= Solution
Fix $x=X_0>0$ and $a>1/2$. For the construction take $0<\varepsilon<x$, the range relevant to localization at zero, and define
$$
g_\varepsilon(y)=\frac a{y\vee\varepsilon}\qquad(y\in\mathbb R).
$$
This drift is bounded by $a/\varepsilon$ and globally Lipschitz with constant at most $a/\varepsilon^2$. Together with the constant diffusion coefficient $1$, the <global existence theorem for stochastic differential equations with Lipschitz coefficients> gives a unique <strong stochastic solution> $U^\varepsilon$ of
$$
dU_t^\varepsilon=dB_t+g_\varepsilon(U_t^\varepsilon)dt,\qquad U_0^\varepsilon=x,
$$
for the prescribed driving <Brownian motion>. Let $T_\varepsilon^\varepsilon=\inf\{t:U_t^\varepsilon=\varepsilon\}$ and choose $X_t^\varepsilon=U_{t\wedge T_\varepsilon^\varepsilon}^\varepsilon$ as the stopped continuation. Before this time the drift is exactly $a/X_t^\varepsilon$. Uniqueness here means uniqueness up to the <stopping time>; the equation imposes no restriction on arbitrary continuations after it.
For $0<\delta<\varepsilon<x$, the two equations have the same coefficients while both solutions exceed $\varepsilon$. By local <pathwise uniqueness> they coincide until that level is reached. Thus the first hit of $\varepsilon$ by $X^\delta$ is $T_\varepsilon^\varepsilon$, and its later first hit of $\delta$ satisfies
$$
\boxed{T_\delta^\delta\geq T_\varepsilon^\varepsilon.}
$$
Patch along a deterministic sequence $\varepsilon_n\downarrow0$: for $t<T:=\lim_nT_{\varepsilon_n}^{\varepsilon_n}$, choose $n$ with $t<T_{\varepsilon_n}^{\varepsilon_n}$ and set $X_t=X_t^{\varepsilon_n}$. 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 $\varepsilon\downarrow0$, 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 $s(X_{t\wedge T_\varepsilon^\varepsilon}^\varepsilon)$ a true <martingale>, because its stochastic integrand is bounded by $\nu\varepsilon^{-\nu-1}$. Positivity therefore implies
$$
\varepsilon^{-\nu}\mathbb P(T_\varepsilon^\varepsilon\leq t)\leq\mathbb E s(X_{t\wedge T_\varepsilon^\varepsilon}^\varepsilon)=x^{-\nu},\qquad\mathbb P(T_\varepsilon^\varepsilon\leq t)\leq\left(\frac\varepsilon x\right)^\nu.
$$
Since $\{T\leq t\}$ is contained in each $\{T_\varepsilon^\varepsilon\leq t\}$, letting $\varepsilon\downarrow0$ gives $\mathbb P(T\leq t)=0$. 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 $\varepsilon$. 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 $\varepsilon$ removes the localization and proves indistinguishability. Thus
$$
\boxed{a>\tfrac12:\quad\text{a global positive strong solution exists for every driving Brownian motion, and pathwise uniqueness holds}.}
$$
The resulting global solution also supplies the stopped solution for any threshold $\varepsilon\geq x$, completing the literal all-threshold existence request. The monotonicity used to construct the zero-boundary lifetime concerns the thresholds $\varepsilon<x$ 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.
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