= Solution
{c}
<Lévy characterization of Brownian motion> says that a <continuous local martingale> $M$ with $M_0=0$ is standard <Brownian motion> with respect to its <filtration> if its <quadratic variation> is $\langle M\rangle_t=t$. Equivalently one may require $M_t^2-t$ to be a <local martingale>.
Any <weak solution> has the <integral> form $Z_t=\int_0^t\operatorname{sgn}(Z_s-2)\,dW_s$. Its bounded <integrand> makes it a <continuous local martingale> starting at zero, and
$$
\langle Z\rangle_t=\int_0^t\operatorname{sgn}(Z_s-2)^2ds=t.
$$
The specified value of the sign at zero has square one as well. <Lévy characterization of Brownian motion> therefore proves that \b[every such <weak solution> is <Brownian motion>]. This is a statement about its law and <filtration>; it does not assert $Z=W$ or provide a strong solution as a function of a specified driver.
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