= Solution
Let $\tau=\inf\{t\geq0:B_t=-2\sqrt{x_0}\}$, the first time $W$ reaches zero. <Recurrence of one-dimensional Brownian motion> implies $\tau<\infty$ <almost surely>. On the common event where the comparison from part (c) holds, $\tau<T$ would imply $0<Y_\tau\leq W_\tau=0$, a contradiction. Therefore
$$
\boxed{T\leq\tau<\infty\quad\text{almost surely},\qquad \mathbb P(T<\infty)=1.}
$$
The argument only uses the comparison at a time strictly before $T$; it does not assume that the singular drift integral is finite at $T$. The given lifetime characterization identifies this finite endpoint with approach to zero. Any extension that absorbs at zero would be a different, globally defined <stochastic process>, not the positive-domain <maximal local solution of a stochastic differential equation> specified here.
Back to article page
