The following question has been puzzling to me and there has been no luck trying to resolve it. Perhaps I am missing something. So, any help would be very much appreciated.
If $f\in C^\infty(\mathbb{R})$ is a smooth deterministic function, from Ito formula, we can have $$ \int_0^t f(s) dB_s = f(t)B_t - \int_0^t B_sdf(s) \,. $$ Therefore, $$ \Big|\int_0^t f(s) dB_s \Big| \leq C_t ( ||f||_\infty + ||f'||_\infty)\sup_{0\leq s \leq t}|B_s| \,. $$ However, the situation is not as good when we make $f$ random. I am wondering if there's a similar result as the above if we're able to have some sort of bounds, say, $$ | f (t,\omega)| + |f'(t,\omega)| \leq C$$ for all $t$ and $\omega$?
I've just corrected a typo in the question.