I'm having a difficulty to understand the difference between the Kolmogorov Forward and Backward Equation in how they describe probability density rather then their mathematical formulation (I know the formulas).
Can I say that both describe the same evolution of density but the only difference is the 'direction' of the evolution they describe?
Assume $0\le t\le s\le T$ and some Ito's process $X$. If I have a particle positioned at $X(t)=x$ and I'm interested in the density of being at $X(s)=y$ can I say that the backward equation describes the (conditional) density of $X(s)$ being at $y$ at time $s$ as seen from position $X(t)=x$ and the forward equation gives me the density of $X(t)$ being at $x$ (i.e. in the past) if it is at $X(s)=y$ in the future time?
Can you please give me an example of what these two equations refer to? The distinction is still not 100 % clear to me.
