Q:

Compute distribution of a stochastic variable

$sign(x)=1$ if $x\geq0$
$sign(x)=-1$ if $x< 0$
Consider
$$
X_t = \int^t_0 sign(W_u)dW_u
$$
where $W_t$ is a wiener proces. 
How can I determine the distribution of $X_t$ and compute $E[\exp(\lambda X_t )]$? 

A:

Note that $\{X_t, \, t \ge 0\}$ is continuous, square-integrable martingale with quadratic
variation process
\begin{align*}
\langle X\rangle_t = \int_0^t {\rm sign}^2(W_s)\, ds =t.
\end{align*}
Then, it is a standard Brownian motion based on Levy’s Characterization of Brownian Motion.  The remaining is straightforward.