What is a valid range of applicability of Ito Lemma?

Question

If I have e.g. such process

$$ Z_{t}=t^{5}B_{t}+10\int_{0}^{t}sB_{s}ds $$

can I take $$ f(t,x):=t^{5}x+10\int_{0}^{t}sB_{s}ds $$

as a function to which I apply Ito formula? I'm concerned about $B_{s}$ term. It would make $f$, the nondeterministic function... Is this a problem?

Answer 1

Since the integral in the definition of $Z$ is with respect to time, it contributes to the drift, not to the diffusion. The drift or diffusion being random is no obstacle.

Answer 2

In order to apply Ito's lemma all you need is that the functions $f(t,x)$ should be twice differentiable in $x$ and differentiable(once) in t i.e. $f(t,x) \in C^{1,2}(\mathbb{R_+ \times R^n)}$ . So yes in your case $f $ is a valid function in order to apply Ito's lemma