Let be $B$ and $D$ independent Brownian motion. Let be $\alpha \in \mathbb{R}$ constant. Define $$M_t = \alpha (\int_0^t B_s dD_s + \int_0^t D_s dB_s)$$ and $$N_t = \frac{\alpha^2}{2} \int_0^t (B_s^2 + D_s^2)ds$$
Let be $X_t = cos(M_t)e^{N_t}$.
First i need to prove that $X_t$ is local martingale. I did it. But now i need to explain that $X_t$ is also martingale.
How can i prove it?