Using of Ito formula with martingales

Question

We have exam test -

$\alpha,\beta \in \mathbb{R}$ and $N(t)=e^{\beta t}cos(\alpha W(t)).$ It is necessary to calculate $\mathbb{E}[cos(\alpha W(t))]$.

I know that $\beta$ can be chosen so that $N$ is a martingale. I need to express it in terms of $\mathbb{E}[cos(\alpha W(t))]$.

Can somebody please help me with the right decision?

Answer 1

Calculate the ito differential:

$$dN=\beta N(t)dt-\alpha e^{\beta t}\sin(\alpha W(t))dW-\alpha^2N(t)dt.$$

It follows that if $\beta=\alpha^2$, then $N(t)$ is a martingale. Thus surely you know how to calculate $E[N(t)]$ and also $\cos(\alpha W(t))=N(t)e^{-\beta t}$, so:

$$E[N(t)]=E[e^{\beta t}\cos(\alpha W(t))]=e^{\beta t}E[\cos(\alpha W(t))]$$