Ito's lemma states that
$$f(X(t))=f(X(0))+\int_0^tf'(X(s))dX(s)+\frac12\int_0^tf''(X(s))d\langle X\rangle(s)$$
Why are we interested specifically in deterministic functions of the integrator $X(t)$? In applications, where do the objects $f$ and $X$ usually come from?