= Cutoff construction of an absorbed square-root diffusion
Replace the square-root coefficient near zero by smooth globally Lipschitz cutoffs that vanish below half the cutoff level. With a common Brownian driver, local <pathwise uniqueness> makes the solutions agree until the larger cutoff level. The exit times increase as the levels decrease. The compatible integrands $\mathbf1_{\{s\leq T_n\}}\sqrt{Z_s}$ have increasing squares and expected time-integrals at most $zt$ by the <nonnegative local martingale> bound. Their limit is square-integrable on every finite horizon. The <Itô isometry> gives a continuous global limiting stochastic integral; at a finite limiting exit time its value is zero, and it remains absorbed afterward.
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