Talk:Feynman–Kac formula

Hi, I'm not very Mathy, but came to this page looking for some clarifications. Can someone explan to me where the "third Term" (described as O(dt du)) comes from in the first place? Chs2048 (talk) 01:20, 4 May 2016 (UTC)

I believe that the proof is now correct. —Preceding unsigned comment added by Eweinber (talk • contribs) 21:55, 9 May 2010 (UTC)

OOPS! The proof stated in earlier versions of this page is actually a derivation of the backward (NOT forward!) Kolmogorov equation. The FORWARD Kolmogorov equation is indeed the linear Fokker-Planck equation, but, in any case, this, in general, isn't it.

The proof given only shows that if you have a solution to the PDE, then it has the stochastic representation. It doesn't actually show that if you define such a function, it solves the PDE.

-- —Preceding unsigned comment added by 68.40.122.241 (talk) 00:32, 17 August 2009 (UTC)
 * I agree with 68.40.122.241 that the article should discuss the issue he raises here! A discussion of non-uniquessness etc would be interesting. Ulner (talk) 22:04, 9 February 2010 (UTC)

I think there should be some mention that the 1st formula is actually the linear Fokker-Planck or also known as the Kolmogorov Forward equation. it is very important to show the links between various formulations. The claim that the expectation of an Ito integral is zero is not true in general, since the Ito integral is in general only a local martingale.

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It would be cool to have references to the original works. —Preceding unsigned comment added by 130.207.104.54 (talk) 18:45, 2 May 2008 (UTC)

ambiguity in notation
Just a little remark ... the PDE is defined with the two-argument function V(x,t), later on this seems to be used in the notation V(x_t), which I found confusing at first view of the article. — Preceding unsigned comment added by OpiSamo (talk • contribs) 10:45, 31 March 2011 (UTC)

proof that u is twice-differentiable
The proof uses Ito's Lemma twice, once on $$Y$$ and once on $$u(t,X)$$, which (it seems to me) implicitly assumes that the solution function $$u$$ is twice-differentiable. Yet there is no argument provided to demonstrate that $$u$$ is indeed twice differentiable. How can that be demonstrated?Ajkirk (talk) 02:59, 6 March 2016 (UTC)

proof appears to contain erroneous step, which needs to either be corrected or flagged
The proof contains the claim

'observing that the right side is an Itô integral, which has expectation zero'

It is not completely clear what is meant by an 'Itô integral', but it appears to mean an integral of an Itô process. If so, the claim appears to be false. An Itô process need only be a semi-martingale, but further constraints are needed, such as being a (full) martingale, for the expectation of the integral to be zero. See this discussion on Mathematics StackExchange.

Perhaps there are some further constraints, that are satisfied by the expression to which the claim refers, that are sufficient to make the expectation of the integral zero. If so, I hope somebody can edit the proof to state those constraints. If that can't be done, I would propose to edit the proof in the main text to point out the flaw and invite readers that know how to fix it to do so. Ajkirk (talk) 00:14, 21 April 2018 (UTC)

Further note 27 April 2018: I have marked the above clause with a 'citation needed' note, pending finding a source for the claim. George Lowther's excellent 'almost sure' blog contains a proof that stochastic integrals of elementary predictable processes have zero expectations (Theorem 5 on this page), but I have yet to find an extension of that to a larger class of integrand that could apply here, such as bounded, integrable Itô processes. Ajkirk (talk) 01:26, 28 April 2018 (UTC)
 * An Itô integral is an integral of an adapted process against the derivative of a Brownian motion, which is always a martingale. CapitalSasha ~ talk 01:25, 10 April 2019 (UTC)

survey article
This looks good. 173.228.123.207 (talk) 03:10, 16 August 2019 (UTC)

the complex case
"The complex case, which occurs when a particle's spin is included, is still[when?] unproven" I am not an expert, but it seems to me that this is a misunderstanding: isn't the "complex" case described the Schrödinger equation with a i in front of the du/dt? If so it has nothing to do with spin, but rather with classical (stochastic) dynamics and QM. This remark should be clarified. — Preceding unsigned comment added by Smeuuh (talk • contribs) 12:36, 9 May 2022 (UTC)