C~r Convergence Of Picard’s Successive Approximations Generated By The Stochastic Differential Equation

The stochastic differential equation has aroused many mathematician’ attention since Ito first aired the concept of stochastic differential equations.As an analysis tool,the stochastic differential equations play more and more important role in many fields such as mathematics,finance,bionomics,etc,and promote the development of our society greatly.So it is necessary to research the related theories of the stochastic differential equations. And among so many theories,the research of the solution to the equation plays an important role.If the coefficients of the stochastic differential equation are C (0<r<∞) function and the first order derivatives of them are bounded,then the solution to the equation is of class Cr-1as a function of the initial condition.Tn general,we proof the existence of the solution to the differential equation firstly,and then proof the smoothness according to feature of the equations.It is easy to proof the existence,however, proofing the smoothness is difficult.This paper gives a proof of smoothness more easily compare to the usual way.In a complete space,we prove that the successive approximation of Picard is well defined and it is a contraction mapping firstly.Then we will show that the picard’s successive approximation converge to the solution in the Cr-sense by the method which is similar to the fixed-point theorem.By this way,we can prove the existence of the solution and at the same time complete the proof of the smoothness.This paper is divided into four parts.We describes the research background and significance of this paper in the first part. The second part introduces the basic concepts, the basic inequalities and the fundamental theorems which are relevant to this paper in order to understand the proof of the theorem in this paper. The third part is the core part,we sum up and analyze on the results that have been obtained firstly. On this basis,we give the main theorem and its process of being proved.In the last part,we summary the whole paper and point out the direction of the research that can try further.