e are considering a one-dimensional process
given by an
SDE
(SDE X)
where the
is the increment of the standard Brownian motion and the
are some smooth real valued functions. We introduce the transitional
distribution
,
according to the
relationship
(SDE X p)
We are interested in evolution of the
as the 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. In addition, every expression of
the form
is replaced by
.
We perform integration by
parts:
By definition of
,
hence
The integration by parts in x-variable does not produce boundary terms due to
fast decay of
at
x-infinities.
Therefore,
We conclude
(Forward Kolmogorov)
for all
and all
.
Theorem
(Forward Kolmogorov equation for diffusion (Ito) process).
Suppose 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
.
