Discretization for backward Kolmogorov equation.

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.

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

(Time-weak 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 ( Time-weak 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.