It? Formula 

By Duong Hao  

In this paper, I try to use Maple to calculate Ito formula for 1 or 2-dimension cases 

1-Dimension It? formula 

Suppose F=F(x,t)where  dx=adt+bdWProcedure for 1-dimension It? formula is 

 

 

 

Example 

 

Proof that  is the solution of the equation y(0)=1We use ito1 procedure as follow 

 

 

	(1.1.1)

 

2-Dimension It? formula 

And here is the procedure for It? formula for 2 - dimensions caseSuppose F=F(x,y,t)where dx=adt+bdw dy=cdt+dw 

 

 

Example 

Considering the equation which has the form   The solution of the linear system (when ε=0) has the formwhere It is easy to show that from the system above, we have  Use ito2 procedure to calculate It? difference of a and φ: 

Calculate da, and d(φ)


 

 

 

	(2.1.1)

 

 

(2.1.2)

 

Calculate d(θ) 

 

 

 

 

(2.1.3)

 

These results have been checked in "Extended first order stochastic averaging method for a class of nonlinear systems"- ND Anh - 1993 

And here is the procedure which gives the Fokker-Planck equation written for the probability density function p of da, d(θ)




 

 

(2.1.4)

 

 

 

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