Dear knowledgeable people of math.stackexchange :) ,
In Stochastic Analysis and Diffusion Processes by Kallianpur on pages 218 to 221 the derivations for the forward and backward Kolmogorov equations are provided.
In essence, the Kolmogorov backward equation (KBE) is derived through the Chapman-Kolmogorov equation and applying a Taylor expansion.
The Kolmogorov forward equation (KFE/Fokker-Planck) is derived from the Chapman-Kolmogorov equation by subtracting the forward variables (again, see pages 218-221 in Kallianpur) and inserting the KBE into the KFE.
What bugs me is that the KFE/Fokker-Planck is different from the KBE.
Why is that and how does the concept of time going forward and backward yield the differences in the forward and backward equations (the drift and diffusion being within the partial derivative)?
See the wikipedia entry https://en.wikipedia.org/wiki/Kolmogorov_backward_equations_(diffusion)
Why can't I apply the same Taylor expansion ansatz that is used in the KBE for the KFE/Fokker-Planck? My hunch is that it is connected to the idea of an adapted stochastic process in the KFE/Fokker-Planck?
I did the mathematical derivations but I'm still missing the holistic idea, which necessitates separate ansatzes for the derivations.
Any help is much appreciated! Thank you. :)