Numerical Solution Of Multidimensional Stochastic Integral Equations

Stochastic integral equation have been widely applied in various fields,such as materials,biology,physics,sociology,economics,medical and so on.However,most of the stochastic Volterra integral equations can't explicit solution.Therefore,it is of great significance to study an effective and accurate numerical algorithm.This paper is the application of the least square method respectively combined with Block pulse functions(BPFs)and Haar wavelets(HWs)approach to solving the multi-dimensional linear stochastic It(?)-Volterra integral equation.The It(?)-Volterra integral equation is transformed into a linear algebraic equation.And the error analysis is proved.At the same time,numerical examples verify the validity and accuracy of the method.The first chapter introduces the research background,research status at home and abroad,research methods and innovation points.The second chapter,the definition of BPFs and HWs,basic properties and related lemmas are introduced.The third chapter,the BPFs and the least square method to solve the multi-dimensional stochastic It(?)-Volterra integral equation.Meanwhile,the error analysis is carried out by isometry property and Doob's inequality.Finally,numerical examples are given to verify the effectiveness and accuracy of the method.The fourth chapter,the HWs combined with the least square method to study the linear stochastic It(?)-Volterra integral equation.The accuracy of the method is proved by error analysis and examples.The fifth chapter combines HWs and the least square to solve the multi-dimensional stochastic It(?)-Volterra integral equation.And Matlab was used to carry out numerical simulation.