Consider the following SDE:
We provide only the ideas of the proofs. Additional assumptions are necessary to justify the use of all the tools.
In what follows, we consider and we let be the density of starting from .
The Forward Kolmogorov equation writes
Set where is with bounded support. By Itô's Formula, Hence, where the last equality is obtained from integration by parts. Differentiating both sides of the equation wrt yields
The Backward Kolmogorov equation writes
Recall that Now, define for , .
Set . By Itô's Formula,
Note that and . Hence, Dividing by and letting , we get Since , we finally obtain which completes the proof.