Motivation of Feynman-Kac formula and its relation to Kolmogorov backward/forward equations?

Solution 1:

First of all, it would be nice if you link the definitions of F-K formula and KBE you have in mind. Anyway:

1. KBE is an equation used to study $P_t f(x):=\mathsf E_x[f(X_t)]$ in terms of the generator $$ \mathcal A :=\lim_{t\downarrow 0}\frac{P_t - P_0}{t}. $$ This is indeed applies to general Markov processes, as its definition depends only such notions as semigroups and generators.

2. F-K formula (at least the one given in Oksendal, Theorem 8.2.1) is devoted to the same problem and appears to be only a generalization to the case of killed diffusions. The term killed refers to the process which is not conservative, e.g. for conservative Markov processes $P_t1 = 1$ for all times $t$, whereas for non-conservative $P_t1\leq 1$ and $P_t1\neq 1$. This refers to the case, when the process "leaves" the state space at some random time $\zeta$ (killing time) and jumps to some auxiliary cemetery state.

   You wrote that F-K formula is a PDE linked to some SDE. I would expect, though, that such SDE has to be Markovian, since otherwise you wouldn't have a nice dependence on states exclusively (as PDE requires). I neither seen F-K formula (as a PDE) for non-Markovian SDEs, so it will be interesting if you provide a link.

Although F-K formula is stated originally for SDE, nothing prevents you from formulating it over general Markov processes. However, I guess in such case it falls down to be a special case of KBE for non-conservative Markov process. Unfortunately, I don't have a monograph of Dynkin "Markov processes" in hand, but I'm pretty sure you can find it there.

Added:

W.r.t. your question on the classification. There are Markovian and non-Markovian SDEs. A class of solutions of Markovian SDEs of a special form is called Ito diffusions. In general, SDEs (in a sense of a martingale problem) can be considered as a superset of Markov processes - but such SDEs do not have to be driven by Brownian motion only.

Killing/non-conservatism is a structure added on top of the structures above. This refers to the case, when $\Bbb P(X_t \in E)<1$ where $E$ is a state space and $t>0$. I'm not an expert on that, though, and I suggest you start reading serious book on stochastic processes, rather than lurking wikipedia.