e aim to derive distribution of the random variable
.
According to the formula
(
Ito_derivative_of_product
) and
rules (
Ito
calculus
)
We
calculate
The integral
was previously calculated, see the formula (
Int
t_dW
):
for some normal variable
.
The variables
and
are jointly normal (because these are sums of linearly dependent components)
with zero mean. It remains to calculate the standard deviation of
.
We
have
The only part that we did not consider yet is
We introduce a uniform mesh
,
,
,
and represent the integral as pre-limit sums with intention to pass back to
the limit after some
transformations:
where the
is a collection of independent standard normal variables,
.
We
continue
We now pass it back to the integral limit
Hence,
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