Question. Apply Ito's formula to calculate the stochastic differential of:
$X_t = e^{at} \cos(b W_t)$
Determine for which values of $a, b \in \mathbb{R}$ the process $X_t$ is a martingale.
Attempt Set $W_t = x$ and applied Itô's formula. I calculate the partial derivatives:
$$\begin{align*} F(t,x) &= e^{at} \cos(bx)\\ \partial_t F(t,x) &= a F(t,x) \\ \partial_x F(t,x) &= b e^{at} (- \sin(bx)) \\ \partial_{xx} F(t,x) &= b^2 e^{at} (-\cos(bx)) \end{align*}$$
The SDE is:
$$dX_t = ae^{at} \cos(bx) dt + be^{at} (-\sin(bx)) dW_t - \frac{1}{2} b^2 e^{at} \cos(bx) dt$$
$$dX_t = e^{at} \cos(bx) (a - \frac{1}{2} b^2) dt - be^{at} \sin(bx) dW_t$$
How do I then calculate the values of $a$ and $b$ for which $X_t$ is a martingale?