background and significance of interest rate model,and then leads to the development of interest rate model.Some classical interest rate models are summarized,and the research status of interest rate model at home and abroad is briefly combed.Finally,the main research content of this paper is related research on the approximate solution of interest rate model.In chapter 2,we introduce some basic theoretical knowledge used in this paper,including Ito lemma,truncated E-M method,BEM method,Gronwall inequality and so on.In chapter 3,we study the properties of its approximate solution based on a stochastic interest rate model.Firstly,two assumptions are put forward under the local Lipschitz and Khasminskii-type conditions.Under the assumptions,the global positive solution is defined,and the existence,uniqueness and boundedness of the global positive solution are proved.Finally,the truncated E-M method is used to prove that the approximate solution converges to the true solution.In chapter 4:when we study the selected stochastic interest rate model,we first define the discrete-time approximate solution of the model,but in order to make the result more accurate,we define the continuous-time approximate solution,and finally prove that the continuous-time approximate solution converges to the true solution under BEM method.Finally,the research contents and results of this paper are summarized,and the future research prospects are given.