iven a filtration
Ito integral is initially defined for "simple" processes of the
where the variables
See the section
for the notations
In other words, the value of
remains constant during
and it is known with certainty at
the stochastic (Ito) integral is defined
is a standard Brownian motion adapted to the filtration
are independent random variables for each
This observation is the key tool when doing anything with stochastic integral.
In particular, we use such observation when proving the following properties.
2. Ito isometry.
With help of the Ito isometry and Chebyshev's inequality we expand the notion
of stochastic integral to the class of
with the property
We approximate (in
sense) any such process with a Cauchy sequence of simple processes. Then the
stochastic integral of the process is a limit in probability of the Cauchy
sequence of the simple integrals.