Truncation construction of a positive Bessel strong solution ($a>\frac{1}{2},dX=dB+a{X}^{-1}dt,X_0=x>0$)

For $0<\epsilon <x$, the drift $g_{\epsilon }(y)=a/(y\vee \epsilon )$ is globally Lipschitz and bounded. Solve its equation strongly for the prescribed Brownian motion, and stop at the first hit of $\epsilon$. Local uniqueness makes these stopped solutions agree up to the larger threshold; their stopping times increase as the threshold decreases. With $\nu =2a-1>0$, Itô's formula for $X^{-\nu }$ gives $\mathbb{P}(T_{\epsilon }\le t)\le (\epsilon /x{)}^{\nu }$ on every finite horizon. Hence the patched lifetime is infinite. Localization away from zero proves pathwise uniqueness of the positive global solution. Uniqueness before a stopping time refers to the stopped solution, not to arbitrary unconstrained continuations afterward.