I have two possibly related questions. Let $\tau:=\min\{t\geq0:B_t=1\}$, where $B_t$ is a standard Brownian motion.
I am supposed to derive the fact that $\mathbf{E}\tau=\infty$ by applying some properties of stochastic integrals to $\int^{\infty}_0\mathbf{1}_{[0,\tau]}(t)dB_t$. I have tried this: $\mathbf{E}\tau=\mathbf{E}\int^{\tau}_0dt=\mathbf{E}\int^{\infty}_0\mathbf{1}_{[0,\tau]}(t)dt=\mathbf{E}\left(\int^{\infty}_0\mathbf{1}_{[0,\tau]}(t)dB_t\right)^2$ And I'm stuck (not even sure if going the right way).
Is Ito's lemma false with stopping times as upper limits of the integrals? For example, is $\int^{\tau}_0dB_t=B_{\tau}$ false? How can I see that?
EDIT: A discussion on possible proofs of $\mathbf{E}\tau=\infty$ is presented in the answers here. I am just curious about the particular suggestion.
And the stochastic integral with a hitting time as an upper limit is defined as follows $\int^{\tau}_0dB_t:=\int^\infty_0\mathbf{1}_{[0,\tau]}(t)dB_t:=L^2-\lim_{N\to\infty}\int^N_0\mathbf{1}_{[0,\tau]}(t)dB_t$ Then I guess the question 2. would be equivalent to asking if Ito's lemma is true for integrals with infinite upper limit.