Consider a twice diferentiable function $F$ on $R$ with bounded
first derivative $F'$ and a Brownian motion $W$. Show that $F(W\_t)-\frac{1}{2} \int\_{0}^{t} F'' (W\_s)ds$ is a true martingale.


I tried do show it using this, but I only got confused and did not find any solution.


1. If $M$ is a local martingale with continues trajectories, then it is a true martingale and $E(M\_t^2) < \infty$ for all $t\geq 0$
Or
2. If $M$ is a local martingale with continues trajectories, then it is a true martingale and $E([M]\_t)<\infty$ for all t.



Ito integral and true martingale
If $F$ were twice continuously differentiable, one could use Ito's Lemma to obtain the stochastic differential equation for $Y\_t = F(W\_t)$ as


\begin{equation}
\mathrm{d}Y\_t = \frac{1}{2}F^{\prime \prime}(W\_t) \mathrm{d}t + F^\prime(W\_t) \mathrm{d}W\_t.
\end{equation}


It follows that
\begin{equation}
F(W\_t)-\int\_0^t \frac{1}{2}F^{\prime \prime}(W\_s) \mathrm{d}s = \int\_0^t F^\prime(W\_s)\mathrm{d}W\_s,
\end{equation}
which is clearly a martingale as $F^\prime$ is bounded.


If $F^{\prime \prime}$ is not continuous, I am not sure the assertion holds anymore. 


