 Solution

Because $W$ is continuous and adapted, the Borel function $\mathrm{sgn}(W)$ is predictable. Its value at zero is $-1$, so its square is exactly one everywhere. Consequently $A$ is a zero-starting continuous local martingale withThe Lévy characterization of Brownian motion therefore makes $A$ a Brownian motion in the same filtration. Here that characterization is the deterministic-clock case of the conditional-exponential argument in question 2.

Apply the Itô formula to $V=W^2$:Since $\sqrt{V_t}=|W_t|$ and $|W_t|\mathrm{sgn}(W_t)=W_t$, including at zero, substitution yieldsThus $W^2$ is a squared Bessel process of dimension one.