I have a question in my homework:
Let $X_t$ and $Y_t$ be two Brownian motions issue de $0$ and define
$S_t=\int_0^tX_s\,dY_s-\int_0^tY_s\,dX_s$
Show that
$E[e^{i\lambda S_t}]=E[\cos(\lambda S_t)]$
Does someone have an idea? I try to show that
$E[\sin(\lambda S_t)]=0$
By Ito's formula applied for $\sin(\lambda S_t)$, we get
$d\sin(\lambda S_t)=\lambda\cos(\lambda S_t)dS_t-\frac{\lambda^2}{2}\sin(\lambda S_t)\,d\langle S,S\rangle_t$
To calculate $d\langle S,S\rangle_t$, I am not sure if my calculs is correct or not:
$d\langle S,S\rangle_t=(X_t^2+Y_t^2)\,dt$
Since $S_t$ is a martingale we have
$E[\sin(\lambda S_t)]=-\frac{\lambda^2}{2}E\left[\int_0^t(X_t^2+Y_t^2)\sin(\lambda S_t)\,dt\right]$
I am not sur whether my previous calculs is correct. Could someone help me? Thanks a lot!