# The most general conditions under which Ito lemma holds

Prompted by a question that came up in the comments here, namely why we can apply the Ito lemma to a function of the form $$f(x)=(x-K)^{+}$$, I would be interested in knowing what are the least restrictive conditions on the smoothness of $$f$$ so that Ito lemma remains applicable. A reference to a book/chapter/theorem would be great! Is this still a topic of mathematical research or do we have an exact characterization of the class of functions to which Ito lemma can be applied?

• @Gabe Another discussion of Tanaka-Meyer for Brownian motion and continuous semimartingales is given in Karatzas and Shreve's Brownian Motion and Stochastic Calculus''. – Kevin Jun 18 '20 at 21:11
• @Magic is in the chain This answer was very helpful. Ultimately I was mainly interested in applicability of Ito Lemma to functions of the form $f(x)=(x-K)^{+}$. – fwd_T Jun 22 '20 at 22:25