= Solution
Since $B_{T_a}=0$, applying <Itô formula> to $B_t^3$ after $T_a$ gives
$$
dX_t=3B_t\,dt+3B_t^2\,dB_t
=3\operatorname{sign}(X_t)|X_t|^{1/3}dt+3|X_t|^{2/3}dB_t.
$$
Before $T_a$, 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 \b[pathwise uniqueness fails].
Back to article page
