I am trying to write a complete proof of the Feynman Kac formula in the multi-dimensional case. My starting point was the proof of the univariate form on wikipedia, at https://en.wikipedia.org/wiki/Feynman–Kac_formula. I've managed to write my multi-dimensional proof (see this link), but it has two gaps in it, which are also present in the wikipedia proof, and which I haven't managed to close. They are:
- The proof uses Ito's Lemma twice on the solution function $u$ of a function thereof. For Ito's Lemma to be applicable, the function must be twice-differentiable. But I cannot see a way to prove that $u$ is twice-differentiable.
- The proof actually shows that, IF there is a solution, it must be of the form shown in the Feynman Kac formula. It does not prove that that formula is necessarily a solution, ie that any solution exists.
Can anybody suggest how to plug these gaps, or post a link to a better proof?
Thank you very much.
PS I searched the site before writing this. There are two other questions on here asking about proofs of Feynman Kac. But one is asking for a multi-dimensional version, whereas I would be happy with a univariate version (I know how to convert it to muti-dimensional myself), and has not received an answer. The other is asking about a specific aspect of the proof rather than for a complete proof.
