The following came up as an additional question in a financial mathematics course:
I am guessing that I should attack this problem with Ito's lemma, which applying to $S(X_t)$ I think gives:
$$ dS(X_t)=\frac{1}{2}\sigma(X_t)^2 S''(X_t)dt + S'(X_t)\sigma(X_t)dW_t $$
This seemed to me to be promising, given that we have the $\frac{1}{2}(\sigma(x))^2S''(x)$ term 'recovered' but I'm not really sure where to go from here, or how the $b(X_t)$ term is going to come into play?
For context, the course this was in doesn't go into very much detail about SDEs, just mentions Ito's lemma and the Feynman-Kac theorem, and solves a few simple SDEs.