Question: Consider a one dimensional $\mathbb{F}$-Wiener process $(W(t))_{t \in [0, T]}$ and consider also the process $(X(t))_{t \in [0, T]}$ given by $$X(t) = f(W(t)) - \frac{1}{2} \int^{t}_{0}f''(W(s))ds$$ for every $t \in [0, T]$, where $f: \mathbb{R} \to \mathbb{R}$ is a $C^2$ function. Assume that there exists a constant $C > 0$ such that $|f''(x)| \leq C$ for all $x \in \mathbb{R}$. Show that $(X(t))_{t \in [0, T]}$ is a martingale with respect to $\mathbb{F}$.
I know how to deal with this type of questions when an explicit form of $f(W(t))$ is given. In that case, I can just apply the Itô formula, and then try to show the process is a stochastic integral. However, in this case, there is no explicit form of $f(W(t))$ given. I also try to show it is a martingale by using the definition of martingale, but did not manage to do it either. Can anyone provide some help please.
Thank you for your attention. I am looking forward to your reply!