Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 202 / 3 / 2 / Solution

Since $B_{T_a}=0$, applying Itô formula to $B_t^3$ after $T_a$ givesBefore $T_a$, both sides vanish. Since $T_a$ is a stopping time determined by $B$, this is a strong solution of a stochastic differential equation.

Taking $a=0$ gives $X_t=B_t^3$, whereas any $a>0$ gives a solution that remains zero until $T_a$; these differ with positive probability while using the same Brownian motion and initial value. Therefore pathwise uniqueness fails.