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Kolmogorov backward/forward equations are pdes, derived for the semigroups constructed from the Markov transition kernels.

Feynman-Kac formula is also a pde corresponding to a stochastic process defined by a SDE. But I was wondering if the stochastic process is also Markovian? I.e., does Feynman-Kac formula apply only to Markov process?

Does the semigroups from the Markov transition kernels also lead to Feynman-kac pde?

  • If not, what leads to Feynman-Kac pde?
  • If yes, Is Feynman-Kac pde also some kind of Kolmogorov backward/forward equation? if not, how is Feynman-Kac pde related to Kolmogorov backward/forward equations?

Thanks and regards!

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up vote 4 down vote accepted

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.


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.

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Thanks! Are diffusions, killed diffusions, and"non-conservative" all Markov processes? – Tim Mar 5 '13 at 6:19
@Tim: in such shape the question is not precise. As much as you didn't provide a source for the F-K formula you have in mind. – Ilya Mar 5 '13 at 8:13
I didn't know there are different Feynman-Kac formulas? I was reading – Tim Mar 5 '13 at 8:19
@Tim: this formulation is backwards in time, whereas the one in Oksendal is forwards in time (as an example of the difference.) – Ilya Mar 5 '13 at 8:19
@Tim: added an answer to your second question – Ilya Mar 5 '13 at 8:28

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