= Truncation construction of a positive Bessel strong solution
{title2=$a>\tfrac12,\quad dX=dB+aX^{-1}dt,\quad X_0=x>0$}
For $0<\varepsilon<x$, the drift $g_\varepsilon(y)=a/(y\vee\varepsilon)$ is globally Lipschitz and bounded. Solve its equation strongly for the prescribed <Brownian motion>, and stop at the first hit of $\varepsilon$. 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 $\mathbb P(T_\varepsilon\leq t)\leq(\varepsilon/x)^\nu$ 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.
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