Square-root stock supermartingale ($dU=\frac{1}{2}U\sigma dW-\frac{1}{8}U{\sigma }^{2}dt$)

For a positive stock satisfying $dS=S\sigma dW$, the Itô formula gives $d\sqrt{S}=\frac{1}{2}\sqrt{S}\sigma dW-\frac{1}{8}\sqrt{S}{\sigma }^{2}dt$. Localization and the conditional Fatou lemma make this nonnegative local supermartingale a true supermartingale.