e adapt
derivations of the section
(
Discretization
with respect to time parameter
) to backward Kolmogorov equation.
Condition
(Backward Kolmogorov PDE setup) We
adopt the setup (
Generic parabolic PDE
setup
) with the following exception. We are considering the
problem


(Backward Kolmogorov PDE)

for some
and
The evolution of the solution
happens in opposite direction: from
at time
to time
.
We adopt notation and constructs of the sections
(
Weak
formulation with respect to time parameter
) and
(
Discretization
with respect to time parameter
). Thus, we introduce the following
definitions.
Definition
(Class
)
We introduce the class of
functions
For any
,
we apply the operation
to the equation (
Backward Kolmogorov
PDE
) and integrate over
:
We integrate the first term by
parts:
Problem
(Weak backward
Kolmogorov with respect to time) In context of the condition
(
Backward Kolmogorov PDE setup
)
we seek
such
that


(Timeweak backward Kolmogorov)

for any
.
We redefine the partition to reflect backward evolution with
time:
Definition of the class
remains the
same:
We introduce the
notations
for
.
We calculate for
and
:
Note that
We
continue
The equation (
Timeweak backward
Kolmogorov
) transforms
into
for any
.
Thus,
.
We replace
with
:
We merge conditions and arrive
to
We fix
and take a
with support within
.
We arrive to the following problem.
If
then
If
then
Note that the solution of the equation
(
Backward Kolmogorov PDE
) satisfies
the above problem.
