@IiroUllin Yeah I know, but this only prove that the $Y(s)$ in this form is a solution to the PDE, I'm just curious why/how do we define $Y(s)$ in this form.

@IiroUllin I mean where does this $Y(s)$ expression come from, How do you know that $Y(s)$ is in this form.

@IiroUllin I see. What really bothers me about the proof in wiki is how process Y was obtained i.e. $Y(s)=e^{-\int _{t}^{s}V(X_{\tau },\tau )\,d\tau }u(X_{s},s)+\int _{t}^{s}e^{-\int _{t}^{r}V(X_{\tau },\tau )\,d\tau }f(X_{r},r)\,dr$.

@IiroUllin Is there any relevant paper that includes this proof?

@IiroUllin Yes, but I didn't find it. From Wikipedia I saw a proof of the correctness of this exact solution, but it wasn't what I wanted. In another hand, the solution to the Feynman-Kac formula might be complicated, but if we knew $f(x,t)=0$, then perhaps the proof would be much simpler.