Suppose
that we are given the probability space
(see the section
(
Definition of
conditional probability
)) and a filtration
(see the section (
Filtration
definition
)). Let
be the expectation associated with the
.
We introduce the new
expectation
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(Definition of change of measure)
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defined for any
and a particular
,
where
and
are continuous processes adapted to the filtration
,
.
The Girsanov's theorem (
Girsanov_theorem
)
hints that
should be a positive martingale.
We would like to establish the expression for
in terms of
.
We noted in the section (
Filtration
and conditional expectation
) that the (
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
(
Definition_of_change_of_measure
)
on the left-hand
side:
On the right-hand side we apply the
(
Definition_of_change_of_measure
)
directly
Hence,
The
is the original measure, hence, the
(
Chain_rule
) holds and the last expression is
Consequently, left and right sides are equal:
for any
.
We
conclude
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(Main property of change of measure)
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for
.
The proof of the last step is the standard analysis argument. We express the
expectations in terms of integrals with respect to the corresponding
distributions and take a sequence of
converging to the delta function around some point where the desired
conclusion may be untrue.
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