e are considering a one-dimensional process
given by the
SDE
(SDE X)
where
is increment of standard Brownian motion and
are some smooth real valued functions. We introduce transitional distribution
,
according to the
relationship
(SDE X p)
We are interested in evolution of
as time
progresses. Let
be any smooth function of both variables rapidly decaying in
.
We
have
We apply the formula (
Ito
formula
).
Since
,
the
term disappears under the expectation sign. We utilize the definition
(
SDE_X_p
):
We perform integration by
parts:
By definition of
,
hence
We substitute this equality into the
relationship
In addition, the integration by parts in
-variable
does not produce boundary terms due to fast decay of
at
-infinities.
Therefore,
We conclude
(Forward Kolmogorov)
for all
and all
.
Theorem
(Forward Kolmogorov equation for diffusion (Ito) process).
If the process
is given by the SDE (
SDE for X
) then the function
(
Distribution of X
) evolves according to the PDE
(
Forward Kolmogorov
) with the initial
condition
.
