Consider the Wikipedia version of Feynman-Kac formula: (https://en.wikipedia.org/wiki/Feynman%E2%80%93Kac_formula) $$f(t,x)=E[f(T,X(T))\mid X(t)=x]$$ for an Ito process $(X(v))$ satisfying the SDE
$dX(v)= \mu(v,X(v))dv + \sigma(v,X(v))dW(v)$ (Eq 1)
There are (at least) three possible ways to formalize the conditional expectation in the above formula, and my question is whether they are all equally rigorous and whether they are actually equivalent (in some sense). These three ways are:
1) The actual definition of the conditional expectation of the random variable $Y$ with respect to the realization of a random variable $X=x$ in the sense of Shiryaev:
A Borel function $m(x)\equiv E[Y\mid X=x]$ such that for any borel set $A\in\mathcal B$, it holds that $\int_{X\in A}YdP = \int_A m(x)dP_X$.
(note that this only defines $m$ $P_X-$a.s. and hence a version needs to be fixed for each $t$ if we want to establish some kind of equivalence with the next two).
2) Assume the existence of a filtered space $(\Omega,\mathcal F,\mathbb F)$, a process $(X(v))$, and a family of probabilities $(P^x)_{x\in\mathbb R}$ such that for each $P^x$ the process $X$ satisfies (Eq 1) and $X(0)=x$ with probability $1$. Denote $E^x[.]$ the expectation under $P^x$.
Then $E[f(T,X(T))∣X(t)=x] \equiv E^x[f(T,X(T-t))]$ rather than "an actual conditional expectation".
(Note that the existence of such a model is proved for example in Karatzas-Shreve, "Brownian motion and stochastic calculus" in the case where $X$ is a Brownian motion)
3) Assume the existence of a filtered probability space $(\Omega,\mathcal F,\mathbb F, P)$ and a family of processes $(X_{t,x}(v))$ satisfying (Eq 1) with initial value $X_{t,x}(t) = x$. Then $E[f(T,X(T))∣X(t)=x] \equiv E[f(T,X_{t,x}(T))]$ rather than "an actual conditional expectation".
So again, the question is whether they are all equally rigorous and whether they are actually equivalent (in some sense).