From the Wikipedia page (https://en.wikipedia.org/wiki/Lemma_(mathematics)), a Lemma is described as: " lemma (pl.: lemmas or lemmata) is a generally minor, proven proposition which is used as a stepping stone to a larger result"
If we look at Ito's Lemma from Stochastic Calculus (https://en.wikipedia.org/wiki/It%C3%B4%27s_lemma):
Let $f(t, X_t)$ be a function such that $f$, $\frac{\partial f}{\partial t}$, $\frac{\partial f}{\partial x}$, and $\frac{\partial^2 f}{\partial x^2}$ are defined and continuous. If $X_t$ satisfies:
$$dX_t = \mu(t, X_t) dt + \sigma(t, X_t) dW_t$$
then the differential $df(t, X_t)$ is given by Ito's Lemma:
$$df(t, X_t) = \left( \frac{\partial f}{\partial t} + \mu(t, X_t) \frac{\partial f}{\partial x} + \frac{1}{2} \sigma^2(t, X_t) \frac{\partial^2 f}{\partial x^2} \right) dt + \sigma(t, X_t) \frac{\partial f}{\partial x} dW_t$$
My Question: It seems like Ito's Lemma is a major and large result – furthermore, I am quite sure that all theorems are eventually used as stepping stones to prove larger results. Given all this – why do we consider the above as a Lemma and not a Theorem? Why is it not Ito's Theorem instead of Ito's Lemma?
Best Answer
As stated in the comments, it is due to historical reasons. The specific reasons are recounted by Jarrow and Protter in a footnote in A short history of stochastic integration and mathematical finance: The early years, 1880–1970. In it they say:
"The book by H. P. McKean, Jr., published in 1969, had a great influence in popularizing the Itô integral, as it was the first explanation of Itô’s and others’ related work in book form. But McKean referred to Itô’s formula as Itô’s lemma, a nomenclature that has persisted in some circles to this day. Obviously this key theorem of Itô is much more important than the status the lowly nomenclature “lemma” affords it, and we prefer Itô’s own description: 'formula'".