An equaion like: Xt=rho*Xt-1 is called Difference Equation of order 1. It is straitforward to write the solution, which is: Xt=rho^t*X0.
When the order is higher, things will be complicated. For example: Xt=rho1*Xt-1+rho2*Xt-2....+rhop*Xt-p
The solution needs a littlle trick. We define C0=1, C1=rho1, C2=C1*rho1+rho2, C3=C2*rho1+C1*rho2+rho3,.....
Cn=Cn-1*rho1+Cn-2*rho2+...Cn-p*rhop.
Xt=rho1*Xt-1+rho2*Xt-2+...+rhop*Xt-p; times C0
Xt-1=rho1*Xt-2+rho2*Xt-3+...+rhop*Xt-1-p; times C1
Xt-2=rho1*Xt-3+rho2*Xt-4+...+rhop*Xt-2-p; times C2
......
X2=rho1*X1+rho2*X0+...+rhop*X2-p; times Ct-2
X1=rho1*X0+rho2*X-1+...+rhop*X1-p; times Ct-1
And then sum them up. The solution of Difference Equation of order p is:
Xt=summation(m=t to t+p-1){[summation(i=m-t+1 to p) Cm-i*rhoi]*Xt-m}
It will be much easier if the companian matrix idea is applied.
Define Yt=(Xt,Xt-1,...Xt-p)'
The difference equation of order p can be rewritten as Yt=A*Yt-1.
A=
| rho1, rho2,......,rhop |
|1,0,0....... 0|
|... |
|0,0,......... 1, 0|
So, Yt=A^tY0
The Difference Equaion has been widely used in Time Series Analysis!
[此贴子已经被作者于2006-9-12 7:09:20编辑过]