Well... I have a problem with Ito integral $\displaystyle \int_{a}^{b} x(t)dw(t) $, where x(t) is a stochasic process, and w(t) is Brownian motion (or Wiener process ). It is said that Ito integral cannot be considered as a Riemann-Stjelties integral because dW has no bounded variation. So for example let us consider sequence of approximating sums $\displaystyle \sum _{i=1} ^{n} (x(t') w(t_{i} - t_{i-1}))$ where $\displaystyle t' \in [t_{i-1}, t_{i}]$ and $\displaystyle a=t_{0}<...<t_{n} =b$ is a division of integral [a,b]. In this case value of this integral does depend of choise of t'! And I've read that this is because Wiener process has no bounded variation.

So my question is... why is this variation very important?

I know how Wiener process look like and know its properties. But I just can't imagine why variation of it has this significant result in choise intermediate points t' ??