Past exam of the stochastic differential equations and Brownian motion in complex analysis course of the University of Oxford / 2007 / A / 2 / c / Solution

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 $⟨M{⟩}_t=t$. Equivalently one may require $M_t^2-t$ to be a local martingale.

Any weak solution has the integral form $Z_t={\int }_0^t\mathrm{sgn}(Z_s-2)d{W}_s$. Its bounded integrand makes it a continuous local martingale starting at zero, andThe specified value of the sign at zero has square one as well. Lévy characterization of Brownian motion therefore proves that 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.