The Function-Separating Method For Some Linear Stochastic Differential Equations

Stochastic differential equation is a branch of mathematics,which is widely applied in many areas.And it has an extremely significant position in the description of objective phenomenon.Because of the addition of uncertainty,stochastic differential equation can be more accurate to describe natural phenomena.In order to utilize the powerful modern tools in many practical fields,experts establish a sort of stochastic model for investigation of the system,which is frequently inevitable.And this tends to signify that the application of stochastic differential equations.However,we merely have model but can't obtain its solution for settling practical issues is pointless.It's such an extensive and practical demand,which prompts us to attach importance to solve stochastic differential equations.This article is mainly to solve linear stochastic differential equations.The main job is as follows:(1)This paper recommends certain essential conceptions about stochastic differential equations.(2)It has demonstrated the existence and uniqueness of solutions for linear stochastic differential equations which include homogeneous and nonhomogeneous linear stochastic differential equations.Firstly,the uniqueness of solution has been verified,and then the existence.We utilize Cauchy-Schwarz inequality,Lipschitz condition and Gronwall lemma to verify the uniqueness,and transformation and separation of function has also been applied during this process.(3)We adopt function-separating method to solve homogeneous linear stochastic differential equations which include ?Ito stochastic differential equations and Stratonovich stochastic differential equations.It firstly introduces the thought of function-separating method,and then makes use of the transformation formula to convert Stratonovich stochastic differential equation to corresponding ?Ito stochastic differential equation and finds its solution.We utilize this method to solve several specific equations.(4)Nonhomogeneous linear stochastic differential equations have also been solved by applying function-separating method.Then we have solved certain linear equations,thereby which made us to comprehend this content.At last,we compare other methods with function-separating method and summarize the advantage of this way.