 Solution

Integrating the stopped stochastic differential equation for $Y$ and then taking a common full-probability event for a localizing sequence gives, simultaneously for every $t<T$,The integral is finite on each compact time interval before $T$: the continuous positive path has a positive minimum there. Its integrand is nonnegative, soThis is a pathwise comparison using the same Brownian motion, not a comparison of independent distributions. Continuity lets the identities established initially at rational stopped times hold on the entire stochastic interval.