e follow the setup of the two previous sections
(
Girsanov setup
) and
(
Girsanov change of measure
) but give the
direct derivation of the expression for the changed probability measure. We
consider a "small time interval" version of the formula
(
Change of measure
):
The last formula is the equation that we need to satisfy by selecting a
process
.
The
,
,
are deterministic functions from point of view of
.
The
and
are random variables from the point of view of
.
These variables represent a small change over the time interval
.
The distribution of
is known,
for a column of iid standard normal variables
in the "original measure". The distribution of
is what we are trying to find from the above equation. We seek
as
a variable adapted to the filtration generated by
.
Hence, we express the last "original measure" expectation as follows
where
is dimensionality of
.
In the last integral the
is the integration parameter over all possible values of
and
is a function of
because
is
adapted.
The last integral is supposed to be equal to
for any smooth and sharply decaying function
.
We seek
such that the
would be a standard Brownian motion in the new (changed) measure:
We need the former and the latter integrals to be equal. This gives us a
recipe for construction of the
.
We introduce the convenience notation
and
and state that we are seeking a
satisfying
for any smooth
.
To make conclusions we need to have the same expression as an argument of
.
Hence, we make a change of variable
in the right integral. It
becomes
Therefore,
has to
satisfy
for any smooth
.
Hence,
must
satisfy
or
Changing to the original variable
we
obtain
or, in the original
notation,
Using the above SDE and the boundary condition
we
conclude


(Girsanov kernel)

We summarize with the following statement.
