# 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?

## 1 Answer

It is a vast topic so my answer wont do justice, but staying within the topic of twice continuously differential settings, Ito's lemma can be applied to generalised functions (derivatives defined in the distribution sense)- examples of such functions are the Heaviside function, dirac delta etc. The particular application you referenced goes by the name Tanaka-Meyer formula - it was developed in the sense of local time, but only a slight tweak was needed to show that that Ito's lemma work for functions of the type mentioned earlier.

Regarding literature, you will find the coverage of this formula in the local times section of stochastic calculus books. For example, Klebanar's Introduction to Stochastic calculus has a couple of pages on the subject. 2nd volume of Rogers and Williams' Diffusion Markov Processes and Martingales has a few pages on the subject. Karatzas and Shreve's Brownian Motion and Stochastic Calculus also covers the topic (as per @KeSchn's comment below).

• Is there a paper or book that contains the statement and proof of the Tanaka-Meyer formula? Any kind of references would be much appreciated! – fwd_T Jun 18 '20 at 20:58
• It will be in the local times section of stochastic calculus books - for example, Klebanar's introduction to stochastic calculus has a couple of pages on the subject. Rogers and Williams have a few pages in the second volume of their book: Diffusion markov processes and martingales – Magic is in the chain Jun 18 '20 at 21:07
• @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
• Thanks! Glad you found it useful! – Magic is in the chain Jun 22 '20 at 22:27