I've been puzzled by the following problem : given B,B' and W three independents (1-D) brownian motions, let X be defined as, for t>1:
$$X_{t}=\sqrt{t} \left(B_{1}\cos \left(W_{\log\left(t\right)} \right) +B^{'}_{1}\sin \left(W_{\log\left(t\right)} \right) \right) $$
I need to compute the Fourier transform of the random variable X(t) (to show that it is normally distributed). The teacher wants us to use the fact that
$$E\left(e^{i\lambda X_{t}}\right) = E\left(E\left(e^{i\lambda X_{t}}\mid W_{\log \left(t\right)}\right)\right) $$
which I completely understand, but I don't know how to work it from here, nor what theorem/lemma to use. Anyone has encountered such a problem ?