We
are given a probability space
(see the section
(
Definition of
conditional probability
)) and a filtration
(see the section (
Filtration
definition
)). Let
be the expectation associated with the measure
.
We introduce another
expectation


(Definition of change of measure)

defined for any
and a particular
,
where
and
are continuous processes adapted to the filtration
and
.
The Girsanov's theorem (
Girsanov_theorem
)
hints that
should be a positive martingale.
We would like to calculate
in terms of
.
We noted in the section (
Filtration
and conditional expectation
) that the formula
(
Chain rule
) may regarded as a definition of
conditional probability. Hence, we
require
for any smooth function
.
Because
is an
adapted
random variable, we apply the formula
(
Definition_of_change_of_measure
)
on the lefthand
side:
On the righthand side we apply the formula
(
Definition_of_change_of_measure
)
directly
Hence,
The operation
is the original measure, hence, the formula
(
Chain_rule
) holds and the last expression is
Left and right sides are equal:
for any
.
We
conclude


(Main property of change of measure)

for
.
The proof of the last step is a standard analysis argument. We express the
expectations in terms of integrals with respect to the corresponding
distributions. If there is a point where the desired property is untrue then
we take a sequence of
converging to the delta function around such point and obtain a contradiction.
