Ito's formula but not given $\mu$ and $\sigma$

Question

I have a little question from one of my worksheets(the solution I was given was almost not even a solution, super brief).

let $f(t,x)=t\cos(x)$. Use Ito's formula to calculate $df(t,W_t)$.

Well, Ito's formula is $df=(\frac{\partial f}{\partial t}+\mu \frac{\partial f}{\partial x}+\frac{\sigma^2}{2} \frac{\partial ^2 f}{\partial x^2})dt + \sigma \frac{\partial f}{\partial x}dW_t$ if I am correct. I can calculate each partial derivative wrp to $f$ but what of $\mu$ and $\sigma$?Do I just leave them as some constant which is unknown?

The solutions simply gives me the derivatives and then it only says "Now apply Ito's formula" and over. Doesn't show me the final result/answer as to what $df$ becomes under Ito.

Can someone tell me if there is a way to find $\mu$ $\sigma$ given only $f$? Or do I really just leave it like so.

Thank you

Answer 1

Itô's formula, as you've written it, is to compute $df(t, X_t)$ where $X_t$ is a process satisfying $dX_t = \sigma\, dW_t + \mu \,dt$.

So in your case, take $\mu = 0$ and $\sigma = 1$, so that $X_t = W_t$.