| As an important branch of probability theory,stochastic differential equation has been widely used in many fields,such as finance,systems science,engineering,biomedicine and so on.However,in most cases,the true solutions of stochastic differential equations are not expressed explicitly,so it is of great significance to discuss the numerical solutions.This paper mainly introduces a global approximation method for solving nonlinear stochastic differential equations and nonlinear stochastic integral differential equations based on block pulse function.The nonlinear stochastic integral equation is transformed into algebraic equation by the improved operator matrix of block pulse function.By solving the undetermined coefficients the numerical solution block pulse basis function representation of the equation is obtained.Furthermore,error analysis is given to prove the error accuracy of this method.Finally,two numerical examples are given to verify the proposed method.The main contents are as follows:The first chapter introduces the background of stochastic differential equation,the research status at home and abroad,and the innovation of this paper.In Chapter two,the definition and properties of block pulse function are introduced,and some important lemmas are given.In Chapter 3,the integral operator matrix and the stochastic integral operator matrix are established by using the correlation properties of the block pulse function,and nonlinear stochastic differential equation is transformed into algebraic equation.The error is analyzed by using Gronwall inequality,and its validity is verified by numerical simulation.In Chapter 4,the nonlinear stochastic Volterra integral differential equation numerical algorithm is studied.The nonlinear function is the analytic function satisfying certain conditions,and its rationality is verified by error analysis and numerical simulation.In chapter 5,the results obtained in this paper are summarized and the future work is prospected. |
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