Hey I have $\tau_n=\sigma_n\wedge\delta_n$ where $\sigma_n=\inf\{t\in[0,T] : \int_0^t|b(s)|^2ds\ge n\}$ and $\delta_n=\inf\{t\in[0,T] : \int_0^tb(s)dW(s)\ge n\}$, $b(s)\in P^2_{[0,T]}=\{f:\Omega\times [0,T]\to \mathbb{R}: f- adapted, P(\int_0^T|f(t)|^j<+\infty)=1\}$ and I have to show that
$P(\bigcup_{n\ge 1} \{\tau_n=T\})=1$
Can anyone prove it?