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Fix $x=X_0>0$ and $a>1/2$. For the construction take $0<\epsilon <x$, the range relevant to localization at zero, and defineThis drift is bounded by $a/\epsilon$ and globally Lipschitz with constant at most $a/{\epsilon }^2$. Together with the constant diffusion coefficient $1$, the global existence theorem for stochastic differential equations with Lipschitz coefficients gives a unique strong stochastic solution $U^{\epsilon }$ offor the prescribed driving Brownian motion. Let $T_{\epsilon }^{\epsilon }=\inf \{t:U_t^{\epsilon }=\epsilon \}$ and choose $X_t^{\epsilon }=U_{t\wedge T_{\epsilon }^{\epsilon }}^{\epsilon }$ as the stopped continuation. Before this time the drift is exactly $a/X_t^{\epsilon }$. Uniqueness here means uniqueness up to the stopping time; the equation imposes no restriction on arbitrary continuations after it.

For $0<\delta <\epsilon <x$, the two equations have the same coefficients while both solutions exceed $\epsilon$. By local pathwise uniqueness they coincide until that level is reached. Thus the first hit of $\epsilon$ by $X^{\delta }$ is $T_{\epsilon }^{\epsilon }$, and its later first hit of $\delta$ satisfiesPatch along a deterministic sequence ${\epsilon }_n\downarrow 0$: for $t<T:={\lim }_n{T}_{{\epsilon }_n}^{{\epsilon }_n}$, choose $n$ with $t<T_{{\epsilon }_n}^{{\epsilon }_n}$ and set $X_t=X_t^{{\epsilon }_n}$. Compatibility makes the definition independent of $n$. It gives a continuous adapted positive process satisfying the original equation on every compact time interval before $T$. The same compatibility holds for other threshold choices, so the lifetime is the limit as $\epsilon \downarrow 0$, not a property of the chosen sequence.

To prove $T=\infty$, put $\nu =2a-1>0$ and $s(y)=y^{-\nu }$. Its generator vanishes. On each finite horizon, Itô's formula makes $s(X_{t\wedge T_{\epsilon }^{\epsilon }}^{\epsilon })$ a true martingale, because its stochastic integrand is bounded by $\nu {\epsilon }^{-\nu -1}$. Positivity therefore impliesSince $\{T\le t\}$ is contained in each $\{T_{\epsilon }^{\epsilon }\le t\}$, letting $\epsilon \downarrow 0$ gives $\mathbb{P}(T\le t)=0$. Taking a countable union over integer $t$ proves $T=\infty$ almost surely. This is the truncation construction of a positive Bessel strong solution.

For two positive solutions driven by the same Brownian motion, localize at their first hits of $\epsilon$. The cutoff equations and pathwise uniqueness make them equal up to those times. On every finite interval both positive continuous paths have positive minima, so decreasing $\epsilon$ removes the localization and proves indistinguishability. ThusThe resulting global solution also supplies the stopped solution for any threshold $\epsilon \ge x$, completing the literal all-threshold existence request. The monotonicity used to construct the zero-boundary lifetime concerns the thresholds $\epsilon <x$ tending to zero; upper-threshold hitting times have the opposite ordering. Strong adaptation is preserved because every cutoff solution is strong and the patch uses their stopping times in the given filtration.