Numerical Solution Of Nonlinear Stochastic Equations Driven By Fractional Brownian Motion

Stochastic integral equations and stochastic differential equations driven by fractional Brownian motion have attracted much attention in the field of stochastic analysis and are widely used in many fields,such as management science,biology,finance,medicine,etc.But only a small number of stochastic equations give exact solutions.Therefore,it is of great theoretical significance to study how to give high precision numerical solutions.Fractional Brownian motionB~H={B ~H(t),t>0}and Hurst parameter 0<H<1 is a kind of central Gaussian process with zero mean.When H?(0,1 2)?(1 2,1),B~H is neither a Markov process nor a semimmartingale.So fractional Brownian motion can describe more processes than Brownian motion.This paper presents a numerical method based on the block pulse function,which provides high-precision approximate solutions for nonlinear stochastic integral equations and stochastic differential equations driven by fractional Brownian motion.The method is that the nonlinear stochastic integral equation is transformed into algebraic equation by using the module impulse function,and then the algebraic equation is solved by using the fsolve function in the software.Error analysis and numerical examples are given to verify the accuracy and effectiveness of the method.The structure of the paper is as follows:The first chapter introduces the research background,foreign research status,and the innovation of this paper.The second chapter,the definition and basic properties of fractional Brownian motion and block pulse functions are introduced,and the integral operator matrix and stochastic integral operator matrix of block pulse functions are given.The third chapter,the nonlinear stochastic It(?)-Volterra integral equation is transformed into nonlinear algebraic equations by using the properties of Block pulse functions,integral operator matrix and stochastic integral operator matrix,and its numerical solution is obtained.At the same time,the error analysis is given by Riemann-Stieltjes integral.Finally,numerical examples are given to illustrate the effectiveness of the proposed method.The fourth chapter,the nonlinear stochastic differential equations driven by fractional Brownian motion are studied by using Block pulse functions,and the rationality and validity of numerical solutions are verified by error analysis and numerical examples.Finally,the results are summarized and the future work is prospected.