to integral is initially defined for "simple" processes of the
form
where the variables
are
measurable
for each
and
In other words (see section
(
Filtration_definition_section
)),
the value of
remains constant during
and it is known with certainty at
.
For such
the stochastic (Ito) integral is defined
as
For each
the
is
measurable
while the
is
measurable
and
-independent.
Hence,
and
are independent random variables for each
.
This observation is the key tool when doing anything with stochastic integral.
In particular, we use this observation when proving the following properties.
Proposition
1.
is an
adapted
martingale.
2. Ito isometry.
.
Proposition
(Chebyshev's inequality).
With help of the Ito isometry and Chebyshev's inequality we expand the notion
of stochastic integral to the set of
adapted processes
with the property
We approximate (in
sense) such a 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 simple integrals.
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