Is there any standard reference for the "modern" Feynman-Kac formula as stated here?
By modern I don't mean the earliest version by Kac (for which the reference seems to be this work: Kac, Mark (1949). "On Distributions of Certain Wiener Functionals". Transactions of the American Mathematical Society. 65 (1): 1–13.), but the version which is cited in the link above or used in most Finance Textbooks when it comes to solving the Black-Scholes PDEs.
The textbooks I read just use it without any reference and a short google-search couldn't reveal any helpful documents.
I'm thankful for any hints and links to this - admittedly not very precise - question.
This should be in any book on stochastic calculus. For example Klebner's book Introduction to Stochastic Calculus with Applications has a proof on page 155, plus all the ingredients for it before. I'm sure other books like Oksendal etc. should likely have it as well. I just like Klebner's book.
If you're interested, I did my undergrad senior thesis on Feynman Kac formula which can be found here https://www.math.ucdavis.edu/undergrad/research/thesis/ or my website https://www.math.purdue.edu/~zselk/
ADDED: as of 22 July 2023 these links are dead.