Fixed Point Approximation Problem For Stochastic Differential Equations

Random phenomena exist everywhere in our lives,and these random factors play an important role in some systems,thus creating the birth of a stochastic dynamical systems.The dynamical systems was originally a mechanical system described by a differential equation derived from classical mechanics.Because the random dynamical systems can handle complex systems,it is favored by many dynamicists.Random dynamical systems also have a wide range of applications in many fields.For example,dynamical systems can be applied in economics,physics,and ecology.In this paper,we study the fixed points of stochastic differential equations with additive noise and stochastic differential equations with multiplicative noise,and give a method to approximate the fixed point.Studies show that the chaotic behavior of a dissipative systems can be explained by an attractor of the progressive system trajectory of time(t tends to infinity).In this paper,we analyze a method of approximating fixed points for stochastic differential equations.For this,we use integral O-U process ?t(?)to approximate Brownian motion Wt in differential equations.In addition,we use stochastic differential equations with ?t? to approximate stochastic differential equations with Brownian motion Wt.The paper mainly studied two aspects.On the one hand,we studied fixed point approximation of multiplicative random dynamic systems that proved by the three theorems.That is,the approximations of the solutions of the two equations and the existence of fixed points that is single point set,and the approximation of the fixed points of the two equations.On the other hand,what is studied is the fixed point approximation of additive stochastic differential equations.The proof of similar multiplicative stochastic differential equations is also proved by three theorems.In addition,in the additive stochastic differential equations,we study the speed of the fixed point approximation.