Proof of Ito isometry

The previous article has proved the Ornstein–Uhlenbeck process, however, it is not a formal proof since some steps is not clear enough such as "defining dWt as Wt+dt - Wt"

Therefore, I shall post the proof of "Ito isometry" here, which is a formal proof for handling expectation and variance of stochastic integral.

We want to proof the followings:





*****************************************************************

To begin with all, we have to learn the concept of "partition".

Most of the reader should have basic concept about differentiation and integration, in fact, the differential itself is a partition:



where h->0 can be understand as "the difference of x2 and x1 tends to 0"

For the integration



What we are actually doing is:



Now, we replace the ambiguous "h->0" to a more vigorous one

Define the partition P to be:



where i  is between 0 to n and xn equals b (the upper bound of the integration range)

Define "the largest" partition to be:



where sup is "supremum" (we will come back to this in after article, now just think supremum is something like "maximum")

Hence, here we have the following definition of integration:



*****************************************************************

Then, we go back to the proof of Ito isometry:

By definition,



Note that



and it is independent of the function f(t,Z), hence



Therefore,



Moreover, we have



Since E(X) part becomes zero,



Substituting f(t,Z) into X,



Because ti - ti-1 is exactly the differential which equals dt,

We have the Ito isometry:



*****************************************************************

The proof is still not perfectly completed yet because we assume the the expectation sign and limit can interchange. We shall come back to this later.