Let $\mathcal{O}$ be an open subset of $\mathbb R$ and $\mu\in\mathcal{P}(\mathcal{O})$. Consider the SDE $$dX_t(\omega)= b(t, X_t(\omega), \omega)dt + \sigma(t, X_t(\omega))dW_t, \quad X_0\sim \mu.$$ Suppose that the coefficients are Lipschitz on $x$ (uniformly on the other variables) and bounded. Let $$\tau_1:=\inf\{t\geq 0: X_t\notin \mathcal{O}\},$$ $$\tau_2:=\inf\{t\geq 0: X_t\notin \bar{\mathcal{O}}\},$$ where $\bar{\mathcal{O}}$ is the closure of $\mathcal{O}$.
My question is, what are the conditions on the coefficients and $\mathcal{O}$ to have $\tau_1=\tau_2$ a.s.?
Any references?
Thanks