Applying Itô’s Formula to Nonsmooth Functions
Summary
The document asks how much smoothness a function needs for Itô’s formula to apply, using the positive-part payoff as its motivating example. The answer points to generalized functions, whose derivatives may be interpreted in the distributional sense, and identifies the Tanaka–Meyer formula as the relevant extension for nonsmooth functions. The discussion connects this extension to local time in stochastic calculus.
The response names several textbooks with coverage of local time and the Tanaka–Meyer formula, providing places to study the topic further. Its scope is limited: it explicitly does not give a full account of the broad subject or state a complete characterization of all admissible functions. It is a conceptual pointer rather than a derivation, proof, or guide to applying the formula in a particular financial model.
Key ideas
- The positive-part payoff motivates a question about applying Itô’s formula beyond smooth functions.
- Generalized derivatives can extend Itô’s formula to certain nonsmooth functions.
- The Tanaka–Meyer formula uses local time to handle functions such as the positive-part payoff.
- The response offers textbook references but does not provide a complete characterization or derivation.
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# The most general conditions under which Ito lemma holds
# 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?
## Answer by Magic is in the chain (score 5, accepted)
https://quant.stackexchange.com/a/55030
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).Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.