Study On Numerical Solutions Of Two Classes Of Stochastic (Delay ) Integral Equations

As an important mathematical model,stochastic integral equations and stochastic delay differential equations are widely used in different fields,such as medicine,physics,biology,finance and so on.However,under certain conditions,it is difficult to obtain the explicit solution even if the analytical solution of the stochastic equation exists,which greatly limits the application of the stochastic equation in the actual situation.Therefore,the research on the numerical solution of stochastic integral equation is an extremely important theoretical and applied subject.In this article,global approach is given to solve stochastic integral equations and stochastic delay differential equations driven by Brownian motion.The basic idea of this method: Firstly,two orthogonal basis functions,namely triangular functions and block pulse functions,are introduced.Then,the integral operator matrixes and random integral operator matrixes are obtained by using the definitions and properties of orthogonal basis functions.Finally,the stochastic(delay)integral equations are converted into a system of algebraic equations.Compared with iterative method,global approach has the advantages of lower computational cost and faster convergence rate.Furthermore,the error analysis of the numerical solution is given,and the accuracy and validity of the numerical method are verified by numerical simulation.The composition of this article is as follows:In the first chapter,the research background and the research status at home and abroad of the project are introduced.In the second chapter,the definitions and related properties of two orthogonal basis functions are introduced,and their integral operator matrices and stochastic integral operator matrices are given.In the third chapter,the numerical solution of stochastic Volterra integral equation is explored by using the definition and related properties of triangular functions.Then the error analysis is given.Finally,the accuracy and validity of the current method are demonstrated by numerical simulations.In the fourth chapter,stochastic delay differential equations are researched by using the definition and related properties of block pulse functions.The correlation lemma of a class of nonlinear analytic functions is given.Meanwhile we gain the error analysis of the current method and the accuracy and validity of the current method are demonstrated by numerical simulations.In the fifth chapter,innovations of the subject and prospects of the following work are presented.