Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 202 / 5 / d / Solution

Let $\tau =\inf \{t\ge 0:B_t=-2\sqrt{x_0}\}$, 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 $0<Y_{\tau }\le W_{\tau }=0$, a contradiction. ThereforeThe 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.