e summarize the situation as follows. We have a 3-dimentional process
given by the
equation
where we use the
notation
We calculate the
increment
For
to be a martingale, we must
have
We introduce the notations
as
follows:
Consequently, we
have
We separate terms with every power and coordinate of
:
We now recover the expressions for
before we substitute it into the above ODEs for
and
.
By comparing
definitions
we
see
Similarly for
we have
definitions
hence
Also,
We now recover the ODEs for
:
We calculate the expression
under assumption that
is the only non-zero
correlation:
Hence,
Equivalently,
The equation for
resolves
to
We summarize the results up to this point as
follows
We would like to compare these results with the results of the paper
[CarrWu2006a]
. Hence, we perform additional
transformations.
We calculated earlier
Hence,
where the
is the notation of the paper. We change variable from
to
,
hence,
The last expressions compare exactly with the paper
[CarrWu2006a]
.
|
