# Expectation of a stopping time of a Wiener process

How can we calculate $\mathbb{E}(\tau)$ when $\tau=\inf\{t\geq0:B^2_t=1-t\}$?

If we can prove that $\tau$ is bounded a.s. (i.e. $\mathbb{E}[\tau]<\infty$), then we can use the fact that $\mathbb{E}[B^2_\tau]=\mathbb{E}[\tau]$. Hence, $1-\mathbb{E}[\tau]=\mathbb{E}[\tau]$ and so $\mathbb{E}[\tau]=\dfrac{1}{2 }$.

Is that right? If so, how can we show that $\mathbb{E}[\tau]<\infty$?

-