uppose
a one-dimensional process
is given
by
with respect to some probability measure that we call "the original measure"
in light of the considerations of the section
(
Definition of change of
measure
). Here and everywhere below
is a column of standard Brownian motions,
is a column of
-adapted
stochastic processes and
is a one-dimensional
-adapted
stochastic process. We introduce some process
given by the
equation
for some processes
.
We would like to find SDE representation of
with respect to the probability measure changed with
in the sense of formula
(
Definition of change of
measure
). To obtain the recipe for such calculation we rephrase the
requirement as follows. We wish to find the processes
and
such
that
where the
is an Ito process with the
properties
where the
is the unit matrix and the operation
is given by the formula
(
Definition of change of
measure
). Hence, we look for
such
that
The
is
-measurable:
.
Hence
Note that
=
=
.
We conclude that the "changed" drift is given by the
recipe
We calculate the diffusion term using a similar approach:
We keep only the terms of
-magnitude,
see the section (
Ito
calculus
).
Hence,
We substitute the (*) and (**) into the
-equation
for
and
conclude
(General change of Brownian motion)
Summary
(Transformation of SDE under change of measure)
Let
be a column of standard Brownian motions adapted to
and a positive-valued process
is given by the
SDE
where the
is a column of
-adapted
positive processes. Under the probability measure given by
,
(see
(
Definition_of_change_of_measure
)),
the process
has the
SDE
for a column
of some standard Brownian motions under the probability measure defined by
.
