I have seen the Paley-Wiener-Zigmund integral given as follows:
$(*) \displaystyle \int_0^1 fdW_t = W_1f(1) -\int_0^1 f'(t)W_tdt $ where $f$ is such that $f'$ is continuous and $W_t$ is Brownian motion.
Further, seems it is sometimes (perhaps often) taken that $f(0) = f(1) = 0$.
Visually it is clear this is an integration-by-parts type formula.
In part 1 of PWZ Integral it states for the above $(*)$ to be proven.
My question is how is this proven?
It is clear it is integration-by-parts motivated, problem is that $W_t$ is nowhere differentiable with probability $1$. Meaning I can't directly just apply IBP.
If either someone has a link to where this is proven or an explanation would be appreciated.
Other places I have seen it stated as a definition rather than something needing to be proved. Hence part of the confusion for me.