uppose the process
is given by the SDE


(martingaleX)

where the
is some adapted stochastic process and
is the standard Brownian motion. We are looking for a function
such that the process
given by the
SDE
would have the same
distributions
for every
We assume that the
SDE
has a solution. Hence, there are
for all
We omit the condition
from notation and
calculate
where the
is the step function and
is the Dirac's delta function. We used the (
Ito
formula
) and the martingale property of
(
martingale X
).
Similarly,
We set
and apply the summary
(
Differentiating
call with respect to maturity 2
) with
,
then
with the same final conditions for
and
.
Then
Consequently
