It is well known that Ito integrals with respect to a Brownian motion cannot be defined pathwise because the Brownian motion has infinite 1st order variation. These integrals are defined as limits of simple integrals in a $L^2$ sense. Now, I was interested in the following: If $f$ is a continuously differentiable function on $[a,b]$, is it possible to bound the integral $\int_a^b f(t) dW_t$ for almost every path $\omega$ (if $W$ is a brownian motion)? I found in the link below on equation (4.33) page 49:
http://www.ems.bbk.ac.uk/for_students/msc_finEng/math_methods/lecture34.pdf
that it is possible to get for almost every $\omega$: \begin{align*} & \left | \int_a^b f(t) dW_t \right| \leq 2(1+b-a) (||f||+||f'||) \sup_{t \in [a,b]} |W(t)|, \end{align*} where $||f||:=\sup_{t \in [a,b]} |f(t)|$, $||f'||:=\sup_{t \in [a,b]} |f'(t)|$.
I am honestly a little dubious about the latter upper bound. I tried to look for some other literature on Ito integrals but they almost always do the same thing, i.e. constructing it as a limit in a $L^2$ sense. I didn't find anything in the specific case of integrals wit respect to a deterministic and smooth integrand $f$.
I'm sorry, I was very stupid. We can just apply Ito's formula and get: \begin{align*} & \int_a^b f(t)dW_t=f(b)W_b-f(a)W_a-\int_a^b W_t f'(t)dt. \end{align*} This yields that indeed we can find the desired pathwise upper bound of $\int_a^b f(t)dW_t$ in terms of $||f||$, $||f'||$, $\sup_{t \in [a,b]}|W(t)|$.