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.
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