Construction of the Ito Integral

Goal

Extend the Ito integral from simple adapted processes to general adapted continuous processes.

Given an adapted continuous process $(Y_t)$ on $[0,T]$, define simple approximations by

$$ Y_t^{(n)}=Y_{t_i} \qquad \text{for } t\in [t_i,t_{i+1}), $$

where

$$ t_i=\frac{iT}{n}. $$

$$ \int_0^T \mathbb E\bigl[|Y_t-Y_t^{(n)}|^2\bigr]\,dt\to 0, $$

then the corresponding simple integrals form an $L^2$ Cauchy sequence.

The Ito integral

$$ \int_0^T Y_t\,dB_t $$

is defined as the $L^2$ limit of these simple-process integrals.