Stopped martingale

Question

Given a probability space $(\Omega,\mathcal{F},\mathbb{P})$ and $\mathbb{P}-$Brownian Motion $B$ starting from 0, consider $\tau$ the first hitting time at 1.

As $\tau<\infty$ $\mathbb{P}$ a.s., the optional stopping theorem tells us that the stopped process $B^\tau$ is a martingale starting from zero and ending at 1.

However, $1=\mathbb{E}[B_\tau]\neq \mathbb{E}[B_{\tau\wedge 0}]=0$, indicating it should not be a martingale. How can we reconcile this? Where the argument goes wrong?

Answer 1

Yes, the process $(B_{t \wedge \tau})_{t \geq 0}$ is a martingale, but you are considering the random variable $B_{\tau}$, which you cannot find among the family $B_{t \wedge \tau}$, since $\tau$ is not bounded.

Answer 2

The martingale property is only concerned with $(B_{\tau \wedge t}: 0 \leq t <\infty)$. You are looking at the limiting random variable $B_{\tau}=\lim_{t \to \infty} B_{\tau \wedge t}$ and there is no reason why its expectation has to be $0$.