# Ito integral and true martingale

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.

• I'm not sure where your statements 1 and 2 came from, but I don't think they're true in general. A local martingale with continuous trajectories does not have to be a true martingale. There is a standard example here. May 28, 2019 at 21:23

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
$$$$\mathrm{d}Y_t = \frac{1}{2}F^{\prime \prime}(W_t) \mathrm{d}t + F^\prime(W_t) \mathrm{d}W_t.$$$$
It follows that $$$$F(W_t)-\int_0^t \frac{1}{2}F^{\prime \prime}(W_s) \mathrm{d}s = \int_0^t F^\prime(W_s)\mathrm{d}W_s,$$$$ which is clearly a martingale as $$F^\prime$$ is bounded.
If $$F^{\prime \prime}$$ is not continuous, I am not sure the assertion holds anymore.