What is Ito's lemma used for in quantitative finance?

Further to my question asked here: prior post

and which left some points unanswered, I have reformulated the question as follows:

What is Ito's lemma used for in quantitative finance? and when is it applicable?

I don't understand for instance if Ito's lemma is used for obtaining a SDE from a stochastic process or the converse: obtain a stochastic process from an SDE.

Furthermore vonjd's reply is a bit confuse to me: does he mean "Ito's lemma can

only

or

also

be used for processes with bounded quadratic variation?

Answer

If you are given a diffusion process $X_t$, and a $C^{1,2}$ transformation $Y_t=f(t,X_t)$ of the process $X_t$.

Then Itô's lemma gives you the SDE followed by the process $Y_t$ in terms of $dX_t$, and $dt$ and partial derivatives of $f$ up to order 1 in time and 2 in $x$.

If you are given the SDE followed by $X_t$ in terms of Brownian motion, drift, and diffusion term then you can write down the SDE of $Y_t$ in terms of Brownian motion, drift, and diffusion term.

This shows in particular that diffusions are stable by those type of transformations.

There is nothing more and nothing less in it.

Of course you can extend this lemma in various fancy and sophisticated ways.

Regards